Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 13 conf., AAECC-13

Lecture Notes in Computer Science Edited by G. Goos, J. Hartmanis and J. van Leeuwen

1719

3 Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo

Marc Fossorier Hideki Imai Shu Lin Alain Poli (Eds.)

Applied Algebra, Algebraic Algorithms and Error-Correcting Codes 13th International Symposium, AAECC-13 Honolulu, Hawaii, USA, November 15-19, 1999 Proceedings

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Series Editors Gerhard Goos, Karlsruhe University, Germany Juris Hartmanis, Cornell University, NY, USA Jan van Leeuwen, Utrecht University, The Netherlands

Volume Editors Marc Fossorier Shu Lin University of Hawaii, Department of Electrical Engineering 2540 Dole St., # 483, Honolulu, HI 96822, USA E-mail: [email protected]; [email protected] Hideki Imai University of Tokyo, Institute of Industrial Science 7-22-1, Roppongi, Minato-ku, Tokyo 106, Japan E-mail: [email protected] Alain Poli Universit´e Paul Sabatier, AAECC/IRIT 118, Route de Narbonne, F31067 Toulouse Cedex, France E-mail: [email protected] Cataloging-in-Publication data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Applied algebra, algebraic algorithms and error-correcting codes : 13th international symposium ; proceedings / AAECC 13, Honolulu, Hawaii, USA, November 15 - 19, 1999. Marc Fossorier . . . (ed.). Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 1999 (Lecture notes in computer science ; Vol. 1719) ISBN 3-540-66723-7

CR Subject Classification (1998): E.4, I.1, E.3, G.2, F.2 ISSN 0302-9743 ISBN 3-540-66723-7 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. c Springer-Verlag Berlin Heidelberg 1999
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Preface

The AAECC symposium was started in June 1983 by Alain Poli (Toulouse), who, together with R. Desq, D. Lazard, and P. Camion, organized the first conference. At the beginning, the acronym AAECC meant “Applied Algebra and Error Correcting Codes”. Over the years this meaning has shifted to “Applied Algebra, Algebraic Algorithms, and Error Correcting Codes”. One reason was the increasing importance of complexity, particularly for decoding algorithms. During the AAECC-12 symposium, after a long discussion and a vote, the conference committee decided to hold the next symposium in Hawaii, with Shu Lin as Chairman. This vote also led to the decision to enforce the theory and practice of the coding side as well as the cryptographic aspects. Algebra is conserved as in the past, but slightly more oriented to finite fields, complexity, polynomials, graphs. The conference committee was modified, passing from 15 to 20 members. For AAECC-13 the main subjects covered are : – – – –

Modulation and codes : communication systems. Combinatorics : graphs and matrices, designs, arithmetic. Cryptography. Codes : iterative decoding, decoding methods, turbo decoding, block codes, convolutional codes, codes construction. – Codes and algebra : algebraic curves, Groebner bases and AG codes. – Algebra : rings and fields, Galois group, differential algebra, polynomials. Six invited speakers characterize the outlines of AAECC-13 : – Hui Jin and Robert McEliece (“RA Codes Achieve AWGN Channel Capacity”). – Bernd Sturmfels (“Monomial Ideals”). – Michael Clausen (“A Near-Optimal Program Generator for Almost Optimal FFT’s”). – Tadao Kasami (“On Integer Programming Problems Related to Iterative Search Type Decoding Algorithms”). – G. David Forney, Jr. (“Codes on Graphs: A Survey for Algebraists”). – Ian Blake (“Applications of Curves with Many Points”). Except for AAECC-1 (Discrete Mathematics, 56,1985) and AAECC-7 (Discrete Mathematics, 33,1991), the proceedings of all the symposia have been published in Springer-Verlag’s Lecture Notes in Computer Science series (vol. 228, 229, 307, 356, 357, 508, 673, 948, 1255). It is a policy of AAECC to maintain a high scientific standard. This has been made possible thanks to the many referees involved. Each submitted paper was evaluated by at least two international researchers.

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Preface

AAECC-13 received 86 submissions; 42 were selected for publication in these proceedings while 33 additional works will contribute to the symposium as oral presentations. The symposium was organized by Marc Fossorier, Hideki Imai, Shu Lin, and Alain Poli, with the help of Stephen Cohen and Marie-Claude Gennero. We express our thanks to the Springer-Verlag staff, especially to Alfred Hofmann and Ruth Abraham, as well as to Jinghu Chen, Ivana Djurdjevic, Yu Kou, Qi Pan, Tsukasa Sugita, and Heng Tang, for their help in the preparation of these proceedings.

September 1999

M. Fossorier, H. Imai, S. Lin, A. Poli

Organization

Steering Committee Conference General Chairman: Conference Co-Chairman: Program Chairman: Publication:

Shu Lin (Univ. of Hawaii, USA) Alain Poli (Univ. of Toulouse, France) Hideki Imai (Univ. of Tokyo, Japan) Marc Fossorier (Univ. of Hawaii, USA)

Conference Committee J. Calmet G. Cohen S.D. Cohen P.G. Farrell G.L. Feng M. Giusti J. Heintz T. Hoehold H. Imai H. Janwa J.M. Jensen

R. Kohno H.W. Lenstra Jr. S. Lin O. Moreno H. Niederreiter A. Poli T.R.N. Rao S. Sakata P. Sole

Program Committee V.K. Bhargava E. Biglieri I.F. Blake M. Bossert J. Calmet P. Charpin D.J. Costello, Jr. T. Ericson C. Heegard P.G. Farrell M. Fossorier M. Giusti A. Glavieux D. Hachenberger J. Hagenauer

J. Heintz E. Kaltofen T. Kasami T. Katsura R. Kohno H.W. Lenstra, Jr. R. McEliece R.H. Morelos-Zaragoza H. Niederreiter S. Sakata B. Vucetic V.K. Wei J.K. Wolf

Table of Contents

Invited Talks Codes on Graphs: A Survey for Algebraists . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. David Forney, Jr.

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RA Codes Achieve AWGN Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Hui Jin and Robert J. McEliece Monomial Ideals and Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Ezra Miller and Bernd Sturmfels A Fast Program Generator of Fast Fourier Transforms . . . . . . . . . . . . . . . . . . 29 Michael Clausen and Meinard M¨ uller On Integer Programming Problems Related to Soft-Decision Iterative Decoding Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Tadao Kasami Curves with Many Points and Their Applications . . . . . . . . . . . . . . . . . . . . . . 55 Ian F. Blake

Codes and Iterative Decoding New Sequences of Linear Time Erasure Codes Approaching the Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 M. Amin Shokrollahi On the Theory of Low-Density Convolutional Codes . . . . . . . . . . . . . . . . . . . . 77 Karin Engdahl,, Michael Lentmaier, and Kamil Sh. Zigangirov

Combinatorics I: Arithmetic On the Distribution of Nonlinear Recursive Congruential Pseudorandom Numbers of Higher Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Frances Griffin, Harald Niederreiter, and Igor E. Shparlinski A New Representation of Boolean Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Claude Carlet and Philippe Guillot

Combinatorics II: Graphs and Matrices An Algorithm to Compute a Nearest Point in the Lattice A∗n . . . . . . . . . . . . 104 I. Vaughan and L. Clarkson

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Sequences from Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Kathy J. Horadam

Block Codes I On the Second Greedy Weight for Binary Linear Codes. . . . . . . . . . . . . . . . . 131 Wende Chen and Torleiv Kløve On the Size of Identifying Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Uri Blass, Iiro Honkala, and Simon Litsyn

Algebra I: Rings and Fields Fast Quantum Fourier Transforms for a Class of Non-abelian Groups . . . . . 148 Markus P¨ uschel, Martin R¨ otteler, and Thomas Beth Linear Codes and Rings of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 M. Greferath and S.E. Schmidt On ZZ 4 -Simplex Codes and Their Gray Images . . . . . . . . . . . . . . . . . . . . . . . . 170 Mahesh C. Bhandari, Manish K. Gupta, and Arbind K. Lal

Decoding Methods Some Results on Generalized Concatenation of Block Codes . . . . . . . . . . . . . 181 M. Bossert, H. Grießer, J. Maucher, and V.V. Zyablov Near Optimal Decoding for TCM Using the BIVA and Trellis Shaping . . . . 191 Qi Wang, Lei Wei, and Rodney A. Kennedy An Optimality Testing Algorithm for a Decoded Codeword of Binary Block Codes and Its Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Yuansheng Tang, Tadao Kasami, and Toru Fujiwara

Algebra II Recursive MDS-Codes and Pseudogeometries . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Elena Couselo, Santos Gonzalez, Victor Markov, and Alexandr Nechaev Strength of MISTY1 without FL Function for Higher Order Differential Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Hidema Tanaka, Kazuyuki Hisamatsu, and Toshinobu Kaneko

Code Construction Quantum Reed–Solomon Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Markus Grassl, Willi Geiselmann, and Thomas Beth Capacity Bounds for the 3-Dimensional (0, 1) Runlength Limited Channel . 245 Zsigmond Nagy and Kenneth Zeger

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Rectangular Codes and Rectangular Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 252 V. Sidorenko, J. Maucher, and M. Bossert

Codes and Algebra I: Algebraic Curves Decoding Hermitian Codes with Sudan’s Algorithm . . . . . . . . . . . . . . . . . . . . 260 T. Høholdt and R. Refslund Nielsen Computing a Basis of L(D) on an Affine Algebraic Curve with One Rational Place at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Ryutaroh Matsumoto and Shinji Miura

Cryptography Critical Noise for Convergence of Iterative Probabilistic Decoding with Belief Propagation in Cryptographic Applications . . . . . . . . . . . . . . . . . . . . . . 282 Marc P.C. Fossorier, Miodrag J. Mihaljevi´c, and Hideki Imai An Authentication Scheme over Non-authentic Public Channel in Information-Theoretic Secret-Key Agreement . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Shengli Liu and Yumin Wang

Codes and Decoding A Systolic Array Architecture for Fast Decoding of One-Point AG Codes and Scheduling of Parallel Processing on It . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Shojiro Sakata and Masazumi Kurihara

Convolutional Codes Computing Weight Distributions of Convolutional Codes via Shift Register Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Mehul Motani and Chris Heegard Properties of Finite Response Input Sequences of Recursive Convolutional Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Didier Le Ruyet, Hong Sun, and Han Vu Thien

Combinatorics III: Designs Lower Bounds for Group Covering Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 K.K.P. Chanduka, Mahesh C. Bhandari, and Arbind K. Lal Characteristic Functions of Relative Difference Sets, Correlated Sequences and Hadamard Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 Garry Hughes

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Decoding of Block Codes Double Circulant Self-Dual Codes Using Finite-Field Wavelet Transforms 355 F. Fekri, S.W. McLaughlin, R.M. Mersereau, and R.W. Schafer

Algebra III: Rings and Fields Linear Codes and Polylinear Recurrences over Finite Rings and Modules (a Survey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 V.L. Kurakin, A.S. Kuzmin, V.T. Markov, A.V. Mikhalev, and A.A. Nechaev Calculating Generators for Invariant Fields of Linear Algebraic Groups . . . 392 J¨ orn M¨ uller–Quade and Thomas Beth Constructing Elements of Large Order in Finite Fields . . . . . . . . . . . . . . . . . . 404 Joachim von zur Gathen and Igor Shparlinski

Modulation and Codes New Lower Bounds on the Periodic Crosscorrelation of QAM Codes with Arbitrary Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 Serdar Bozta¸s Conjectures on the Size of Constellations Constructed from Direct Sums of PSK Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 Matthew G. Parker

Codes and Algebra II: Gr¨ obner Bases and AG Codes A New Criterion for Normal Form Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 430 B. Mourrain Discrete Fourier Transform and Gr¨ obner Bases . . . . . . . . . . . . . . . . . . . . . . . . 444 A. Poli, M.C. Gennero, and D. Xin

Block Codes II On the State Complexities of Ternary Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Sylvia Encheva and G´erard Cohen Binary Optimal Linear Rate 1/2 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Koichi Betsumiya, T. Aaron Gulliver, and Masaaki Harada On Binary/Ternary Error-Correcting Codes with Minimum Distance 4 . . . . 472 ¨ Patric R.J. Osterg˚ ard

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Algebra IV: Polynomials The Euclidean Algorithm and Primitive Polynomials over Finite Fields . . . 482 James W. Bond, Stefen Hui, and Hank Schmidt On the Computational Hardness of Testing Square-Freeness of Sparse Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 Marek Karpinski and Igor Shparlinski Mastrovito Multiplier for General Irreducible Polynomials . . . . . . . . . . . . . . . 498 A. Halbutoˇgulları and C ¸ .K. Ko¸c

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

Codes on Graphs: A Survey for Algebraists G. David Forney, Jr. Massachusetts Institute of Technology, Cambridge MA 02139 USA [email protected] Abstract. There is intense current interest in the subject of “codes on graphs.” Graphical models for codes arise from state realizations. Normal realizations may be assumed without loss of generality. A normal realization has a dual which necessarily generates the dual code. Conventional trellises and tail-biting realizations are inherently normal. Efficient realizations of Reed-Muller codes are presented. Low-density parity-check codes use generic parity-check realizations. Algebraic tools may improve such graph-based codes.

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Introduction

The subject of “codes on graphs” was founded by Tanner [Tan81], who was inspired by Gallager’s low-density parity-check (LDPC) codes [Gal62]. In recent years graphical models and their associated decoding algorithms have come into prominence as a common intellectual foundation for the study of capacity-approaching codes such as turbo codes and LDPC codes [AM98,KFL98]. This paper is a survey of “codes on graphs” for readers with a background in algebraic coding theory. There are many interesting problems in this emerging field for which algebraic tools may be helpful, both in constructing efficient representations of known codes and in finding new families of graph-based codes. 1.1

Codes, Realizations, and Graphs
A block code C is any subset of A = k∈IA Ak , where the symbol index set IA is any finite discrete set, not necessarily ordered, and the symbol configuration space A is the Cartesian product of any collection of symbol variables Ak , k ∈ IA . The code thus consists of a subset of configurations a = {ak , k ∈ IA } ∈ A, called valid configurations, or codewords. A group code is a code C with
a componentwise group property; i.e., the symbol configuration space A = k∈IA Ak is the direct product of a collection of symbol groups Ak , k ∈ IA , and the code is a subgroup C ⊆ A. Any linear code is a group code. In coding for memoryless channels, the ordering of the symbol index set IA is immaterial; two codes that differ only by a permutation of IA have the same distance profile and the same performance on a memoryless channel, and are usually taken to be equivalent. We may use this freedom to order IA however we like, or even to let IA be unordered. In the field of “codes on graphs,” the “time axis” IA is represented by a graph rather than by a subinterval of the integers. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 1–9, 1999. c Springer-Verlag Berlin Heidelberg 1999 

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A state realization of a code C involves two further types of elements: 1. A collection of state variables (“state spaces”) Sj , j ∈ IS ; 2. A collection of local codes or local constraints Ci , i ∈ IC , where IS and IC are finite discrete symbol
and constraint index sets, respectively. The state configuration space is S = j∈IS Sj . In general, the three index sets IA , IS and IC need not bear any relationship to one another. Whereas symbol variables are specified a priori as part of the code description, state variables are introduced by the designer into a realization to improve it in some sense. Symbol variables are visible, whereas state variables are hidden. In coding, symbol variables are transmitted over a channel and are observed in the presence of noise, whereas state variables are part of the internal representation of the code and are unobserved. Each local code Ci is a subset of a certain local Cartesian product set,   Ak × Sj = A(i) × S(i), Ci ⊆ k∈IA (i)

j∈IS (i)

where IA (i) ⊆ IA and IS (i) ⊆ IS are the subsets of the corresponding index sets that are involved in the local code Ci . The local code thus defines a set of valid local codewords (a|IA (i), s|IS (i) ) = {{ak , k ∈ IA (i)}, {sj , j ∈ IS (i)}} ∈ Ci . The degree of a local code Ci is defined as the number of symbol and state variables that are involved in Ci . Conversely, the degree of a symbol or state variable is defined as the number of local codes in which it is involved. In a group realization, each symbol variable Ak and state variable Sj is a group, and each local code Ci is a subgroup of the direct product A(i) × S(i). The full behavior B of a state realization is the subset of all configurations (a, s) ∈ A × S such that all local constraints are satisfied; i.e., such that (a|IA (i), s|IS (i) ) ∈ Ci for all i ∈ IC . In a group state realization, B is a subgroup of A × S. The code C that is realized by such a state realization is the projection C = B|A of the full behavior B onto the symbol configuration space A; i.e., the set all a ∈ A that occur as part of some valid configuration (a, s) ∈ B. In a group state realization, C is a subgroup of A; i.e., a group code. A normal realization is a state realization in which all symbol variables have degree 1 and all state variables have degree 2. In other words, each symbol variable is involved in precisely one local code, and each state variable is involved in precisely two local codes. A normal graph is a graphical model of a normal realization with the following elements: 1. 2. 3. 4.

Each symbol variable Ak is represented by a leaf edge of degree 1; Each state variable Sj is represented by an ordinary edge of degree 2; Each local code Ci is represented by a vertex; An edge is incident on a vertex if the corresponding state or symbol variable is involved in the corresponding local code.

Codes on Graphs: A Survey for Algebraists

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A normal graph is thus a graph with leaves; i.e., a simple hypergraph in which edge degrees can be only 1 or 2. A normal realization may be characterized by the properties of its associated normal graph; e.g., we may say that a realization is “connected” or “cycle-free.” It is shown in [For98] that any state realization can be “normalized” to become a normal realization; i.e., given any realization of a code C, one can find an equivalent normal realization of C. A group normal realization of a group code C may be “dualized” by replacing each state or symbol variable by its character group, replacing each local code by its dual (orthogonal) code (in the sense of Pontryagin duality), and inverting the sign of each state variable in one of the two local codes in which it is involved. In the case of linear codes over a field F, symbol and state variables are unchanged by this procedure, and local codes are replaced by their dual codes in the usual sense. Moreover, if the field has characteristic 2, then sign inversions are unnecessary. The dual realization theorem of [For98] shows that such a dual realization necessarily realizes the dual code C ⊥ . 1.2

Decoding Codes on Graphs

From a practical viewpoint, the most important property of a graphical model is that it specifies a family of associated “fast” decoding algorithms, called generically the sum-product algorithm. See [AM98,KFL98,For98] for expositions of this algorithm. The essential features of the sum-product algorithm are local computations at each vertex using the sum-product update rule, and “message-passing” of results along edges. Two principal variants of the sum-product algorithm can perform APP decoding, which computes the a posteriori probability of every value of every symbol and state variable, or ML decoding, which computes the maximum-likelihood value and its likelihood for every symbol and state variable. On a cycle-free graph, the sum-product algorithm completes in a time of the order of the diameter of the graph, and is exact. On a graph with cycles, the sum-product update rule may still be used locally, but the global algorithm becomes iterative and approximate. Nonetheless, it often performs very well; e.g., it is the standard decoding algorithm for capacity-approaching codes such as turbo codes and low-density parity-check codes.

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Examples

Examples will now be given of four types of normal realizations: conventional state realizations (trellis realizations), tail-biting realizations, realizations of ReedMuller codes, and parity-check realizations such as are used for LDPC codes.

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Conventional State Realizations (Trellises)

A conventional state realization of a code C— i.e., a trellis realization— is obtained by ordering the symbol index set IA and identifying it with a subinterval of the integers; e.g., IA = [0, n). Notice that the ordering of IA is a design choice. The state index set IS is taken to be essentially the same time axis, e.g., IS = [0, n]. The state spaces Sk , k ∈ IS are generally chosen to be as small as possible. For group codes, the State Space Theorem [FT93] defines an essentially unique minimal state space Sk for each time k, namely the quotient group Sk =

C , C|k− × C|k+

where C|k− and C|k+ are the subcodes of C that are supported entirely on the “past” k − = [0, k) or on the “future” k + = [k, n), respectively. In particular, the initial and final state variables may always be taken to be trivial unary variables: i.e., |S0 | = |Sn| = 1. The local code index set is also the same, e.g., IC = [0, n). The local codes Ck specify the configurations (sk , ak , sk+1 ) ∈ Sk × Ak × Sk+1 that can actually occur; i.e., which state transitions (sk , sk+1 ) ∈ Sk × Sk+1 can occur, and which (output) symbols ak ∈ Ak may be associated with each possible transition. In other words, the local codes specify trellis sections. Thus every symbol variable Ak is involved in one local code, namely Ck , and every state variable Sk other than S0 or Sn is involved in two local codes, namely Ck and Ck−1 . Actually S0 and Sn are not involved in the realization, because a function of a unary variable does not actually depend on that variable. Therefore, excluding S0 and Sn , a conventional state realization is a normal realization. Figure 1 depicts the normal graph of a conventional state realization of a block code of length 6. State variables other than S0 or S6 are represented by ordinary edges, symbol variables are represented by leaf edges (indicated by a special symbol), and local constraints (trellis sections) are represented by vertices.

Figure 1. Normal graph of a conventional state realization (trellis). A conventional state realization is evidently cycle-free, so the sum-product algorithm is exact and completes in a finite time of the order of the length of the code. In this case the sum-product algorithm for APP decoding is known as the “forward-backward” or “BCJR” algorithm, while for ML decoding it is equivalent to the Viterbi algorithm [AM98,KFL98]. The complexity of the algorithm is proportional to the maximum state space size, or more precisely to the maximum local constraint size (“branch complexity”). In the past decade there has been a great deal of research into minimizing the trellis complexity of block codes, which for linear codes is essentially a matter of

Codes on Graphs: A Survey for Algebraists

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finding the best ordering of the symbol index set IA . This research is summarized in [Var98]. For many of the best-known families of codes, such as Reed-Muller codes and the (24, 12, 8) Golay code, the optimal coordinate ordering is known. If the trellis of Figure 1 is a minimal realization of a linear code C for a given ordering of IA , then the dual realization theorem implies that the dual trellis shown in Figure 2 is a minimal realization of the dual linear code C ⊥ for the same coordinate ordering. (The small circles in Figure 2 represent sign inverters.) This result not only confirms the well-known fact that dual codes have identical minimum state complexity profiles in a given coordinate ordering [For88], but also gives an explicit specification for the dual trellis, as found previously by Mittelholzer [Mit95].

Figure 2. Dual trellis, realizing dual code. 2.2

Tail-Biting Realizations

A tail-biting realization of a code C is obtained by identifying the symbol index set IA with a circular time axis, which may be represented by the cyclic group Zn of integers modulo n. Again, the mapping from IA to Zn is a design choice. The state index set IS and the local code index set IC are also taken as Zn . Again, the local codes Ck specify trellis sections; i.e., the configurations (sk , ak , sk+1 ) ∈ Sk × Ak × Sk+1 that can actually occur, where the indices are evaluated in Zn (i.e., modulo n). Thus again every symbol variable Ak is involved in one local code Ck , and every state variable Sk is involved in two local codes, Ck and Ck−1 . Therefore a tail-biting realization is a normal realization. Figure 3 shows the normal graph of a tail-biting realization of a block code of length 8.

Figure 3. Normal graph of a tail-biting realization (trellis). Given a tail-biting realization of a group code C, the dual realization theorem again implies that the dual code C ⊥ is realized by the dual tail-biting realization, which in particular has the same state space sizes. Again, the state spaces Sk should be chosen to be as small as possible. On a graph with cycles, no definite minimum for |Sk | can be given in general. However, there is a useful lower bound based on cut sets.

6

G. David Forney, Jr.

A cut set of a connected graph is a minimal set of edges such that removal of that set partitions the graph into two disconnected subgraphs. The symbol variables are correspondingly partitioned into two disjoint subsets, say “past” and “future.” The Cut Set Lower Bound [WLK95,CFV99] lowerbounds the product of the sizes of the state spaces corresponding to the cut set by the minimum size of the state space in any conventional trellis for the same code that has the same “past” and “future” symbol subsets. In a tail-biting trellis, every cut set comprises two edges. For two diametrically opposite edges, the Cut Set Lower Bound implies that the product of the two corresponding state space sizes cannot be less than the minimum state space size Smid at the midpoint of any conventional trellis for the same code. This in turn implies √ that the minimal maximum state space size in any tail-biting trellis is at least Smid . For example, for the (24, 12, 8) Golay code, it is known that Smid = 256, so the minimal maximum state space size is lowerbounded by 16. In fact, there exists a 16-state tail-biting trellis for this code [CFV99]. Sum-product decoding is inexact, but on an additive white Gaussian noise (AWGN) channel is within 0.1 dB of ML decoding performance. In this example and in general, the Cut Set Lower Bound shows that dramatic reductions in state complexity are possible in realizations with cycles. On the other hand, in cycle-free realizations, every edge is a cut set, and therefore the size of every state space is lowerbounded by the size of some state space in some conventional trellis. Therefore most research on “codes on graphs” has focused on realizations with cycles, even though sum-product decoding is then inexact. 2.3

Realizations of Reed-Muller Codes

Reed-Muller (RM) codes are an infinite family of binary linear block codes that include many of the best shorter codes. RM codes are known to have rather simple conventional state realizations (trellises) [For88,Var98]. The Reed-Muller code RM(r, m) of length 2m and minimum distance d = 2m−r is defined as the code generated by the rows of the 2m × 2m binary Hadamard matrix H2m that have weight d or greater. In other words, the codewords are given by a = AH2m , where the components of the information vector A corresponding to rows of weight less than d are zero, and the remaining components are free. This definition leads to an efficient normal realization for every RM code. For example, Figure 4 is the graph of a realization of the (8, 4, 4) RM code. The codeword a is represented by the 8 binary symbol variables at the top. The information vector A, which is not visible, is represented by the 8 binary state variables at the bottom, 4 of which are constrained to be zero and 4 of which are free (indicated by “0” and “F” constraints, respectively). The remainder of the realization enforces the relation a = AH8 with a “fast Hadamard transform,” whose basic building block is a 2×2 Hadamard transform. All state spaces (edges) are binary, and all local constraints are either (3, 1, 3) repetition codes or (3, 2, 2) single-parity-check (SPC) codes, indicated by “=” or “+,” respectively.

Codes on Graphs: A Survey for Algebraists

7

Figure 4. Realization of (8, 4, 4) Reed-Muller code. The dual realization is obtained by interchanging zero and free constraints, as well as repetition and SPC constraints. Starting from a graph for an RM(r, m) code as above, dualization produces a graph which is the left-right mirror image of the graph for the RM(m − r − 1, m) code. The dual realization theorem then implies the well-known result that these two codes are duals. For any particular code, the generic realization above can be reduced by propagating the effects of the free and zero constraints upward in the realization. For example, a reduced realization of the (8, 4, 4) RM code is shown in Figure 5.

Figure 5. Reduced realization of (8, 4, 4) RM code. These realizations are very efficient, with binary state spaces and of the order of n log2 n constraints of the simplest type, namely (3, 1, 3) or (3, 2, 2). However, in general they have cycles, which degrade decoding performance. For example, on an AWGN channel, decoding of the (8, 4, 4) code using the realization of Figure 5 is about 0.2 dB inferior to ML decoding, and for larger codes the performance loss increases rapidly. Cycles may be removed by clustering (“sectionalization”). For example, with the clusters indicated by dotted lines in Figure 5, the cycle-free realization of the

8

G. David Forney, Jr.

(8, 4, 4) code shown in Figure 6 is obtained. All state and symbol variables are now quaternary. The rectangular constraints represent isomorphisms between quaternary variables, which require no computation in a sum-product update. The two constraints denoted by “(6, 3)” represent (6, 3) binary linear codes.

Figure 6. Cycle-free realization of (8, 4, 4) code. This realization is in fact equivalent to the standard 4-state sectionalized trellis realization of this code with 2-bit sections. Similar cycle-free realizations with a similar binary tree structure can be constructed for all RM codes. Their maximum constraint complexity (“branch complexity”) is always equal to that of the minimal standard trellis realization for the same code. However, they specify exact ML decoding algorithms that are typically somewhat more efficient than Viterbi algorithm decoding of the standard trellis, because of their “divide-by-2” tree structure. In fact, these algorithms are precisely those that were proposed for ML decoding of RM codes in [For88]. As a final simple example, the reduced realization of the (8, 7, 2) SPC code is shown in Figure 7, as well as the standard 2-state trellis. Both realizations involve 6 (3, 2, 2) constraints, so both have the same “space complexity.” However, because of its binary tree structure, the diameter of the former realization is less, so its “time complexity” is less (2 log2 n/2 vs. n − 2, in the general case). This shows how the complexity of even such an apparently irreducible realization as a 2-state SPC trellis can be reduced by a more general graphical model.

Figure 7. Realizations of (8, 7, 2) code: (a) tree; (b) trellis. 2.4

Low-Density Parity-Check Codes

A binary linear block code C may be specified by an n × r parity-check matrix H T , as follows: C = {a ∈ Fn2 | aH T = 0}. The normal graph of such a parity-check realization is bipartite, and is shown in Figure 8(a). It is essentially equivalent to a Tanner graph [Tan81]. The dual bipartite graph, representing a dual generator-matrix realization of a code C ⊥ with generator matrix H, is shown in Figure 8(b).

Codes on Graphs: A Survey for Algebraists

9

Figure 8. Realizations based on (a) parity-check matrix; (b) generator matrix. Low-density parity-check (LDPC) codes, originally proposed by Gallager [Gal62], are codes in which the matrix H T (and therefore the adjacency matrix in the graph of Figure 8(a)) is sparse. Although such a graph has cycles, their girth can be made large. Sum-product decoding performance closely approaching channel capacity has now been demonstrated with long LDPC codes [RSU99]. For very long blocks, a pseudo-random choice of H T appears to be satisfactory. For moderate block lengths of the order of 1000 bits, however, it is expected that algebraic techniques may be helpful in specifying an optimal H T . There may also be some advantage to using nonbinary “states” (edges).

References AM98. S. M. Aji and R. J. McEliece, “The generalized distributive law,” submitted to IEEE Trans. Inform. Theory (1998). CFV99. A. R. Calderbank, G. D. Forney, Jr. and A. Vardy, “Minimal tail-biting trellises: The Golay code and more,” IEEE Trans. Inform. Theory 45 (1999) 1435–1455. For88. G. D. Forney, Jr., “Coset codes II: Binary lattices and related codes,” IEEE Trans. Inform. Theory 34 (1988) 1152–1187. For98. G. D. Forney, Jr., “Codes on graphs: Normal realizations,” submitted to IEEE Trans. Inform. Theory (1998). FT93. G. D. Forney, Jr. and M. D. Trott, “The dynamics of group codes: State spaces, trellis diagrams and canonical encoders,” IEEE Trans. Inform. Theory 39 (1993) 1491–1513. Gal62. R. G. Gallager, “Low-density parity-check codes,” IRE Trans. Inform. Theory IT-8 (1962) 21–28. KFL98. F. R. Kschischang, B. J. Frey and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” submitted to IEEE Trans. Inform. Theory (1998). Mit95. T. Mittelholzer, “Convolutional codes over groups: A pragmatic approach,” Proc. 33d Allerton Conf. Commun. Control Comput. (1995) 380–381. RSU99. T. J. Richardson, A. Shokrollahi and R. Urbanke, “Design of provably good low-density parity-check codes,” submitted to IEEE Trans. Inform. Theory (1999). Tan81. R. M. Tanner, “A recursive approach to low-complexity codes,” IEEE Trans. Inform. Theory IT-27 (1981) 533–547. Var98. A. Vardy, “Trellis structure of codes,” in Handbook of Coding Theory (V. Pless and W. C. Huffman, eds.), Elsevier, Amsterdam (1998). WLK95. N. Wiberg, H.-A. Loeliger and R. K¨ otter, “Codes and iterative decoding on general graphs,” Euro. Trans. Telecomm. 6 (1995) 513–525.

RA Codes Achieve AWGN Channel Capacity
Hui Jin and Robert J. McEliece California Institute of Technology Pasadena, California USA {hui,rjm}@systems.caltech.edu

Abstract. In ref. [3] we introduced a simplified ensemble of serially concatenated “turbo-like” codes which we called repeat-accumulate, or RA codes. These codes are very easy to decode using an iterative decoding algorithm derived from belief propagation on the appropriate Tanner graph, yet their performance is scarcely inferior to that of full-fledged turbo codes. In this paper, we prove that on the AWGN channel, RA codes have the potential for achieving channel capacity. That is, as the rate of the RA code approaches zero, the average required bit Eb /N0 for arbitrarily small error probability with maximum-likelihood decoding approaches log 2, which is the Shannon limit. In view of the extreme simplicity of RA codes, this result is both surprising and suggestive.

1

Introduction

In ref. [3] we introduced a class of turbo-like codes, the repeat and accumulate (RA) codes, which are simple enough to allow a fairly complete theoretical analysis, yet powerful enough to perform nearly as well as full-fledged turbo codes. The general idea is shown in Figure 1. An information block of length k is repeated q times, scrambled by a pseudorandom permutation (interleaver) of size qk, and then encoded by a rate 1 accumulator. The accumulator can be viewed as a truncated rate-1 recursive convolutional encoder with transfer function 1/(1 + D), but we prefer to think of it as a block encoder whose input block [x1 , . . . , xn ] and output block [y1 , . . . , yn ] are related by the formula y1 = x1 y2 = x1 + x2 y3 = x1 + x2 + x3 .. . yn = x1 + x2 + x3 + · · · + xn . In other words, the outputs of the accumulator are the mod-2 partial sums of the inputs. An RA code as just described is therefore a (qk, k) linear block code, but because of the unspecified interleaver, there are a large number, (qk)! to be precise, RA codes with these parameters. 

This work was supported by NSF grant no. CCR-9804793, and grants from Sony and Qualcomm.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 10–18, 1999. c Springer-Verlag Berlin Heidelberg 1999


RA Codes Achieve AWGN Channel Capacity

k

qk

rep. q

P

qk

acc.

11

qk

Fig. 1. Encoder for a (qk, k) RA code. The “rep. q” component repeats its kbit input block q times; the “P ” block represents an arbitrary permutation of its qk-bit input block; and the “acc.” is an accumulator, whose outputs are the mod-2 partial sums of its inputs.

For the theorist, the most important thing about RA codes is that their combinatorial properties are reasonably well understood [3],[4]. For the practitioner, the most important thing is the experimentally verified fact [3] that if an iterative decoding algorithm derived from belief propagation on the appropriate Tanner graph is applied to them, their AWGN channel performance is scarcely inferior to that of full-fledged turbo codes. In this paper, we will give a partial explanation of the latter property, by using the former property to show that on the AWGN channel, RA codes have the potential for achieving channel capacity. That is, as the rate of the RA code approaches zero, the average required bit Eb /N0 for arbitrarily small error probability with maximum-likelihood decoding approaches log 2, which is the Shannon limit. The remaining problem, of course, is to explain why the low-complexity but suboptimal iterative decoding algorithm performs so well.

2

An Ensemble AWGN Coding Theorem for RA Codes

Our main theorem deals not with any particular RA code, but with the average, or ensemble performance, as the block length approaches infinity. Therefore we begin with a brief description of what we mean by an ensemble of codes. An ensemble of linear codes is a sequence Cn1 , Cn2 , . . . of sets of linear codes of a common rate R, where Cni is a set of (ni , ki ) codes with ki /ni = R. We assume that the sequence n1 , n2 , . . . approaches infinity. If C is an (n, k) code in the ensemble, the weight enumerator of C is the list A0 (C), A1 (C), . . . , An (C), where Ah (C) is the number of words of weight h in C. The average weight enumerator for the set Cn is defined as the list (n)

(n)

(n)

A0 (C), A1 (C), . . . , An (C), where (n) def

Ah

=

1
Ah (C) |Ck | C∈Ck

for h = 0, 1, . . . , n.

(2.1)

12

Hui Jin and Robert J. McEliece

Also, we define the spectral shape of Cn , def

rn (δ) =

1 (n) log Aδn , n

for 0 < δ < 1,

(2.2)

and the ensemble spectral shape : def

r(δ) = lim rn (δ)

for 0 < δ < 1,

n→∞

(2.3)

where we are implicitly assuming that the limit exists. As a special case, consider the ensemble of rate R = 1/q RA codes. Here the sequence n1 , n2 , n3 . . . is q, 2q, 3q, . . . , and the set Cqk consists of (qk)! linear (n, k) block codes, all with n = qk. In [3][4] it was shown that the spectral shape for this ensemble of codes is given by the formula   H(u) , (2.4) f(u, δ) + rq (δ) = max 0≤u≤min(2δ,2−2δ) q where def

f(u, δ) = −H(u) + (1 − δ)H(

u u ) + δH( ), 2(1 − δ) 2δ

(2.5)

and H(u) = −(u log u + (1 − u) log(1 − u)) is the (natural) entropy function. Figure 2 shows the function rq (δ) for q = 4.

Weight Spectrum of 4−RA code

0.2

r(δ)

0.1

0

−0.1

0

0.1

0.2

0.3

0.4

0.5 δ=h/N

0.6

0.7

Fig. 2. The function of r4 (δ)

0.8

0.9

1

RA Codes Achieve AWGN Channel Capacity

13

In [4], using a bound derived in [2], we proved that if we define, for each integer q ≥ 2, def

1 − δ 1 − e−2rq (δ) δ 2 0 0.792 dB, then as n → ∞, the word error probability for the ensemble of rate 1/3 RA codes approaches zero.1 On the other hand, for Eb /N0 < −0.495 dB, (the Shannon limit for binary codes of rate 1/3) as n → ∞, the word error probability for any sequence of codes of rate 1/3 must approach 1. In the table, and more clearly in Figure 3, we see that as the rate R approaches zero, the Shannon limit approaches log 2 = −1.592 dB. This is of course well known, and in fact the value −1.592 dB is usually referred to as the Shannon limit for the AWGN channel. The interesting thing for us, however, is that the RA code thresholds also seem to approach −1.592 dB. This empirical observation is in fact true, and it is the object of this paper to prove the following theorem. Theorem 21. We have limq→∞ γq = log 2, i.e., RA codes achieve the Shannon limit for the AWGN channel. We will give a proof of Theorem 21 in Section 3. But here we note that it is easy to prove that the limit cannot be smaller than log 2:2 1

2

We have included the threshold for q = 2 in the table because it can be shown that for rate 1/2 RA codes, if Eb /N0 > γ2 , the ensemble bit error probability approaches zero, although the word error probability does not. Of course, Theorem 22 follows from the fact that log 2 is the Shannon limit for the channel, but it is interesting to have an “elementary” proof.

14

Hui Jin and Robert J. McEliece

8

Binary Shannon Limit RA code threshold (GD bound)

7

6

5

Eb/No (dB)

4

3

2

1

0

−1

−2

0

0.1

0.2

0.3

0.4

0.5 0.6 Code Rate R

0.7

0.8

0.9

1

Fig. 3. The RA thresholds

Theorem 22. With the threshold γq defined as above, lim inf q→∞ γq ≥ log 2. Proof: First note that in (2.5), we have f(u, 1/2) = 0, so that rq (1/2) = max H(u)/q = H(1/2)/q = 0≤u≤1

log 2 . q

Thus by taking δ = 1/2 in the definition (2.6) of c0 (q), we have the lower bound c0 (q) ≥ =

1 − (1/2) 1 − e−2rq (1/2) (1/2) 2 1 − e−2 log 2/q . 2

It therefore follows that γq = qc0 (q) ≥

1 − e−(2/q) log 2 . (2/q)

(2.8)

The desired result now follows by observing that the limit, as q → ∞, of the right side of (2.8) is log 2. ✷

RA Codes Achieve AWGN Channel Capacity

3

15

Proof of Main Theorem

In this section, we present the proof of Theorem 21. We begin with some preliminary technical results which concern the spectral shape functions rq (δ). Lemma 31. The function f(u, δ) defined in (2.5) is nonpositive, i.e., for any (u, δ) ∈ [0, 1] × [0, 1], f(u, δ) ≤ 0. Furthermore, f(u, δ) = 0 if and only if δ = 1/2 or u = 0. Proof: Jensen’s inequality, and the fact that H(u) is strictly convex ∩, implies that for any δ ∈ [0, 1], and any two numbers u1 , u2 ∈ [0, 1], we have H(δu1 + (1 − δ)u2 ) ≥ δH(u1 ) + (1 − δ)H(u2 ), with equality if and only if u1 = u2 and/or δ = 0 or 1. Letting u1 = u/(2δ) and u2 = u/(2(1 − δ)), we obtain the desired result. ✷ Corollary 32. We have, for each q ≥ 2 and δ ∈ [0, 1], rq (δ) ≤

log 2 , q

with equality if and only if δ = 1/2. Proof: Let u(q, δ) denote the optimizing value of u in the computation of rq (δ), i.e.,   H(u) f(u, q) + . (3.1) u(q, δ) = arg max q 0≤u≤min(2δ,2−2δ) Then by the definition of rq (δ), we have rq (δ) = f(u(q, δ), δ) + H(u(q, δ))/q. But since by Lemma 31, f(u, δ) ≤ 0, we have rq (δ) ≤ H(u(q, δ))/q ≤

log 2 . q

(3.2)

For equality to hold in (3.2), we must have, for u = u(q, δ), both f(u, δ) = 0 and u = 1/2. But by Lemma 31, if f(u, δ) = 0, then either u = 0 or δ = 1/2. Thus equality can hold only when δ = 1/2, as asserted. ✷ Lemma 33. If u(q, δ) is defined as in (3.1), then   q u(q, δ) · (1 − u(q, δ))q−1 ≤ 2 δ(1 − δ) .

(3.3)

16

Hui Jin and Robert J. McEliece

Proof: It is easy to check that the “max” in (2.4) does not occur at either endpoint (at u = 0 the function is zero and its derivative is infinite, and at u = min(2δ, 2(1 − δ)), the function is negative), so at the point u = u(q, δ), the partial derivative of the function in braces on the right side of (2.4), with respect to u, must be zero. This condition works out to be −(1 − 1/q) log(1 − u) − (1/q) log u + (1/2) log(2 − 2δ − u) + (1/2) log(2δ − u) = 0,

or in exponential form, 

(2 − 2δ − u)(2δ − u) = 1. (1 − u)1−1/q u1/q

(3.4)

But since 0 ≤ u ≤ min(2δ, 2(1 − δ)), the numerator on the left side of (3.4) is ≤ 2 δ(1 − δ). Therefore at the maximizing point u = u(q, δ), we must have  2 δ(1 − δ) ≥ 1. (3.5) (1 − u)1−1/q u1/q ✷

Rearranging this, we get (3.3).

Corollary 34. If (δq ) is a sequence of real numbers in the range [0, 1], and limq→∞ δq = δ ∗ = 1/2, then limq→∞ u(q, δq ) = 0.  Proof: If δ ∗ = 1/2, then 2 δ ∗ (1 − δ ∗ ) < 1, and so the right hand side of (3.3) approaches zero as q → ∞. Thus by (3.3), u(q, δ) must approach either zero or one. But since u ≤ min(2δ ∗ , 2 − 2δ ∗ ) < 1, the only possibility is u(q, δq ) → 0. ✷ Corollary 35. There exists a q0 ≥ 2 and δ0 > 0 such that if q ≥ q0 and δ ≤ δ0 , then u(q, δ) < δ 2 . Proof: Let δ < 1/2. Then u < min(2δ, 2 − 2δ) = 2δ, so that the left side of (3.3) is lower bounded by the quantity u(1 − 2δ)q . Thus from (3.3), we obtain q   2 δ(1 − δ) . (3.6) u(q, δ) ≤ 1 − 2δ If the quantity in parenthesis in (3.6) is less than one, which is true for δ < 0.14, then the right hand side is decreasing in q. For example, if q ≥ 6, we have   6 2 δ(1 − δ) , (3.7) u(q, δ) ≤ 1 − 2δ which, being of order δ 3 , will plainly be ≤ δ 2 for small enough δ. ✷ In the definition of c0 (q) in (2.6), let δ(q) be the optimizing value of δ, i.e., def

δ(q) = arg sup 0

xy 7

< x , y 10 > y 10

y

Fig. 1. The monomial ideal M = x4 , x3 y 4 , xy 7 , y 10 , with its generators (white circles), standard monomials (black dots), and irreducible components (shaded circles). At any stage in Buchberger’s algorithm for computing Gr¨ obner bases, one considers the S-pairs among the current polynomials and removes those which are redundant [7]. The minimal S-pairs define a graph GM on the generators of any monomial ideal M. Our aim is to study this graph, and to give an elementary introduction to recent work on computing the following objects associated to M: Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 19–28, 1999. c Springer-Verlag Berlin Heidelberg 1999


20

Ezra Miller and Bernd Sturmfels

i. the Hilbert series of M, i.e. the formal sum of all monomials not in M; ii. the minimal resolution of M by free modules over the polynomial ring; iii. the primary decomposition and the irreducible decomposition of M. For a first introduction to these problems see [10, §9.2] and [11, Exercises 3.8 and 17.11]. Their importance for Buchberger’s Algorithm was emphasized by M¨ oller and Mora in [16]. Research papers describing effective algorithms are [4] and [6] for problem (i), [9] and [14] for problem (ii), and [8] for problem (iii). Our point of view differs from these sources. We regard (i), (ii), (iii) as the same problem and we develop combinatorial tools for presenting structured solutions. It is our belief that this approach will ultimately lead to faster algorithms. This paper is organized as follows. The easy solution for n = 2 variables will be reviewed in Section 2. In Sections 3 and 4 we present the generalization to the case of three variables. We shall see that the S-pair graphs GM for n = 3 are precisely the planar graphs. Section 5 summarizes what is known for n ≥ 4.

2

Two Variables

Let us begin by answering the three computational questions for our example. 1. The Hilbert series of the monomial ideal M =  x4 , x3 y 4 , xy 7 , y 10  equals 1 − x4 − x3 y 4 − xy 7 − y 10 + x4 y 4 + x3 y 7 + xy 10 . (1 − x)(1 − y)

(2.1)

This expression equals the sum of the 28 standard monomials in Figure 1. 2. The minimal free resolution of M is the following exact sequence of modules, ∂

∂

1 0 K[x, y]4 −→ M −→ 0 , 0 −→ K[x, y]3 −→  

y4

0

−x y3 where, in matrix notation, ∂1 =   0 −x2 0

0

0 y

3

(2.2)

   and ∂0 = x4 

x3 y4 xy7 y10



.

0 −x

3. The ideal M is x, y-primary and its irreducible decomposition equals M

=

x4 , y 4  ∩ x3 , y 7  ∩ x, y 10  .

(see Figure 1) (2.3)

It is not difficult to extend this to an arbitrary monomial ideal in two variables, M =  xa1 y b1 , xa2 y b2 , . . . , xar y br , where a1 > · · · > ar and b1 < · · · < br . Proposition 1. The following holds for any monomial ideal M in two variables: 1. The Hilbert series of M equals

i j

{x y

i j

: x y ∈ M}

=

1−

r

j=1

xaj y bj +

r−1 i=1

(1 − x)(1 − y)

xai y bi+1

. (2.4)

Monomial Ideals and Planar Graphs

21

2. The minimal free resolution of M equals ∂

∂

1 0 0 −→ K[x, y]r−1 −→ K[x, y]r −→ M −→ 0,

(2.5)

where ∂0 is the canonical map and ∂1 is given on standard basis vectors by ∂1 (ei ) = y bi+1 −bi · ei − xai −ai+1 · ei+1 . 3. The irreducible decomposition of M equals M = y b1  ∩ xa1 , y b2  ∩ xa2 , y b3  ∩ · · · ∩ xar−1 , y br  ∩ xar  , (2.6) where the first or last component are to be removed if b1 = 0 or ar = 0. This implies that for monomials in two variables only consecutive S-pairs matter. Corollary 1. When applying Buchberger’s Algorithm to r polynomials  in two variables, it suffices to form the r − 1 consecutive S-pairs, instead of all r2 pairs.

3

Three Variables: The Generic Case

This section gives a combinatorial introduction to the results in the article [3]. The Hilbert series of any monomial ideal M =  m1 , m2 , . . . , mr  in the polynomial ring K[x1 , . . . , xn ] can be written using Inclusion–Exclusion as follows. For I ⊆ {1, 2, . . . , r} let mI denote the least common multiple of {mi : i ∈ I}. Proposition 2. The Hilbert series of M equals 1 {xi11 xi22 · · · xinn ∈ M} = n · (1 − xi ) i=1



(−1)|I| · mI .

(3.1)

I⊆{1,2,... ,r}

Unfortunately this formula is useless for our problem (i), because the number of summands is an exponential function in r. But almost all summands cancel. In this section we show that for n = 3 the true number of summands in the numerator of the Hilbert series of M is a linear function in r. We write x, y, z for the variables. To simplify our discussion, we first assume that r ≥ 4 and M is generic in the following sense: if mi = xai y bi z ci and mj = xaj y bj z cj are minimal generators of M neither of which is a power of a single variable, then ai = aj , bi = bj , ci = cj . In Section 4 we shall remove this hypothesis. We define a graph GM with vertices 1, 2, . . . , r by declaring {i < j} an edge whenever mij = lcm(mi , mj ) is not a multiple of mk for any k ∈ {1, . . . , r}\{i, j}. Let TM be the set of triples {i < j < k} which form a triangle {i, j}, {i, k}, {j, k} in GM , and which do not consist entirely of powers xai , y bj , z ck of the variables. Consider the free module K[x, y, z]TM with basis { eijk : {i, j, k} ∈ TM }, the free module K[x, y, z]GM with basis { eij : {i, j} ∈ GM }, and the linear maps ∂1 : K[x, y, z]GM → K[x, y, z]r

mij mij mj · ej − mi · ei , m m eijk → mijk · ejk − mijk · eik jk ik

, eij →

∂2 : K[x, y, z]T M → K[x, y, z]GM ,

+

mijk mij

· eij . (3.2)

22

Ezra Miller and Bernd Sturmfels

Theorem 1. Let M be a generic monomial ideal in K[x, y, z]. 1. The Hilbert series of M equals

r



1 − i=1 mi + {i,j}∈GM mij − {i,j,k}∈TM mijk (1 − x)(1 − y)(1 − z)

.

(3.3)

2. The minimal free resolution of M equals ∂

∂

∂

2 1 0 K[x, y, z]GM −→ K[x, y, z]r −→ M −→ 0 . 0 −→ K[x, y, z]TM −→

(3.4)

3. If M is artinian then the irreducible decomposition equals M =  xdeg x (mijk ) , y degy (mijk ) , z degz (mijk )  .

(3.5)

{i,j,k}∈TM

Here M is called artinian if some power of each variable is in M, or equivalently, if the number of standard monomials is finite. The non-artinian case can be reduced to the artinian case by considering M + xm , y m , z m  for m  0. We illustrate Theorem 4 for the generic artinian monomial ideal 1 2 3 4 5 6 7 8 9 10 11 12 x10 , y 10 , z 10 , x8 y 6 z, x9 y 3 z 2 , x4 y 8 z 3 , x7 y 5 z 4 , xy 9 z 5 , x3 y 7 z 6 , x5 y 4 z 7 , x6 yz 8 , x2 y 2 z 9  =

x10 , y 10 , z x10 , y 3 , z 8  x2 , y 9 , z 10  x4 , y 9 , z 6 

∩ ∩ ∩ ∩

x8 , y 10 , z 3  x9 , y 4 , z 8  x, y 10 , z 10  x6 , y 4 , z 9 

∩ ∩ ∩ ∩

x7 , y 8 , z 6  x5 , y 7 , z 9  x8 , y 8 , z 4  x10 , y 6 , z 2 

∩ ∩ ∩ ∩

x9 , y 5 , z 7  ∩ x7 , y 7 , z 7  ∩ x3 , y 9 , z 9  ∩ x10 , y, z 10  ∩ x9 , y 6 , z 4  ∩ x4 , y 10 , z 5  ∩ x6 , y 2 , z 10 

3 12 9 11

10 7

8

6

2

5 1

4

Fig. 2. A generic monomial ideal with 12 generators (white circles), 30 minimal S-pairs (black dots/edges), and 19 irreducible components (shaded circles/triangular regions).

Monomial Ideals and Planar Graphs

23

The right picture shows the graph GM . Its triangles correspond to irreducible components; for instance, the last component x6 , y 2 , z 10  arises from the triangle {3, 11, 12} ∈ TM . The numerator of the Hilbert series has 62 = 1+12+30+19 summands, one for each vertex, edge and region of the planar graph GM . We are now ready to establish the connection promised in this paper’s title. Theorem 2. The S-pair graph GM of a generic monomial ideal M in three variables is planar. If M is artinian then (GM , TM ) is triangulation of a triangle. Proof sketch. It suffices to consider artinian ideals M, since one can throw in high powers xm , y m , z m , compute the S-pair graph, and then obtain GM by deleting any edges involving xm , y m , or z m . We may further assume that each other generator of M is divisible by xyz. The idea is to construct a 3-dimensional polytope whose edge graph equals GM . Our claim follows because edge graphs of 3-polytopes are planar. For instance, Figure 2 is the edge graph of an icosahedron. Fix a large real number t  0. Let Pt denote the convex hull of the points (tai , tbi , tci ) ∈ R3 as (ai , bi , ci ) runs over the exponents on the minimal generators xai y bi z ci of M. Each such point is a vertex of Pt , and the edge graph of Pt is independent of t. We call Pt the hull polytope of M. One shows that every edge E of GM is an edge of the hull polytope by producing a linear functional which is minimized on Pt along E (this does not use genericity of M). Finally, the genericity of M is used to show that every edge of Pt is obtained in this fashion. Theorem 2 gives an answer to our questions (i), (ii), (iii) even if M is not generic, but that answer is typically nonminimal. (For a minimal solution see Section 4). The answer is found by deforming the non-generic ideal M to a “nearby” generic ideal M , where  represents a small positive real. For instance, M = x2 , xy, xz, y 2 , yz, z 2  deforms to M = x2 , x1+ y, xz 1− , y 2 , y 1+ z, z 2 . We can apply Theorem 1 to the generic ideal M and then set  = 0 to get the Hilbert series, a choice of minimal S-pairs, and the irreducible decomposition of M. For instance, the irreducible decomposition of the deformation equals M

=

x2 , y, z 1−  ∩ x1+ , y 2 , z 1−  ∩ x, y 2 , z ∩ x, y 1+ , z 2 .

We invite the reader to draw the graph GM and see what happens for  → 0. The preceding discussion allows Theorem 2 to provide the following complexity result for all monomial ideals in K[x, y, z]. Recall that a planar graph GM on r vertices has at most 3r − 6 edges and at most 2r − 5 bounded regions. Corollary 2. 1. The numerator of the Hilbert series of an ideal generated by r monomials in three variables x, y, z has at most 6r − 10 summands. 2. When applying Buchberger’s Criterion to r polynomials in three variables, it suffices to form at most 3r − 6 S-pairs instead of all r2 possible S-pairs.

24

4

Ezra Miller and Bernd Sturmfels

Three Variables: The Non-generic Case

In this section we outline a proof for the following solution to (i), (ii), and (iii): Theorem 3. Every monomial ideal M in K[x, y, z] has a minimal resolution by the bounded regions of a planar graph. That resolution gives irredundant formulas for the numerator of the Hilbert series and the irreducible decomposition of M. For generic monomial ideals M this was established in Theorem 1. Before discussing the proof for arbitrary M, we must first explain the meaning of “minimal resolution by a planar graph” G. Suppose the vertices of G are labeled by the minimal generators m1 , . . . mr of M. Label each edge {i < j} of G by mij = lcm(mi , mj ). Now G, determines a set T of bounded regions, with each region R ∈ T having a certain set of vertices {i1 < · · · < it } on its boundary. We define the label mR = lcm(mi1 , . . . , mit ). Finally, for each region R and each edge {i, j} define the sign ε(R, ij) to be 0 unless {i, j} is on the boundary of R, then 1 if R is on the left as one goes from i to j and −1 if R is on the right. In analogy with Eq. (3.2), consider the following maps of free modules: mij mij · ej − · ei mj mi mR → ε(R, ij) · · eij mij

∂1 : K[x, y, z]G → K[x, y, z]r ,

eij →

∂2 : K[x, y, z]T → K[x, y, z]G ,

eR

(4.1)

{i,j}∈G

which define a complex FG of free K[x, y, z]-modules in analogy with Eq. (3.4). Theorem 3 says that it is always possible to choose a labeled planar graph G so that FG is exact and minimal, i.e., mij = mR for any edge ij of a region R. The graph G will be far from unique for monomial ideals M which are not generic. Example. Consider the mth power of the maximal ideal x, y, z, that is, M

=

x, y, zm

=

 xi y j z k : i, j, k ∈ N, i + j + k = m  .

(4.2)

The staircase of M is depicted in Figure 3(a) for m = 5. The graph in Figure 3(b) represents a free resolution of M for m = 5. This graph is essentially the edge graph of the hull polytope PM of M, as defined in Section 3. In fact, the hull polytope makes sense for any monomial ideal (in any number of variables) and gives a nice but nonminimal solution to our problems (i), (ii), and (iii): The resolution defined by the graph in Figure 3(b) is nonminimal because every downward-pointing triangle has the same label as all three of its edges. We get a graph G satisfying Theorem 3 by deleting precisely one edge from each downward-pointing triangle. When an edge is deleted, the two adjacent regions join together, as the one with the larger label “swallows” the one with the smaller label. Notice how many distinct graphs G result that minimally resolve M ! Since the monomial ideal M is invariant under the symmetric group on x, y, z, we may ask whether there exists a minimal planar graph resolution which is

Monomial Ideals and Planar Graphs

25

(c) (b) (d)

(a) (e) (f)

Fig. 3. Symmetric resolutions of various powers of the maximal ideal x, y, z. invariant. The answer is “yes” if and only if m is congruent to 0 or 1 modulo 3. Examples of these symmetric choices are shown for various m in Figure 3: (c) is m = 3; (d) is, without the dotted edges, m = 6; (e) is m = 4; and (f) is m = 7. The general construction for the case m ≡ 0 (mod 3) is gotten by putting together m2 /9 of the graphs in Figure 3(c) in the pattern of Figure 3(b), erasing the dotted edges as in Figure 3(d) when everything is put together. In the case m ≡ 1 (mod 3), one radiates upside-down side-length 2 triangles outward from the center, cutting them off at the boundary as in Figures 3(e,f). Of course, the upside-down side-length 2 “triangles” are really hexagonal regions in the graph. Proof Sketch of Theorem 3. As in the generic case, we may assume that M is artinian. We can always find some planar graph G such that FG is a (possibly nonminimal) free resolution of M, e.g. by using the hull polytope or by deforming to nearby generic monomial ideal. Since the labels on the vertices of G are minimal generators, the matrix ∂1 never has ±1 entries. The matrix ∂2 has a ±1 entry if and only if some region has the same label as one of its boundary edges. One shows that the graph obtained from G by deleting this edge and joining the two adjacent regions still defines a free resolution of M. A graph G satisfying Theorem 3 is obtained when there are no nonminimal edges left to delete. Once we know that FG is a minimal free resolution then the irredundant irreducible decomposition of M can be read off the labels on the regions of G, in analogy with Eq. (3.5). The Hilbert series of M can be written down in analogy

26

Ezra Miller and Bernd Sturmfels

with Eq. (3.3). None of the monomials in the resulting formula for the numerator cancel. This very last fact relies on n ≤ 3 and may be false for n ≥ 4 variables. We claim that the following converse to Theorem 3 holds. Theorem 4. For every 3-connected planar graph G there exists a monomial ideal M in 3 variables which is minimally resolved by the bounded regions of G. This result is a variant of Steinitz’ Theorem [19, Theorem 4.1] which states that 3-connected planar graphs are the edge graphs of 3-dimensional convex polytopes. If G is a planar triangulation then M can be chosen generic, and, in this case, Theorem 4 follows from Schnyder’s Theorem on order dimension [17, Theorem 6.2.1, pp. 128], as explained in [3, §6]. The general non-triangulated case is more difficult. It does not immediately follow from its order-theoretic analogue due to Brightwell and Trotter [17, Theorem 6.3.1]. The complete proof of Theorem 4 is “under construction” and will be published elsewhere. In Figure 4 is a non-trivial example illustrating the encoding of a planar graph G by a monomial ideal M. Note that the order 8 (dihedral group) symmetry of the graph cannot be reproduced in the monomial ideal. The square is realized by having an irreducible component which is determined by four surrounding generators, two of which have one coordinate in common. Similarly, the hexagons have six generators spread around the corresponding irreducible component, with each such generator sharing one coordinate with the irreducible component. Only the artinian components—those with generators on all three sides—define regions in G. After M has been chosen to realize G, it may be that G can be altered while still remaining a minimal resolution for M. For instance, edge a can have its exterior vertex changed from 1 to 2, or even to 3, making the left hexagon into a heptagon or octagon. Independently, edge b can have its vertex 4 moved to 5, making the right hexagon into a pentagon. What must remain constant are the numbers of vertices, edges, and regions. 2

3

1 a

5

b 4

Fig. 4. A monomial ideal constructed to have the given graph as minimal resolution.

Monomial Ideals and Planar Graphs

5

27

Four and More Variables

The ideas presented in Sections 3 and 4 make perfect sense also for n ≥ 4 variables. There is the hull polytope PM in Rn , and there are deformations to generic monomial ideals M → M . Resolutions by planar graphs can be generalized to resolutions by cell complexes, as defined by Bayer and Sturmfels [5]. The familiar duality of planar graphs now becomes Alexander duality for monomial ideals as defined by Miller [15], and yields an efficient algorithm for problem (iii). However, it is more difficult to construct cellular resolutions which are minimal. There seems to be no n-dimensional analogue to Theorem 3. One obstruction is that the minimal resolution may depend on the characteristic of the field K. Any general formula for minimal resolutions is expected to involve homological algebra over the lcm-lattice of Gasharov, Peeva and Welker [12]. See the recent work of Yuzvinsky [18] for a possible approach. Cellular resolutions provide efficient formulas for Hilbert series and related questions even if they are nonminimal. The following complexity result holds. Theorem 5. Fix n ≥ 2. The numerator of the Hilbert series of an ideal M n generated by r monomials in n variables can have order r 2 many terms for r  0. The same upper and lower bound holds for the number of irreducible components of M and the ranks of the modules in a minimal resolution of M. This upper bound was derived by Bayer, Peeva and Sturmfels [3, Theorem 6.3] from the Upper Bound Theorem for Convex Polytopes [19, Theorem 8.23]. The Upper Bound Theorem states, roughly speaking, that the maximum numn ber of faces of an n-dimensional polytope with r vertices has order r 2 . The matching lower bound for Theorem 5 was established by Agnarsson [1]. There seems to be no analogue to Theorem 4 is higher dimensions. For instance, there is a triangulation of a tetrahedron with three interior vertices [3, §6] which does not support a minimal resolution of a monomial ideal M in R = K[a, b, c, d]. A detailed study of the graphs GM for n = 4 appears in recent work of Agnarsson, Felsner and Trotter [2]. The following example is instructive: M = a9 b9 c9 , d9 , a6 b7 c4 d, a2 b3 c8d5 , a5 b8 c3 d2 , ab4 c7 d6 , a8 b5 c2 d3 , a4 bc6 d7 , a7 b6 cd4 , a3 b2 c5 d8 . Its S-pair graph GM is the complete graph on 12 vertices. The minimal resolution of M is cellular and looks like 0 → R53 → R108 → R66 → R12 → M → 0. It is a triangulation of a tetrahedron with 8 interior vertices, which is neighborly in the sense that any two vertices form an edge. Polytope combinatorics [19, §8] tells us that such a neighborly triangulation has 53 tetrahedra and 108 triangles. Let φ(n) denote the maximum of generators of a monomial ideal M in n variables whose S-pair graph GM equals the complete graph. It is a non-trivial fact that φ(n) is actually finite. However, it grows doubly-exponentially in n; see [17, Theorem 7.2.13]. Note that Corollary 1 implies φ(2) = 2, Corollary 2 implies φ(3) = 4, and φ(4) = 12 is attained by the neighborly monomial ideal M above. The specific values in Table 1 are derived from the following theorem.

28

Ezra Miller and Bernd Sturmfels

Theorem 6. (Hosten and Morris [13]). The number φ(n) equals the number of distinct simplicial complexes T on the set {1, 2, . . . , n − 1} with the property that no two faces of T have their union equal to {1, 2, . . . , n − 1}. Table 1. The maximum number φ(n) of generators of a neighborly monomial ideal. variables = n 2 generators = φ(n) 2

3 4

4 12

5 81

6 2,646

7 1,422,564

8 229,809,982,112

References 1. G. Agnarsson. On the number of outside corners of monomial ideals, Journal of Pure and Applied Algebra, 117 (1997) 3–22. 2. G. Agnarsson, S. Felsner and W. Trotter. The maximum number of edges in a graph of bounded dimension, with applications to ring theory, Discrete Mathematics (201) (1999) 5–19 3. D. Bayer, I. Peeva and B. Sturmfels. Monomial resolutions, Mathematical Research Letters 5 (1998) 31-46. 4. D. Bayer and M. Stillman. Computation of Hilbert functions, Journal of Symbolic Computation 14 (1992), 31–50. 5. D. Bayer and B. Sturmfels. Cellular resolutions of monomial modules. Journal f¨ ur die reine und angewandte Mathematik, 502 123–140, 1998. 6. A. Bigatti, M. Caboara and L. Robbiano. On the computation of Hilbert-Poincar´e series. Appl. Algebra Engrg. Comm. Comput. 2 (1991), no. 1, 21–33. 7. B. Buchberger, A criterion for detecting unnecessary reductions in the construction of Gr¨ obner bases, Proc. EUROSAM 79, Marseille, W. Ng (ed.), Lecture Notes in Computer Science, Vol. 72, Springer 1979, pp. 3–21. 8. M. Caboara, P. Conti and C. Traverso, Yet another ideal decomposition algorithm. Applied algebra, algebraic algorithms and error-correcting codes (Toulouse, 1997), Lecture Notes in Computer Science, Vol. 1255, Springer, 1997, pp. 39–54, 9. A. Capani, G. de Dominicis, G. Niesi and L. Robbiano. Computing minimal finite free resolutions, Journal of Pure and Applied Algebra 117/118 (1997), 105–117. 10. D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms, 2nd edition, Springer-Verlag, New York, 1997 11. D. Eisenbud. Introduction to Commutative Algebra with a View towards Algebraic Geometry, Springer Verlag, New York, 1995. 12. V. Gasharov, I. Peeva and V. Welker, The lcm-lattice, preprint, 1998. 13. S. Hosten and W. Morris Jr. The order dimension of the complete graph, Discrete Mathematics (201) (1999), 133–139. 14. R. La Scala and M. Stillman. Strategies for computing minimal free resolutions, Journal of Symbolic Computation 26 (1998), no. 4, 409–431. 15. E. Miller. Alexander duality for monomial ideals and their resolutions, preprint, 1998, math.AG/9812095. 16. H.M. M¨ oller and F. Mora. New constructive methods in classical ideal theory, Journal of Algebra 100 (1986) 138–178. 17. W. Trotter. Combinatorics and Partially Ordered Sets. Dimension Theory, The Johns Hopkins University Press, Baltimore, 1992. 18. S. Yuzvinsky, Taylor and minimal resolutions of homogeneous polynomial ideals, preprint, 1999. 19. G. Ziegler. Lectures on Polytopes, Springer Verlag, New York, 1994.

A Fast Program Generator of Fast Fourier Transforms Michael Clausen and Meinard M¨ uller Universit¨ at Bonn, Institut f¨ ur Informatik V, R¨ omerstr. 164, D-53117 Bonn, Germany [email protected], [email protected] Abstract. Let G be a finite group of order n. By Wedderburn’s Theorem, the complex group algebra CG is isomorphic to an algebra of block diagonal matrices: CG
⊕hk=1 Cdk ×dk . Every such isomorphism D, a so-called discrete Fourier transform of CG, consists of a full set of pairwise inequivalent irreducible representations Dk of CG. A result of Morgenstern combined with the well-known Schur relations in representation theory show that (under mild conditions) any straight line program for evaluating a DFT needs at least Ω(n log n) operations. Thus in this model, every O(n log n) FFT of CG is optimal up to a constant factor. For the class of supersolvable groups we will discuss a program that from a pc-presentation of G constructs a DFT D = ⊕Dk of CG and generates an O(n log n) FFT of CG. The running time to construct D is essentially proportional to the time to write down all the monomial (!) twiddle factors Dk (gi ) where the gi are the generators corresponding to the pc-presentation. Finally, we sketch some applications.

1

Introduction

This paper is concerned with fast discrete Fourier transforms. From an engineering point of view, there are two types of domains: a signal domain and a spectral domain. Algebraically, both domains are finite dimensional vector spaces (over the complex numbers, say) and, in addition, they are equipped with a multiplication which turns both domains into associative C-algebras. The multiplication in the signal domain, called convolution, comes from the multiplication in a finite group, whereas the multiplication in the spectral domain is closely related to matrix multiplication. Fourier transforms isomorphically link these two domains and thus link convolution and matrix multiplication. To be more specific, let G be a finite group. The set CG := {a|a : G → C} of all C-valued functions (signals) on G becomes a vector space over C by pointwise addition and scalar multiplication. A natural basis is given by the indicator functions (G  h → δgh ) of the group elements g ∈ G. Identifying each group element with its indicator function, CG
can be viewed as the C-linear span of G, i.e., the span of all formal sums g∈G ag g with complex coefficients. The multiplication in G extends to the so-called convolution in CG:       ag g · bh h = ag bg−1 k k. g∈G

h∈G

k∈G

g∈G

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 29–42, 1999. c Springer-Verlag Berlin Heidelberg 1999 

30

Michael Clausen and Meinard M¨ uller

In this way, CG becomes a C-algebra, the so-called group algebra of G over C. For example, if G = Cn = X | X n = 1 is the cyclic group of order n, then CG can be identified with the polynomial ring C[X] modulo the ideal generated by X n − 1. In this case, convolution in CG means ordinary polynomial n = 1. If ω is a primitive nth root of unity, multiplication modulo the relation X n−1 n then the factorization X − 1 = j =0 (X − ω j ) combined with the Chinese 1×1 remainder theorem shows that CCn is isomorphic to the algebra ⊕n−1 of j =0 C n-square diagonal matrices. With respect to natural bases in both spaces this isomorphism is described by the classical DFT matrix D = (ω jk )0≤j,k 14 |G| log |G|. Thus performing a DFT with only O(|G| log |G|) additions, subtractions, and scalar multiplications by small program constants is almost optimal in the L2 -model. This justifies the name Fast Fourier Transform. (For more details on lower complexity bounds, see Chapter 1

In the classical FFT algorithms these program constants are the so-called twiddle factors.

A Fast Program Generator of Fast Fourier Transforms

31

13 of [4].) As an example, the Cooley-Tukey FFT of CG, G a cyclic 2-group, uses only roots of unity as program constants and is thus essentially 2-optimal in the L2 -model. Prior to performing a DFT one has to solve a fundamental problem in representation theory: up to equivalence, all irreducible representations of CG have to be generated. In general not an easy task! Even worse: as we are interested in a fast evaluation of D = ⊕Dk we should choose the representatives Dk in the equivalence classes very carefully. In other words, we have to choose the right Xk above. At least for the class of supersolvable groups, it turns out that choosing the right bases is quite easy. Moreover both problems (generating a DFT, performing a DFT) can be solved in an essentially optimal way. Furthermore, the results lead to fast algorithms not only in theory but also in a practical sense. The aim of this paper is to present these results and some of its consequences without giving too much technical details. For more information we refer to [1,2,4,5,8,12]. The rest of this paper is organized as follows. After some preparations, we describe in Section 2 geometrically a monomial DFT for supersolvable groups and indicate how this yields FFTs for supersolvable groups. Section 3 presents the main ideas of the algorithm that constructs monomial DFTs. Furthermore, some implementation details and running times are shown. Section 4 sketches some applications.

2

Monomial DFTs for Supersolvable Groups

A finite group G is called supersolvable iff there exists a chain T = (G = Gn ⊃ Gn−1 ⊃ . . . ⊃ G1 ⊃ G0 = {1}) such that each Gi is a normal subgroup in G and all indices [Gi : Gi−1 ] =: pi are prime. Thus T is a chief series of G with chief factors Gi /Gi−1 of prime order. For example, all nilpotent groups (especially all groups of prime power order) are supersolvable. In this section we are going to describe the irreducible representations of supersolvable groups in a geometric way. This approach will be the theoretical basis of the algorithmic approach shown in the next section. We need some preparations. 2.1

Basic Definitions and Tools

The character χ of a representation D of CG is defined by χ(g) := Trace(D(g)), for g ∈ G. Characters are constant on conjugacy classes and two representations are equivalent iff their characters are equal. Characters corresponding to irreducible representations are called irreducible characters. A character is called linear iff it corresponds to a representation of degree 1. By Irr(G) we denote the set of all irreducible characters of G. The space CF (G, C) of all complex-valued
class functions on G becomes an inner product space by χ|ψ := |G|−1 g∈G χ(g)ψ(g). For a proof of the following facts we refer to [9]:

32

Michael Clausen and Meinard M¨ uller

Theorem 1. For a finite group G with h conjugacy classes the following is true: (1) Irr(G) = {χ1 , . . . , χh } is an orthonormal basis of CF (G, C). (2) Let F and Dk be representations of CG with characters χ and χk , respectively. If Dk is irreducible, then the multiplicity Dk |F with which Dk occurs in F equals χk |χ .
k (1) −1 )g, then e1 , . . . , eh are a basis of the (3) If ek := eχk := χ|G| g∈G χk (g center of CG. Moreover, 1 = e1 + . . . + eh and ek ej = δkj ek . (The ek are called central primitive idempotents in CG.) (4) If M is a left CG-module affording the representation F with character χ, then M = ⊕hk=1 ek M (isotypic decomposition). If Mk is a simple module affording the character χk , then ek M, the isotypic component of type χk , is isomorphic to the direct sum of χk |χ copies of Mk . Every simple submodule of M affording the character χk is contained in ek M. Let H be a subgroup of G. Then CH can be viewed as a subalgebra of CG. If D is a representation of CG, then its restriction to CH is a representation of CH denoted by D ↓ H =: F . In turn, D is called an extension of F . Similarly, χ ↓ H denotes the restriction of the character χ. One important tool in constructing representations is the process of induction, where a representation of a group G is constructed from a representation of a subgroup H. In terms of modules, the construction is straightforward: let L be a left ideal in CH. Then CGL is a left ideal in CG and with G = ri=1 gi H one obtains the decomposition CGL = ⊕ri=1 gi L as a C-space. In particular, CGL has dimension [G : H] · dim L. The left CG-module CGL is said to be induced by L. A look at the corresponding matrix representations leads to the following definition. Let H be a subgroup of the group G, T := (g1 , . . . , gr ) a transversal of the left cosets of H in G and let F be a representation of CH of degree f. The induced representation F ↑T G of CG of degree f · r is defined for x ∈ G by F ↑T G(x) := (F˙ (gi−1 xgj ))1≤i,j≤r ∈ (Cf ×f )r×r , where F˙ (y) := F (y) if y ∈ H, and F˙ (y) is the f-square all zero matrix, if y ∈ G \ H. It is easily checked that this defines a representation of CG. Taking different transversals gives possibly different, but equivalent representations. Thus in non-critical situations we sometimes write F ↑G instead of F ↑T G. Note that F ↑T G(x), for x ∈ G, is a block matrix, with exactly one non-zero block in each block row and in each block column. In particular, if F is of degree 1, then, for all x ∈ G, the matrix F ↑T G(x) is monomial. (Recall that a matrix is called monomial iff it has exactly one non-zero entry in each row and in each column. A representation D of CG is said to be monomial iff D(g) is monomial, for all g ∈ G.) A group G is called an M-group if every irreducible representation is equivalent to a monomial one. Below we will give an alternative proof to the well-known fact that supersolvable groups are M-groups. There is a close connection between restriction and induction. A more precise statement reads as follows.

A Fast Program Generator of Fast Fourier Transforms

33

Theorem 2 (Frobenius Reciprocity Theorem). Let H be a subgroup of G. Furthermore, let F and D be irreducible representations of CH and CG, respectively. Then the multiplicity F |D↓H of F in D↓H equals the multiplicity D|F ↑G of D in F ↑G: F |D↓H = D|F ↑G . If N is a normal subgroup of G and F a representation of CN , then for g ∈ G we define a new representation F g of CN by F g (n) := F (g −1 ng) for all n ∈ N . F and F g are called conjugate representations. As {f(n)|n ∈ N } = {F g (n)|n ∈ N }, F is irreducible iff F g is irreducible, and G acts on Irr(N ) by conjugation via gχ := (N  n → χ(g −1 ng)). The last tool needed for our geometric approach is the following special case of Clifford theory. Theorem 3 (Clifford’s Theorem). Let N be a normal subgroup in G of prime index p, and let F be an irreducible representation of CN . For a fixed g ∈ G \ N , let T denote the transversal (1, g, g 2 , . . . , g p−1 ) of the cosets of N in G. Then exactly one of the following two cases applies. i

(1) All F g are equivalent. Then there are exactly p irreducible representations D0 , . . . , Dp−1 of CG extending F . The Dk are pairwise inequivalent and satisfy F ↑G ∼ D0 ⊕ . . . ⊕ Dp−1 . Moreover, if χ0 , χ1 , . . . , χp−1 are the linear characters of the cyclic group G/N in a suitable order, we have Dk = χk ⊗D0 for all k, i.e., Dk (x) = χk (xN)D0 (x), for all x ∈ G. i (2) The representations F g are pairwise inequivalent. In this case, the induced representation F ↑G is irreducible. To decide which case in Clifford’s Theorem applies, we work with intertwining spaces. Recall that for two representations D and ∆ of CG, both of degree d, the intertwining space is defined by Int(D, ∆) := {X ∈ Cd×d |XD(g) = ∆(g)X, for all g ∈ G}. 2.2

A Geometric Approach

Let T = (Gn ⊃ Gn−1 ⊃ . . . ⊃ G0 ) be a chief series of the supersolvable group G = Gn of exponent e and let χ ∈ Irr(G) be a fixed irreducible character of G. We are going to associate to χ a simple left CG-module M affording χ and a basis in M such that the resulting representation D is monomial, i.e., D is monomial and each non-zero entry of D(g), g ∈ G, is an e-th root of unity. To this end, we consider all sequences w = (χ0 , . . . , χn = χ), with irreducible characters χi ∈ Irr(Gi ). By Theorem 1 (4) and Frobenius Reciprocity the product e(w) := eχ0 · . . . · eχn of the corresponding central primitive idempotents is non-zero iff all multiplicities χi |χi+1 := χi |χi+1 ↓Gi are positive. The set of all those sequences will be denoted by W (χ): W (χ) := {(χ0 , . . . , χn = χ) | χi ∈ Irr(Gi ), χi |χi+1 > 0}.

34

Michael Clausen and Meinard M¨ uller

Theorem 4 ([6]). Let G be a supersolvable group of exponent e, χ an irreducible character of G, and T a chief series of G. Then the following holds. (1) Let w ∈ W (χ). Then e(w) is a primitive idempotent in CG and CGe(w) is a simple CG-module affording the character χ. The dimension of CGe(w) equals χ(1) = |W (χ)|. (2) Let v ∈ W (χ) and set M := CGe(v). Then M = ⊕w∈W (χ) e(w)M is a decomposition of M into 1-dimensional linear subspaces e(w)M. (3) G acts transitively on W (χ) by g(χ0 , . . . , χn ) := (gχ0 , . . . , gχn ). (4) For all g ∈ G and all w ∈ W (χ) we have ge(w)g −1 = e(gw). (5) Let U be the stabilizer of v ∈ W (χ) and L a transversal of the left cosets of U in G. Then {&e(v)|& ∈ L} is a C-basis of CGe(v) and the corresponding representation D of CG is e-monomial. More precisely, if the 1-dimensional CU-module e(v)CGe(v) affords the linear character λ, then e(v) equals the central primitive idempotent corresponding to λ: e(v) = eλ , and D = λ↑L G.  

Proof. (1). eχ = ( i ρ − i, for 1 ≤ i ≤ ρ}. (4.5)

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Tadao Kasami

Example 4.2: We introduce “Multiple-GMD”. For a positive integer h, h-GMD consists of successive GMD(u(i)) for 1 ≤ i ≤ h, where u(1)
z and u(i)
v[

i−1

j=1 (V

N

\ RGMD(u(j) ) )], for 1 < i ≤ h. (2)

(2)

(4.6)

(2)

It is shown in [17] that u(2) + z = (u1 , u2 , . . . , uN ), where  1, if h + p is even and 1 ≤ (h + p)/2 ≤ ρ, (2) uh = 0, otherwise.

(4.7)

Example 4.3: Generalized Chase Algorithm [18]: Bounded distance-t0 decoding around an input word v(j) ∈ V N , called the j-th search center, is used for generating c(j), where 0 < t0 ≤ (dmin − 1)/2. Then, ¯ d(j) (u(j)), C \ R(j) ⊆ O

(4.8)

where  u(j)
 d(j)


c(j), if c(j) = ∗, v(j), if c(j) = ∗,

(4.9)

dmin , if c(j) = ∗, t0 + 1, if c(j) = ∗.

(4.10)

For choosing v(j), the following method is proposed in [15]: Choose v(j + 1) such that v(j + 1) = v[VdN1 ,d2 ,... ,dh (u1 , u2 , . . . , uh )],

(4.11)

where u1 , u2 , . . . , uh are chosen from u(i) with 1 ≤ i ≤ j defined by (4.9). Example 4.4: Multiple Chase-Like Decoding Algorithm: For 1 ≤ τ < N and u ∈ V N , Chase(u) denotes a 2τ stage decoding algorithm whose j-th stage is bounded distance-t0 decoding with input u + e(j), where t0 = (dmin − 1)/2 and e(j) is the j-th test error pattern in Eτ
{x ∈ V N : pτ +1,N (x) = 0}. Chase(z) is the original Chase decoding algorithm [11]. Then V N \ RChase(u) = {x ∈ V N : dH,τ +1,N (x, u) ≥ t0 + 1}.

(4.12)

For a positive integer h, h-Chase consists of successive Chase(u(i) ) with 1 ≤ i ≤ h, where u(i) , called the i-th search center, is given by u(1)
z and for 1 < i ≤ h i−1 u(i)
v[ j=1 (V N \ RChase(u(j) ) )] = v[{x ∈ V N : dH,τ +1,N (x, u(j) ) ≥ t0 + 1, for 1 ≤ j < i}].

(4.13)

Example 4.5: Decoding Based on Ordered Statistics [7]: Let C be a binary linear (N, K) code, and let MK = {i1 , i2 , . . . , iK }, where 1 ≤ i1 < i2 < · · · < iK ≤ N − dmin + 1, be the most reliable basis (MRB) [7]. For

On Integer Programming Problems

49

x = (x1 , x2 , . . . , xK ) ∈ V K , there is a unique codeword γ(x) = (c1 , c2 , . . . , cN ) in C such that cih = zih + xh , for 1 ≤ h ≤ K.

(4.14)

For 0 ≤ h ≤ K, define VhK be the set of those binary K-tuples whose weights are h or less. The order-h decoding [7] generates candidate codewords {γ(x) : x ∈ VhK } iteratively in an increasing order of the weight of x. The search region, denoted Ros-h , is the subcode of C defined by Ros-h
{γ(x) : x ∈ VhK } = {v ∈ C : dH,MK (v, z) ≤ h}.

(4.15)

Similarly, we can treat the decoding algorithm in [19]. Example 4.6: Iterative MWTS Decoding Algorithm: In the iterative minimum-weight subtrellis search (MWTS) decoding algorithm [6] for binary linear codes, the first candidate codeword c(1) is obtained by the ordered statistics decoding, the zero-th order or the first order [7]. At the j-th stage with j > 1, the next candidate codeword c(j) is the best codeword in {v ∈ C : dH (v, c(j − 1)) = w1 } which is obtained by MWTS around c(j − 1), where w1 (= dmin ) is the minimum weight. Let w2 be the second smallest weight of C. Then, R(j) = Ow1 (c(j − 1)) \ {c(j − 1)}, for j > 1,

(4.16)

¯w1 (c(1)) and for j > 1, and we have that C \ R(1) ⊆ O C \ R(j) = C \ {Ow1 (c(j − 1)) \ {c(j − 1)}}   ¯ w1 (c(j)). ¯ w2 (c(j − 1)) {c(j − 1)}) O ⊆ (O

5

(4.17)

Formulation as Integer Programming Problem

For the decoding algorithms stated in Sec. 4 and in [13,14], the derivation of the lower bounds LS , LRO and LET and the selection of search centers can be uniformly formulated as the following optimization problem. For a received sequence r = (r1 , r2 , . . . , rN ), a set Σ ⊆ [1, h], nonempty sets Ij ⊆ [1, N ], uj ∈ V N , and positive integers dj ≤ N for 1 ≤ j ≤ h, find the following lower bound min L(x), where x ∈ V N subjects to the inequalities  ≥ dj , for j ∈ Σ, dH,Ij (x, uj ) = dj , for j ∈ / Σ,

(5.1)

(5.2)

where u1 , u2 , . . . , uh are called reference words. Define 1
(1, 1, . . . , 1). Note that dH,Ij (x, uj ) ≤ dj if and only if dH,Ij (x, uj + 1) ≥ |Ij | − dj , for 1 ≤ j ≤ h.

(5.3)

50

Tadao Kasami

Define y
x + z by yi = xi + zi (mod 2) for 1 ≤ i ≤ N . Then L(x) =

N 

|ri |yi .

(5.4)

i=1

For v ∈ V N , 1 ≤ j ≤ h and I ⊆ [1, N ], define D0,I (v)
{1, 2, . . . , N} \ I, D1,I (v)
{i ∈ I : vi = zi },

(5.5)

D−1,I (v)
{i ∈ I : vi = zi }, n−1,I (v)
|D−1,I (v)|.

(5.6)

Since dH,I (x, uj ) = dH,Ij (y, uj + z),  yi − dH,Ij (x, uj ) = i∈D1,Ij (uj )



yi + n−1,Ij (uj ).

(5.7)

i∈D−1,Ij (uj )

For a vector α = (α1 , α2 , . . . , αh ) ∈ {−1, 0, 1}h , define h Dα
j=1 Dαj ,Ij (uj ), nα
|Dα |.

(5.8)

Define A
{α ∈ {−1, 0, 1}h : Dα = φ}. Since ∪α∈A Dα = [1, N ] and Dα ∩Dα = ¯
{Dα : α ∈ A} is a partition of [1, N ] and therefore |A| ≤ N . φ for α = α , D For 1 ≤ i ≤ N and 1 ≤ j ≤ h, let aji be the j-th component of the vector α ∈ A with i ∈ Dα . Then Lemma 4 follows from (5.4) to (5.8). Lemma 4. The optimization problem (5.1) and (5.2) can be expressed as the following integer programming problem: min

N 

|ri |yi ,

(5.9)

i=1

where yi ∈ {0, 1} subjects to the equalities N 

aji yi = δj + σj , for 1 ≤ j ≤ h,

(5.10)

i=1

where δj
dj − n−1,Ij (uj ), and σj is a slack variable such that σj ≥ 0 for j ∈ Σ and σj = 0 for j ∈ Σ. For the above integer programming problem, denoted IPP, a binary N -tuple y which meets the constraints is called feasible and a feasible y at which the object function takes the minimum is called optimal. For α ∈ A, define  yi . (5.11) qα = i∈Dα

Then 0 ≤ |qα | ≤ nα .

(5.12)

On Integer Programming Problems

51

Equalities (5.10)can be expressed as 

αj qα = δj + σj , for 1 ≤ j ≤ h,

(5.13)

α∈A

where αj denotes the j-th component of α. Let Q denote the set of those |A|tuples over nonnegative integers (qα : α ∈ A)’s which satisfy (5.12) and (5.13). By using vector representation, equalities (5.13) are expressed as 

qα α = δ + σ,

(5.14)

α∈A

where δ = (δ1 , δ2 , . . . , δh ) and σ = (σ1 , σ2 , . . . , σh ). For a subset X ⊆ [1, N ] and an integer m with m ≤ |X|, let X[m] denote the set of m smallest integers in X for 1 ≤ m ≤ |X| and the empty set φ for m ≤ 0. Then, from (4.3), (5.9) and (5.11), the optimal value of the object function of the IPP can be expressed as  min |ri |, (5.15) i∈



α∈A

Dα [qα ]

where the minimum is taken over Q. Define q
(qα : α ∈ A), where the components are ordered in a fixed order. Then q defined by (5.11) from an optimal solution y of IPP is also called optimal. Conversely, for an optimal q, the solution y whose i-th bit is 1 if and only if i ∈ Dα [qα ] for α ∈ A such that i ∈ Dα is optimal from (4.3) and (5.11). Hereafter, we consider the IPP in terms of qα with α ∈ Q. For S ⊆ A, Σ  ⊆ Σ and an h-tuple c = (c1 , c2 , . . . , ch ) over nonnegative integers such that cj = 0 for j ∈ / Σ, let IPP(S, Σ  , c), called a sub-IPP, denote the integer programming problem whose constraints are those of the IPP and the following additional ones: / S, qα = 0, for α ∈  ≥ cj , for j ∈ Σ  , σj = = cj , for j ∈ Σ  .

(5.16) (5.17)

We say that the IPP can be reduced into a set of sub-IPP’s if and only if there exists an optimal solution q of the IPP which is an optimal solution of one of the sub-IPP’s. For example, suppose that the j-th restriction is dH (x, uj ) ≥ dmin with uj ∈ C and it is enough to consider x to be a codeword of C. Since the restriction is equivalent to dH (x, uj ) = w1 or dH (x, uj ) ≥ w2 , IPP can be reduced to subIPP’s. Define S(q)
{α ∈ A : qα > 0}. The following lemma and simple corollary are useful for the reduction of IPP. Lemma 5 follows from (4.3), (5.14) and (5.15).

52

Tadao Kasami

Lemma 5. Let q be optimal. Then (i) for any nonempty subset X of S(q),
α ≤ σ and (ii) for α ∈ S(q) and α ∈ A such that nα > qα , if the qα α∈X th integer of Dα is greater than the (qα + 1)-th integer of Dα , then α − α ≤ σ. Corollary 1. Let q be optimal. Then (i) α ∈ / S(q), for α ∈ {0, −1}h , (ii) {α, −α} ⊆ S(q), for any α ∈ A, hence |S(q)| ≤ min{N, 3h /2}, (iii) at least one component of σ is zero, and (iv) if there are α ∈ S(q) and i ∈ [1, h] such that αi = 1 and αj = −1 for j = i, then σi = 0 and βi = 1 for all β ∈ S(q). Some further details can be found in [14]. As the number of the reference words grows, the computational complexity for solving IPP grows. We may choose a relatively small subset of reference words to derive a little weaker but still useful lower bound. The reference words u1 , u2 , . . . , uh in (5.2) can be partitioned into two blocks, denoted RW1 and RW2 , in such a way that uj ∈RW1 if and only if uj is chosen independently of any candidate codeword generated already. A block RW1 or RW2 may be empty. Examples of RW1 are {z} for LRO in [13], {z, z +1} for LET in [14], {v(1), . . . , v(j) : decoding failures continue up to the j-th stage} for finding v(j +1) in Example 4.3, {u(1), . . . , u(i−1) } for finding u(i) in Example 4.4 and {z} for LET or LS in Example 4.5. Assume that the first h1 reference words are in RW1 . For 1 ≤ i ≤ h1 , define Ai
{p1,i (α) : α ∈ A} and for β = (β1 , β2 , . . . , βi ) ∈ Ai , define Dβ
∩ij=1 Dβj ,Ij (uj ). For v ∈ V N and I ⊆ [1, N ], if Da,I (v) with a ∈ {−1, 0, 1} is empty or an interval, then (v, I) is called interval type. For an interval I or I = φ, (z, I) and (z + 1, I) are interval type. If ui with i > 1 is an optimal solution of IPP with reference words u1 , u2 , . . . , ui−1 such that (uj , Ij ) is interval type for 1 ≤ j < i, then it follows from (4.3) that ui is also interval type. Suppose (uj , Ij ) is interval type for 1 ≤ j ≤ i. Then Dβ is an interval for β ∈ Ai , that is, there are two integers ν1 (β) and ν2 (β) such that Dβ = [ν1 (β), ν2 (β)]. For different β and β in Ai , we write β < β  if and only if ν2 (β) < ν1 (β ). Since Dβ ∩ Dβ = φ, β < β  or β  < β holds. This simplifies the solution of IPP. For instance, (ii) of Lemma 5 becomes the following more useful version: (ii’) For 1 ≤ i ≤ h, let β and β be in Ai such that β < β. Then, for α and α in A such that p1,i (α) = β and p1,i (α ) = β  , either qα = 0, or qα = nα , or α − α ≤ σ. Thus, it is better to choose a relatively small number of reference words in RW2 . Reasonable choice rules are: i) give priority to a reference word ui with a larger di or a smaller L(ui ); and ii) to renew the test condition, the candidate codeword generated most recently must be chosen. Example 5.1: Consider the evaluation of LS,h
L[VdN1 ,d2 ,... ,dh (u1 , u2 , . . . , uh )]. For this case, RW1 = φ, Σ = [1, N ] and Ij = [1, N ] for 1 ≤ j ≤ h. Formulas for LS,2 and LS,3 were presented in [3,4] and an efficient algorithm for computing LS,4 is presented in [5]. Upper bounds of the number of real number additions and comparisons for evaluating LS,h with 2 ≤ h ≤ 4 have been derived in [20] for 2 ≤ h ≤ 3 and [5] for h = 4. Table 1 lists the upper bounds under the assumption of (4.3) and the number of sub-IPP’s.

On Integer Programming Problems

53

Tab. 1 Upper bounds of the number of real number additions and comparisons for evaluating LS,h and the number of sub-IPP’s, where δ
min1≤i≤h {δi }. h Upper Bound The number of sub-IPP’s 2 δ−1 1 3 10δ − 2 2 4 310δ 2 + 184δ − 1 9

Acknowledgments The author is grateful to Mr. Y. Tang for his help in preparing this paper, and to Prof. S. Lin, Prof. M. P. C. Fossorier, Prof. T. Fujiwara and Dr. T. Koumoto for their helpful suggestions.

References 1. D. J. Taipale and M. B. Pursley, “An Improvement to Generalized MinimumDistance Decoding,” IEEE Trans. Inform. Theory, vol. 37, pp. 167–172, Jan. 1991. 2. T. Kaneko, T. Nishijima, H. Inazumi and S. Hirasawa, “An Efficient MaximumLikelihood-Decoding Algorithm for Linear Block Codes with Algebraic Decoder,” IEEE Trans. Inform. Theory, vol. 40, pp. 320–327, Mar. 1994. 3. T. Kasami, T. Takata, T. Koumoto, T. Fujiwara, H. Yamamoto and S. Lin, “The Least Stringent Sufficient Condition on Optimality of Suboptimal Decoded Codewords,” Technical Report of IEICE, Japan, IT94-82, Jan. 1995. 4. T. Kasami, T. Koumoto, T. Takata and S. Lin, “The Effectiveness of the Least Stringent Sufficient Condition on the Optimality of Decoded Codewords,” Proc. of the 3rd Int. Symp. on Commu. Theory & Appl., pp. 324–333, Ambleside, UK, July 1995. 5. Y. Tang, T. Kasami and T. Fujiwara, “An Optimality Testing Algorithm for a Decoded Codeword of Binary Block Codes and Its Computational Complexity,” to appear in Proc. of the 13th Int. Symp. AAECC, Honolulu, HI, USA, Nov. 1999. 6. T. Koumoto, T. Takata, T. Kasami and S. Lin, “A Low-Weight Trellis Based Iterative Soft-Decision Decoding Algorithm for Binary Linear Block Codes,” IEEE Trans. Inform. Theory, vol. 45, pp. 731–741, Mar. 1999. 7. M. P. C. Fossorier and S. Lin, “Soft-Decision Decoding of Linear Block Codes Based on Ordered Statistics,” IEEE Trans. Inform. Theory, vol. 41. pp. 1379–1396, Sept. 1995. 8. D. Gazelle and J. Snyders, “Reliability-Based Code-Search Algorithms for Maximum-Likelihood Decoding of Block Codes,” IEEE Trans. Inform. Theory, vol. 43. pp. 239–249, Jan. 1995. 9. C. X. Chen, “Sufficient Conditions for the Optimality of a Codeword in SoftDecision Decoding of Binary Linear Block Codes,” Master Thesis, Univ. of Hawaii, Oct. 1998. 10. M. P. C. Fossorier, T. Koumoto, T. Takata, T. Kasami and S. Lin, “The Least Stringent Sufficient Condition on the Optimality of a Suboptimally Decoded Codeword Using the Most Reliable Basis,” Proc. of ISIT, pp. 430, Ulm, Germany, June 1997.

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11. D. Chase, “A New Class for Decoding Block Codes with Channel Measurement Information,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 170–182, Jan. 1972. 12. T. Kaneko, T. Nishijima and S. Hirasawa, “An Improvement of Soft-Decision Maximum-Likelihood Docoding Algorithm Using Hard-Decision Bounded-Distance Decoding,” IEEE Trans. Inform. Theory, vol. 43, pp. 1314–1319, July 1997. 13. T. Koumoto, T. Kasami and S. Lin, “A Sufficient Condition for Ruling Out Some Useless Test Error Patterns in Iterative Decoding Algorithms,” IEICE Trans. on Fundamentals, vol. E81-A, No. 2, pp. 321–326, Feb. 1998. 14. T. Kasami, Y. Tang, T. Koumoto and T. Fujiwara, “Sufficient Conditions for Ruling-Out Useless Iterative Steps in a Class of Iterative Decoding Algorithms,” to appear in IEICE Trans. Fundamentals, vol. E82-A, Oct. 1999. 15. T. Koumoto and T. Kasami, “An Iterative Decoding Algorithm Based on Information of Decoding Failure,” Proc. of the 20th SITA, pp. 325–328, Matsuyama, Japan, Dec. 1997. 16. G. D. Forney, Jr., “Generalized Minimum Distance Decoding,” IEEE Trans. Inform. Theory, vol. IT-2, pp. 125–181, Apr. 1966. 17. T.Koumoto and T. Kasami, “Analysis and Improvement on GMD-like Decoding Algorithms,” Proc. of ISITA, pp. 419–422, Mexico City, Mexico, Oct. 1998. 18. N. Tendolkar and C. P. Hartmann, “Generalization of Chase Algorithms for Soft Decision Decoding of Binary Linear Codes,” IEEE Trans. Inform. Theory, vol. IT30, pp. 714–721, Sept. 1984. 19. Y. H. Han, C. R. P. Hartman, and C. C. Chen, “Efficient Priority-First Search Maximum-Likelihood Soft-Decision Decoding of Linear Block Codes,” IEEE Trans. Inform. Theory, vol. 39, pp. 1514–1523, Sept. 1993. 20. T. Kasami and T. Koumoto, “Computational Complexity for Computing Sufficient Conditions on the Optimality of a Decoded Codeword,” Technical Report of NAIST, Japan, NAIST-IS-TR98008, July 1998.

Curves with Many Points and Their Applications Ian F. Blake Hewlett-Packard Laboratories 1501 Page Mill Road Palo Alto, CA 94304 USA [email protected]

Abstract. The use of curves in algebraic geometries in applications such as coding and cryptography is now extensive. This note reviews recent developments in the construction of such curves and their application to these subjects.

1

Introduction

The use of curves in algebraic geometries, in applications such as coding and cryptography, has now an extensive literature and has been responsible for dramatic developments in both subjects. In coding theory, the use of modular curves to construct asymptotically good codes i.e., to construct an infinite sequence of codes whose parameters asymptotically exceed the Varshamov-Gilbert bound over alphabets of size ≥ 49, was a remarkable achievement. Likewise, the introduction of the theory of elliptic curves to public key cryptography has opened up a new area of both practical and theoretical significance. This note reviews some of the current developments and directions of research in areas related to the construction of curves with many points. Of necessity it is highly selective. The term ‘curves with many points’ refers to curves whose set of points has order close to the maximum allowed by the Hasse-Weil theorem, √ namely q + 1 + 2g q. The construction of such curves is an interesting subject in its own right. The use of such curves in coding theory results in codes of long length, relative to the dimension and distance, in comparison to codes obtained from other curves of the same genus and alphabet size. The application of elliptic curves to cryptography is to replace the more usual finite groups, such as the multiplicative group of a finite field, with the group of points on the curve under point addition. As such, the construction of curves with many points (which in the case of elliptic curves means the upper √ bound of q + 1 + 2 q) is less important than the size of the largest prime order subgroup of points. However, by recourse to the prime number theorem and √ the fairly uniform distribution of curve orders over the interval q + 1 ± 2 q, as the curve parameters range over all allowable values, one can argue that curves with many points are of interest in this case as well. The reason for interest in such cryptosystems is that the same level of security can be achieved with much smaller key lengths compared to the more conventional systems, resulting in more efficient implementations. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 55–64, 1999. c Springer-Verlag Berlin Heidelberg 1999


56

Ian F. Blake

This work reviews the subject and its applications, with a particular emphasis on recent developments and directions. The next section presents some background material needed for a discussion of the subject and its applications to coding theory and cryptography, which are considered in the succeeding sections.

2

Preliminaries

A very brief introduction to the background needed to discuss the problems addressed in this work is given. More complete introductions to the material are given in [32], [13], [43] and [3]. ¯ q its algebraic closure. Let f(x, y) ∈ Fq [x] be a Let Fq be a finite field and F bivariate polynomial over Fq and consider the set ¯ q |f(u, v) = 0} ¯q × F C = {(u, v) ∈ F ¯ q . Each such solution is i.e. the set of solutions of the equation f(x, y) = 0 over F referred to as a point of the curve. The subset of these points with coordinates in Fq is referred to as the set of rational points. We will only be interested in the case where f(x, y) ∈ Fq [x, y] is an irreducible polynomial over Fq and where the curve is smooth i.e. contains no non-singular points, points where at least one of the formal derivatives vanishes. There is an equivalent approach to the subject in terms of function fields for which we refer to [43]. A divisor of the curve C is a finite formal sum
m i Pi , m i ∈ Z D= Pi ∈C

where only a finite number of the integers mi are nonzero. The degree of the divisor is i mi . Denote by D the set of all divisors of C, an abelian group, and by D0 the set of divisors of degree 0, a subgroup of D. Denote D ≥ 0 if mi ≥ 0 for all i.   greatest common divisor (gcd) ofdivisors D = i mi Pi and D =  The    to be gcd(D, D ) = i min(mi , m i )Pi − kO where k = i m i Pi is defined  min(m , m ) i.e. the gcd is a divisor of degree 0.To a bivariate polynomial i i i g(u, v) ∈ F¯q [u, v] can be associated the divisor (g) = i mi Pi − kO ∈ D0 where k is chosen so the divisor has degree 0, and the integer mi represents the order of vanishing of g at Pi . Associated with the rational function g(u, v)/h(u, v) is the divisor (g) − (h). A divisor of this form is referred to as a principal divisor. Clearly all such principal divisors have degree 0 and they form a subgroup P of D0 . The factor group D0 /P is referred to as the jacobian J of the curve and will play a role in the formulation of cryptosystems using hyperelliptic curves. Notice that notions of divisors discussed above hold also for curves and their corresponding divisors defined with polynomials of a single variable. Associated with a curve is an invariant g, the genus of the curve, which is a non-negative

Curves with Many Points and Their Applications

57

integer which is related to the dimension of certain subspaces associated with divisors of the curve. A fundamental result of curves is the Hasse-Weil theorem which states that the number of rational points, N1 , on a curve of genus g is bounded by the relation |N1 − (q + 1)| ≤ 2gq 1/2 .  The right hand side of this bound was improved by Serre [42] to g[ 2q 1/2 ]. When the genus of the curve is large compared to q, this bound is not very effective. A more accurate bound is given in this case in a theorem by Oesterle (and reported by Schoof [38]). It can be shown that the number of solutions (of the curve defined over Fq ) over Fqr , r > g, is determined by N1 , N2 , · · · , Ng where Ni is the number of solutions over Fqi . The relationship is made explicit via the zeta function. Define a curve of genus g over Fq with N1 points to be maximal if N1 = q + 1 + 2gq 1/2. To achieve this upper bound it is clearly necessary that q be a square, q = r2 . For a given genus g and finite field size q, let Nq (g) be the maximum number of points on any curve of genus g over Fq , Nq (g) ≤ q+1+2gq 1/2 . A curve with this maximum number of points Nq (g) will be called optimal. Some limited, specific results on the existence of certain classes of optimal and maximal codes are known (e.g. [12]). A curve that contains ‘almost’ q + 1 + 2gq 1/2 points will be referred to as ”a curve with many points”. Subsequent sections survey recent contributions to the problem of constructing curves with many points. Asymptotics: Define the quantity A(q) = limsupg→∞ Nq (g)/g and notice from the Hasse-Weil bound that A(q) ≤ 2q 1/2 or, using the Serre improvement, A(q) ≤ [2q 1/2 ]. A variety of upper and lower bounds on this quantity are known ([25], [44], [8], [42], [50], [40]). by exhibiting a suitable infinite family of modular curves. Niederreiter and Xing [37] have recently established a variety of lower bounds on A(q) including the fact that, for q odd and m ≥ 3, A(q m ) ≥ 2q/2(2q + 1)1/2  + 1. Additionally they show that A(2) ≥ 81/317, A(3) ≥ 62/163, and A(5) ≥ 2/3 improving on the previously best bounds of these cases of 2/9, 1/3 and 1/2 respectively. Many other bounds on A(q), both upper and lower exist, but are not pursued here. Construction Techniques: A variety of techniques have been used to establish curves with many points and a web site containing many of the ‘best’ curves available is maintained [17]. The techniques have been categorized as: I Methods from general class field theory; II Methods from class field theory based on Drinfeld modules of rank 1; III Fibre products of Artin-Schreier curves; IV Towers of curves with many points; V Miscellaneous methods such as : (i) formulas for Nq (1) and N2 (q); (ii) explicit curves, curves, (iii) elliptic modular curves (iv) Deligne-Lusztig curves; (v) quotients of curves with many points.

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Numerous techniques to construct curves with many points that depend on coding theory are given in [18]. In essence, the codewords lead to families of curves and low weight codewords lead to curves with many points. The role of coding theory to yield linear spaces of curves with many points is then of interest. Elliptic Curves It can be shown [31] that an elliptic curve over Fq can always be reduced to the form: E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 ,

ai ∈ Fq .

¯ q , together with the point at infinity The set of solutions of the equation over F ¯ O, form the points of the curve E(Fq ). All such curves have genus 1. The above (Weierstrass) equation can be further reduced to one of two forms, depending on whether the characteristic of the field is even or odd. The maximum number of points such a curve of genus 1 can have over Fq is completely determined ([41], [45]). For q = pa if a is an odd integer, a ≥ 3 and p|[2q 1/2 ], then q is called exceptional. Then  Nq (1) =

q + 1 + [2q 1/2 ], q nonexceptional q + [2q 1/2 ], q exceptional.

Partial results are available for g = 2. There also exists necessary and sufficient conditions [45] for the existence of a curve with exactly N points when the characteristic does not divide (N − q − 1). Hyperelliptic Curves As a generalization of elliptic curves, consider the hyperelliptic curves, defined by an equation of the form: y 2 + h(x)y = k(x) ,

h(x), k(x) ∈ Fq [x]

where h(x) is a polynomial of degree at most g and k(x) is monic of degree exactly 2g + 1. We require the curve to be smooth i.e. have no singular points ¯ 2 where both of the partial derivatives 2y+h(x) = 0 and h y−k  (x) = 0 (x, y) ∈ F q vanish([27]). In such a case the genus of the curve is g. Notice that for elliptic curves h(x) is at most a linear polynomial and k(x) a monic cubic, and the curve is of genus 1. If char(Fq ) = 2 then the change of variables x → x, y → y − (h(x)/2) transforms the equation to y 2 = f(x) , degf(x) = 2g + 1 . In this case, if P = (x, y) is a point on the curve, then (x, −y − h(x)) is also a point on the curve, the sum of P and the point at infinity. If char(Fq ) = 2 and (x, y) is on the curve, then so also is (x, y + h(x)). This point will be designated P. Just as the zeta function leads to the enumeration of points on a curve over extensions of the base field, given the number on the first g extensions, a slight modification leads to the enumeration of the number of points in the Jacobian J(Fqr ), r > g, given the number of points on the curve for r = 1, 2, . . . , g [27].

Curves with Many Points and Their Applications

3

59

Curves in Coding Theory

For a given divisor G on a curve C define a subset of rational functions on the curve by L(G) = {f ∈ Fq (x, y)|(f) + G ≥ 0} ∪ {O} . Clearly L(G) is a finite dimensional vector space over Fq . Denote its dimension by l(G). A consequence of the Riemann-Roch theorem [10] is then that l(G) ≥ deg(G) + 1 − g where g is the genus of the curve. Indeed for a divisor G with deg(G) > 2g − 2, this equation will hold with equality. To construct the class of codes that will be of interest, let C be a nonsingular irreducible curve over Fq of genus g and let P1 , P2 , · · · , Pn be the rational points on C and D = P1 + P2 + · · · + Pn , a divisor. Let G be a divisor with support disjoint from D and assume that 2g − 2 < deg(G) < n. Define the linear code C(D, G) over Fq as the image of the linear map ([43]) α : L(G) −→ Fnq f

−→ (f(P1 ), f(P2 ), · · · , f(Pn )) .

This is often referred to as the evaluation construction of codes from curves in the literature. The parameters of the code C(D, G) are established by using the properties discussed above. The kernel of the map is the set L(G − D) and, for the restrictions on the parameters noted, k = dimC(D, G) = dimL(G) − dim(G − D) = deg(G) − g + 1. The minimum distance of C(D, G) is d ≥ d∗ = n − deg(G), the designed minimum distance. The other standard construction for codes, the residue construction, uses the idea of differentials of the curve, and will not be considered. Using the above evaluation construction of codes for elliptic curves, one can √ define a class of codes over Fq with the parameters length n ≤ q + 1 + 2 q, dimension k and minimum distance n − k. Since the distance is one away from the Singleton bound, one often refers to this as a code with ‘defect’ of 1. Further information on codes from elliptic curves is given in [21]. One can also use the construction on hyperelliptic curves and Xing [46] determines certain classes of such codes whose minimum distance can be determined exactly. Other aspects of the relationship between algebraic curves and codes are touched upon. The first establishes the connection between codewords of low weight in certain trace codes and curves with many points. The emphasis here will be on Reed-Muller (RM) codes and follow [20]. The generalized RM code Rq (s, m) can be defined as follows. Consider the vector space (over Fq ) Ps = {f ∈ Fq [X1 , X2 , . . . , Xm ]|deg(f) ≤ s}

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and denote the evaluation map by β : Ps −→ Fnq , f −→ (f(v)v∈Fm . The q kernel of this map is the ideal generated by the polynomials Xiq − Xi and a polynomial f ∈ Ps is called reduced if f is an Fq linear combination of monomials dm with 0 ≤ di ≤ q − 1. The image β(Ps ) is the RM code Rq (s, m) X1d1 X2d2 · · · Xm of order s in m variables. It can be shown that the evaluation codeword corresponding to certain polynomials f can in fact be written in the form: cf = (T r(R(x)))x∈Fm

for some R(x) ∈ Fqm [x].

An element in Fqm has trace 0 if and only if it can be written in the form y q −y for some y ∈ Fqm . Now if we associate to the codeword cf the irreducible complete smooth (Artin-Schreier) curve Cf over Fqm given by the equation y q − y = R(x), then the curve has genus g(Cf ) = (q − 1)(deg(R) − 1)/2. It follows immediately that w(cf ) = q m − (|Cf (Fqm )| − 1)/q, giving an interesting relationship between codewords of low weight and curves with many points. In [20] these considerations are also shown to lead in some cases to maximal curves, achieving the Hasse-Weil upper bound for the case of √ √ small genus i.e. ([16], [20]) g ≤ ( q − 1)2 /4 or g = (q − q)/2 (see [49]). Much of this work exploits the connection between generalized Hamming weights and curves with many points. Another contribution to the relationship between codes and curves, is the work on Niederreiter and Xing [37] on the existence of asymptotically good codes, i.e., sequences of codes whose rate/distance profile asymptotically exceeds the Varshamov-Gilbert bound. They are able to show a result equivalent to the celebrated one [44] for q a square. Specifically it is shown that for m ≥ 3 an odd integer and r a prime power ≥ 100m3 for odd r and ≥ 576m3 for even r then there is an open interval on (0, 1) which contains the point (rm − 1)/(2rm − 1) where the Varshamov-Gilbert bound can be exceeded. The result is achieved by establishing lower bounds on A(q m ), some of which were mentioned earlier.

4

Curves in Cryptography

The notion from public key cryptography that will be required for our presentation is that of a one-way function, on which cryptographic protocols , such as the Diffie-Hellman key exchange, depend. Other protocols, such as digital signatures, authentication and others follow readily, depending on the function in use. The particular one-way function of interest here is that of the discrete logarithm. In the multiplicative group of the finite fields F2m or Fp , p a prime, the order of complexity of the discrete logarithm problem in both time and space, is L(1/3, c, N) where L(v, c, N) = exp{c(log N )v (log log N )1−v }

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for some constant c where N is the largest prime divisor of the multiplicative group order. For a prime field, the constant c is approximately 1.6. This complexity function represents subexponential growth in log N . Establishing this complexity for the discrete logarithm problem uses a technique known as index calculus, which is facilitated by the group actually having two operations, residing as it does in a finite field. To ensure the discrete logarithm problem is as difficult as possible, the cyclic group order, q − 1, should have as large a prime divisor as possible. The essence of the application of curves, either elliptic or hyperelliptic, to the discrete logarithm problem, is to replace the groups previously used with additive groups defined from the curves. The presentation of these groups and the group operations, is the focus of the cryptographic aspects of the problem. On an elliptic curve it is possible to define a point addition such that the sum of any two points on the curve uniquely defines a third turning the set of curve points into a commutative additive group. The identity or neutral element of the group is the point at infinity O. The discrete logarithm problem in the additive group of points on the curve is: given a point P of large prime order N and a point multiple c · P = P + · · · + P (c times), determine the integer c. No subexponential algorithm is known to attack this problem. The most efficient algorithm known is the so called square root attack (or baby-step-giant-step) which has complexity on the order of the square root of the largest prime divisor of the group order. The essence of the problem appears to be the lack of an index calculus for it, as was possible for F∗p . A problem of some importance for the use of elliptic curves in cryptography, is the ability to determine precisely the number of points on the curve, or equivalently to determine curve parameters that yield a curve whose group order contains a large prime divisor. This so-called ‘point counting’ problem is both a theoretically and practically challenging one. The more general problem of counting points on curves of higher genus is even more so [9]. With the success of using elliptic curves for public key cryptosystems, it is natural to consider the use of other curves. A group structure is required in order to uniquely carry out the public key operations and no such addition seems possible on more general curves. However, it is possible to define an addition on the jacobian of a hyperelliptic curve and we give a few comments on this situation here. The definition of the jacobian of a curve given previously is valid for any curve. There are two reasons however why the jacobian of a hyperelliptic curve is especially interesting [27]. Since the definition of the jacobian is as a quotient group of two infinite groups, it is necessary to be able to describe in an efficient manner, representatives of cosets or, to be able to determine when two divisors are in the same coset, and hence identified. Secondly it is important to be able to add two elements of the jacobian and, again, identify the coset the result is in. For the jacobian of a hyperelliptic curve, these two problems have effective algorithms. For the first problem, it can be shown that every element in J can be uniquely represented as a reduced divisor defined as follows:

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 Definition 1. A divisor D = mi Pi − (∗)∞ ∈ D0 is said to be reduced if: (i) All the mi are non-negative and m1 ≤ 1 if Pi = Pi . (ii) IfPi = Pi then Pi and Pi do not both occur in the sum. (iii) mi ≤ g. Indeed any reduced divisor can be uniquely represented as the gcd of polynomials of a certain form. Likewise, an algorithm to add two divisors is also possible (see [27], Section 5. and Appendix) although it is too detailed to give here. Hyperelliptic curve cryptosystems have been seriously investigated by several prominent researchers. To date they seem to have disadvantages with respect to efficiency of operations, compared to elliptic curve systems at the same level of security and they have not been developed very far. At the same time a √ subexponential algorithm of complexity O( g log g), for fixed prime p, has been found for hyperelliptic curves of large genus, although no such algorithm has been found for the lower genus case.

5

Comments

The construction of curves with many points is an interesting area in its own right. Not only have these constructions given insights to the structure of such curves, they have also influenced developments in areas such as coding theory and cryptography, as touched upon in this article. New applications continue to be found such as low discrepancy sequences, sequences with low correlation and authentication codes, largely by Xing, Niederreiter and their co-authors. The continued development of constructions of such curves and their applications will be followed with interest. The set of references included here goes beyond the topics described, as a convenience for the reader wishing to pursue a topic.

References 1. L.M. Adleman, J. DeMarrais and M.-D. Huang, A subexponential algorithm for discrete logarithms over the rational subgroup of the jacobians of large genus hyperelliptic curves over finite fields, in Algorithmic Number Theory, L.M. Adleman and M.-D. Huang eds., Lecture Notes in Computer Science, vol. 877, Spriger-Verlag, pp. 28-40, 1994. 2. A. M. Barg, S. L. Katsman and M. A. Tsfasman, Algebraic Geometric codes from curves of small genus, Problems of Information Transmission, vol. 23, pp. 34-38, 1987. 3. Ian F. Blake, Gadiel Seroussi and Nigel Smart, Elliptic Curves in Cryptography, Cambridge University Press, 1999. 4. D. le Brigand, Decoding of codes on hyperelliptic curves, Lecture Notes in Computer Science, G. Cohen and P. Charpin eds., vol. 514, Berlin: Springer-Verlag, pp. 126-134, 1990. 5. A. Brouwer, Bounds on the minimum distance of a linear code, http://win.tue.nl/~aeb/voorlincod.html.

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6. Y. Driencourt and J.F. Michon, Elliptic codes over fields of characteristic 2, Jour. Pure and Appl. Algebra, vol. 45, pp. 15-39, 1987. 7. Y. Driencourt, Some properties of elliptic codes over a field of characteristic 2, in Lecture Notes in Computer Science, Berlin: Springer-Varlag, vol. 229, 1985. 8. V.G. Drinfeld and S. Vladut, Sur le nombre de points d’une courbe alg´ebrique, Anal. Fonct. et Appl., vol. 17, pp. 68-69, 1983. 9. N.D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in D.A. Buell and J.T. Teitelbaum, editors, Computational Perspectives on Number Theory: Proceedings of a Conference in Honor of A.O.L. Atkin, American Mathematical Society International Press, 7, 21–76, 1998. 10. W. Fulton, Algebraic Curves: An Introduction to Algebraic Geometry, Reading, MA: Addison-Wesley Publishing Company, 1969. 11. Number of rational points of a singular curve, Proc. Am. Math. Soc., vol. 126, pp. 2549 - 2556, 1998. 12. A. Garcia and H. Stichtenoth, A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound, Inventiones Math., vol. 121, pp. 211222, 1995. 13. A. Garcia and H. Stichtenoth, Algebraic function fields over finite fields with many rational points, IEEE Trans. Information Theory, vol. 41, pp. 1548-1563, 1995. 14. A. Garcia and J.F. Voloch, Fermat curves over finite fields, Jour. Number Theory, vol. 30, pp. 345- 356, 1988. 15. A. Garcia and H. Stichtenoth, On the asymptotic behaviour of some towers of function fields over finite fields, Jour. Number Theory, vol. 61, pp. 248-273, 1996. 16. G. van der Geer and M. van der Vlugt, Curves over finite fields of characteristic 2 with many rational points, C.R. Acad. Sci., Paris, S´erie I, vol. 317, pp. 593-507, 1993. 17. G. van der Geer and M. van der Vlugt, Tables of cuves with many points, preprint, available at http://www.wins.uva.nl/~geer. 18. G. van der Geer and M. van der Vlugt, How to construct curves over finite fields with many points, available at http://www.wins.uva.nl/~geer. 19. G. van der Geer and M. van der Vlugt, Constructing curves over finite fields with many points by solving linear equations, preprint. 20. G. van der Geer and M. van der Vlugt, Generalized Reed-Muller codes and curves with many points, J. Number Theory, vol. 72, pp. 257-268, 1998. 21. G. van der Geer, Codes and elliptic curves, in Effective Methods in Algebraic Geometry, T. Mora and C. Traverso eds., Birkh¨ auser, pp. 160-168, 1991. 22. V.D. Goppa, Codes associated with divisors, Prob. Info. Trans., vol. 13, pp. 22-27, 1977. 23. V. D. Goppa, Codes on algebraic curves, Soviet Math. Dokl., vol. 24, pp. 170-172, 1981. 24. M.-D. Huang and D. Ierardi, Counting points on curves over finite fields, J. Symbolic Conp., vol. 25, pp. 1-21, 1998. 25. Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Tokyo, vol. 28, pp. 721-724, 1981. 26. N. Koblitz, A course in number theory and cryptography, Graduate Texts in Mathematics, vol. 114, Berlin: Springer-Verlag, 1994. 27. N. Koblitz, Algebraic aspects of Cryptography, Algorithms and Computation in Mathematics, Vol. 3, Springer, 1998. 28. Kristin Lauter, Ray class field constructions of curves over finite fields with many rational points, in Algorithmic Number Theory, Henri Cohen ed., Lecture Notes in Computer Science, vol. 1122, Spriger-Verlag, 187-195 1996.

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29. Kristin Lauter, A formula for constructing curves over finite fields with many rational points, J. Number Theory, vol. 74, pp. 56-72, 1999. 30. Yu. I. Manin, What is the maximum number of points on a curve over F2 , J. Fac. Sci. Univ. Tokyo, Sec. IA, vol. 28, pp. 715-720, 1981. 31. A. Menezes, Elliptic Curve Public Key Cryptosystems, Dordrecht, The Netherlands: Kluwer, 1993. 32. C. Moreno, Algebraic curves over finite fields, Cambridge Tracts in Mathematics 97, Cambridge University Press, 1993. 33. H. Niederreiter and C. Xing, Low-discrepancy sequences and global function fields with many rational places, Finite Fields and their Applications, vol. 2, pp. 241-273, 1996. 34. H. Niederreiter and C. Xing, Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places, Acta Math., vol. 79, pp. 53-76, 1997. 35. H. Niederreiter and C. Xing, Explicit global function fields over the binary field with many rational places, Acta Math., vol. 75, pp. 383-396, 1996. 36. H. Niederreiter and C. Xing, Drinfeld modules of rank 1 and algebraic curves with many rational points II, Acta Math., vol. 81, pp. 81-100, 1997. 37. H. Niederreiter and C. Xing, Towers of global function fields with asymptotically many rational places and an improvement on the Gilbert-Varshamov, Math. Nachr. vol. 195, pp. 171-186, 1998. 38. Ren´e Schoof, Algebraic curves and coding theory, UTM 336, Univ. of Trento, 1990. 39. R. Schoof, Families of curves and weight distributions of codes, Bull. Amer. Math. Soc., vol. 32, pp. 171-183, 1996. 40. R. Schoof, Algebraic curves over F2 with many rational points, Jour. Number Theory, vol. 1, pp. 6-14, 1992. 41. J.-P. Serre, Nombres des points des courbes alg´ebriques sur Fq , S´eminaire de Th´eorie des Nombres de Bordeaux, 1982/83, no. 22. 42. J.-P. Serre, Sur le nombre des points rationnels d’une courbe alg´ebrique sur un corps fini, C.R. Acad. Sci., Paris, S´erie I, vol. 296, pp. 397-402, 1983. 43. H. Stichtenoth, Algebraic Function Fields and Codes, Berlin: Springer-Verlag, 1991. 44. M.A. Tsfasman, S.G. Vladut and T. Zink, Modular curves, Shimura curves and Goppa codes better than the Varshamov-Gilbert bound, Math. Nach., vol. 109, pp. 21-28, 1982. 45. W.C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. E.N.S., vol. 2, pp. 521-560, 1969. 46. C. Xing, Hyperelliptic function fields and codes, J. Pure and Appl. Alg., vol. 74, pp. 109-118, 1991. 47. C. Xing and H. Stichtenoth, The genus of maximal function fields, Manuscripta Math., vol. 86, pp. 217-224, 1995. 48. C. Xing and H. Niederreiter, Modules de Drinfeld et courbes alg´ebriques ayant beaucoup de points rationnels, C.R. 156 Acad. Sci., Paris, S´erie I, vol. 322, pp. 651- 654, 1996. 49. C. Xing and H. Stichtenoth, The genus of maximal function fields over finite fields, Manuscripta Math., vol. 86, pp. 217-224, 1995. 50. T. Zink, Degeneration of Shimura surfaces and a problem in coding theory, in Proceedings, Fundamentals of Computation Theory, Lecture Notes in Computer Science, vol. 199, L. Budach, ed., Springer-Verlag, 1986.

New Sequences of Linear Time Erasure Codes Approaching the Channel Capacity M. Amin Shokrollahi Bell Labs, Room 2C-353, 700 Mountain Ave, Murray Hill, NJ 07974, USA [email protected]

Abstract. We will introduce a new class of erasure codes built from irregular bipartite graphs that have linear time encoding and decoding algorithms and can transmit over an erasure channel at rates arbitrarily close to the channel capacity. We also show that these codes are close to optimal with respect to the trade-off between the proximity to the channel capacity and the running time of the recovery algorithm.

1

Introduction

A linear error-correcting code of block length n and dimension k over a finite field IFq —an [n, k]q -code for short—is a k-dimensional linear subspace of the standard vector space IFnq . The elements of the code are called codewords. To the code C there corresponds an encoding map Enc which is an isomorphism of the vector spaces IFkq and C. A sender, who wishes to transmit a vector of k elements in IFq to a receiver uses the mapping Enc to encode that vector into a codeword. The rate k/n of the code is a measure for the amount of real information in each codeword. The minimum distance of the code is the minimum Hamming distance between two distinct codewords. A linear code of block length n, dimension k, and minimum distance d over IFq is called an [n, k, d]q -code. Linear codes can be used to reliably transmit information over a noisy channel. Depending on the nature of the errors imposed on the codeword during the transmission, the receiver then applies appropriate algorithms to decode the received word. In this paper, we assume that the receiver knows the position of each received symbol within the stream of all encoding symbols. We adopt as our model of losses the erasure channel, introduced by Elias [3], in which each encoding symbol is lost with a fixed constant probability p in transit independent of all the other symbols. As was shown by Elias [3], the capacity of this channel equals 1 − p. It is easy to see that a code of minimum distance d is capable of recovering d−1 or less erasures. In the best case, it can recover from any set of k coordinates of the encoding which means that d − 1 = n − k. Such codes are called MDScodes. A standard class of MDS-codes is given by Reed-Solomon codes [10]. The connection of these codes with polynomial arithmetic allows for encoding and decoding in time O(n log 2 n log log n). (See, [2, Chapter 11.7] and [10, p. 369]). However, these codes do not reach the capacity of the erasure channel, since there is no infinite sequence of such codes over a fixed field. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 65–76, 1999. c Springer-Verlag Berlin Heidelberg 1999


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Elias [3] showed that a random linear code can be used to transmit over the erasure channel at any rate R < 1 − p, and that encoding and decoding can be accomplished with O(n2 ) and O(n3 ) arithmetic operations, respectively. Hence, we have on the one hand codes that can be encoded and decoded faster than general linear codes, but do not reach the capacity of the erasure channel; and on the other hand we have random codes which reach the capacity but have encoding and decoding algorithms of higher complexity. The paper [1] was the first to design codes that could come arbitrarily close to the channel capacity while having linear time encoding and decoding algorithms. Improving these results, the authors of [8] took a different approach and designed fast linear-time algorithms for transmitting just below channel capacity. For all  > 0 they were able to produce rate R = 1 − p(1 + ) codes along with decoding algorithms that could recover from the random loss of a p fraction of the transmitted symbols in time proportional to n ln(1/) with high probability, where n is the block length. These codes could also be encoded in time proportional to n ln(1/). They belong to the class of low-density parity check codes of Gallager [4]. In contrast to Gallager codes, however, the graphs used to construct the asymptotically good codes obtained in [8] are highly irregular. The purpose of the present paper is twofold. First, we prove a general trade-off theorem between the proximity of a given Gallager code to the channel capacity in terms of the loss fraction and the running time of the recovery algorithm of [8]. We show that in this respect, the codes constructed in that paper are close to optimal. Next, we exhibit a different sequence of asymptotically close to optimal codes which have better parameters than the codes in [8]. An interesting feature of these codes is that the underlying bipartite graphs are right regular, i.e., all nodes on the right hand side of the graph have the same degree. Since they are theoretically better than their peers, we expect them to also perform better in practice. The organization of the paper is as follows. In the next section we will review the construction of Gallager codes. Next, we prove upper bounds on the maximum tolerable loss fraction in terms of the running time of the decoding algorithm. The last two sections are concerned with the derivation of the sequence of right regular erasure codes.

2

Codes from Bipartite Graphs

In this section, we will briefly review the class of codes we are interested in, and the erasure recovery algorithm associated to them. Our codes are similar to the Gallager codes [4] in that they are built from sparse bipartite graphs. In contrast to Gallager codes, however, our codes will be constructed from graphs that have a highly irregular degree pattern on the left. Let G be a bipartite graph with n nodes on the left and n − k nodes on the right. G gives rise to a binary code of block-length n and dimension ≥ k in the

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67

following way: let the adjacency matrix of the graph G be given as 
H 0 , A= H 0 where H is some (n − k) × n matrix describing the connections in the graph. The code defined by the graph is the code with parity check matrix H. A different way of describing the code is as follows: we index the coordinate positions of the code with the n nodes on the left hand side of the graph. The code consists of all binary vectors (c1 , . . . , cn) such that for each right node in the graph the sum of the coordinate places adjacent to it equals zero. The block-length of this code equals n, and its dimension is at least k since we are imposing n − k linear conditions on the coordinates of a codeword. Expressed in terms of the graph, the fraction of redundant symbols in a codeword is at most aL /aR where aL and aR are the average node degrees on the left and the right hand side of the graph, respectively. In other words, the rate of the code is at least 1 − aL /aR . This description of the rate will be useful in later analysis. In the following, we will assume that the rate is in fact equal to this value. This is because the statements we will prove below will become even stronger if the rate is larger. The above construction needs asymptotically O(n2 ) arithmetic operations to find the encoding of a message of length k, if the graph is sparse. One can apply a trick to reduce the running time to O(n) by a modification of the construction. Details can be found in [8]. Suppose now that a codeword (c1 , . . . , cn ) is sent and that certain erasures have occurred. The erasure recovery algorithm works as follows. We first initialize the contents of the right hand nodes of G with zero. Then we collect the nonerased coordinate positions, add their value to the current value of their right neighbors, and delete the left node and all edges emanating from it from the graph. After this stage, the graph consists of the erased nodes on the left and the edges emanating from these nodes. In the next step we look for a right node in the graph of degree one, i.e., a node that has only one edge coming out of it. We transport the value of this node to its unique left neighbor , thereby recovering the value of c . We add c to the current value of all the right neighbors of , delete the edges emanating from , and repeat the process until we cannot find a node of degree one on the right, or until all nodes on the left have been recovered. It is obvious that, on a RAM with unit cost measure, the amount of arithmetic operations to finish the algorithm is at most proportional to the number of edges in the graph, i.e., to naL , where aL is the average node degree on the left. The aim is thus to find graphs with constant aL for which the recovery algorithm finishes successfully. The main contribution of [8] was to give an analytic condition on the maximum fraction of tolerable losses in terms of the degree distribution of the graph. More precisely, define the left and the right degree of an edge in the graph as the degree of the left, resp. right node it is emanating from. Further, denote by λi and ρi the fraction of edges degree i, and consider the  of left, resp. right generating functions λ(x) := i λi xi−1 and ρ(x) := i ρi xi−1 . In the following we will call the pair (λ, ρ) a degree distribution.

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Table 1. Rate 1/2 codes built from heavy tail/Poisson distribution. N 8 16 27 47 79 132 221

aR 5.9266 7.0788 8.0054 9.0256 10.007 10.996 12.000

θ 5.9105 7.0729 8.0027 9.0243 10.007 10.996 12.000

δ/(1 − R) 0.91968 0.95592 0.97256 0.98354 0.98990 0.99378 0.99626

δ 0.45984 0.47796 0.48628 0.49177 0.49495 0.49689 0.49813

δˆ 0.49085 0.49609 0.49799 0.49902 0.49951 0.49975 0.49988

δ/δˆ 0.93682 0.96345 0.97648 0.98547 0.99087 0.99427 0.99650

The main theorem of [8] states that if the graph is chosen at random with degree distribution (λ, ρ) and if the erasures occur at random positions, then the above erasure recovery algorithm can correct a δ-fraction of losses if ρ(1 − δλ(x)) ≥ 1 − x for x ∈ [0, 1]. In a later paper [6], this condition was slightly relaxed to δλ (1 − ρ(1 − x)) < x

for x ∈ (0, δ].

(1)

The paper [8] further exhibited for any  > 0 and any rate R an infinite sequence of degree distributions (λ, ρ) giving rise to codes of rate at least R such that the above inequality is valid for δ = (1−R)(1−), and such that the average left degree of the graph is O(log(1/)). In other words, δ can get arbitrarily close to its optimal value (1 − R) with a “logarithmic” sacrifice in the running time of the algorithm. Explicitly, for any given 0 <  < 1, one chooses an integer N close to 1/ and considers the pair (λN , ρN ) with λN (x) =

N −1 k  1 x , H(N − 1) k

ρN (x) = eθN ·(x−1),

(2)

k=1

where θN is chosen to make the fraction of the nodeson the left and the right N −1 equal to 1 − R, and H(N − 1) is the harmonic sum k=1 1/k. This sequence is referred to as the heavy tail/Poisson sequence in the following. Table 1 gives an example of the performance of this sequence for some codes of rate 1/2. In that table, aR denotes the average right degree, δ is the maximum tolerable loss fraction, and δˆ is a theoretical upper bound on δ, see Remark 1 following Corollary 1. In the following we will show that this sort of relationship between the average degree and the maximum tolerable loss fraction is the best possible. Furthermore, we will exhibit a new sequence of degree distributions for which the same type of relationship holds between the average degree and the maximum tolerable loss fraction.

New Sequences of Erasure Codes

3

69

Upper Bounds for the Maximum Tolerable Loss Fraction

In this section we will prove some upper bounds on the maximum tolerable loss fraction δ for which the algorithm given in the previous section is successful when applied to a random graph with a given degree distribution. Our main tool will be the inequality (1). We will first need a preliminary result. Lemma 1. Let G be a bipartite graph with edge degree distribution (λ, ρ). Then, the average left, resp. right, node degree of G equal 1 0

1 λ(x)dx

,

1 0

1 ρ(x)dx

,

respectively. Let the polynomials Λ and R be defined by x x λ(t)dt ρ(t)dt 0 , R(x) := 01 . Λ(x) :=  1 λ(t)dt ρ(t)dt 0 0 Then the coefficient of xi in Λ(x), resp. R(x) equals the fraction of nodes of degree i on the LHS, resp. RHS of G. Proof. Let L denote the number of nodes on the LHS of G. Then the number of edges in G equals aL L, and the number of edges of degree i equals λi aL L. Hence, the number of nodes of degree i on the LHS of the graph equals λi aL L/i, since each such node contributes to exactly i edges of degree i. Hence, the fraction of nodes of degree i equals λi aL /i. Since the sum of all these fractions equals 1, we 1  obtain a−1 L = i λi /i = 0 λ(x)dx and this yields both assertions for the LHS of the graph. The assertions on the RHS follow similarly.   The following theorem shows a rather strong upper bound on δ. Theorem 1. Let λ and ρ denote the right and left edge degree distributions of a bipartite graph. Let δ be a positive real number such that δλ (1 − ρ(1 − x)) ≤ x for 0 < x ≤ δ. Then we have δ≤

aL (1 − (1 − δ)aR ) , aR

where aL and aR are the average node degrees of the graph on the left, resp. right, hand side. Proof. As a real valued function the polynomial λ(x) is strictly increasing for positive values of x. Hence it has a unique inverse λ−1 which is also strictly increasing. The first of the above inequalities is thus equivalent to 1 − ρ(1 − x) ≤ λ−1 (x/δ)

70

M. Amin Shokrollahi

for 0 < x ≤ δ. Integrating both sides from 0 to δ and simplifying the expressions, we obtain  1−δ  1  1 ρ(x)dx − ρ(x)dx ≥ δ λ(x)dx. (Note that have

1 0

0

λ−1 (x)dx = 1 −

aL δ≤ − aL aR



1−δ 0

1 0

0

0

λ(x)dx.) Invoking the previous lemma, we thus

aL ρ(x)dx = aR



 1−δ

1− 1 0

0

ρ(x)dx



ρ(x)dx

=

aL (1 − R(1 − δ)) , aR

where the polynomial R is defined as in the statement of Lemma 1. To finish the proof we only need to show that R(1 − δ) ≥ (1 − δ)aR . To see this, first note that if a1 , . . . , aM are nonnegative real numbers adding up to 1 and µ is any nonnegative real number, then   ai µi ≥ µ i iai . (3) i

This follows from taking logarithms of both sides and noting that the log-function is concave. Denoting by ai the coefficients of R(x) and setting µ := 1 − δ, this gives our desired inequality:  the ai are by the previous lemma the fractions of nodes of degree i, and hence i iai is the average right degree aR .   Corollary 1. With the same assumptions as in the previous theorem, we have δ≤

aL (1 − (1 − aL /aR )aR ) . aR

Proof. Follows from δ ≤ aL /aR .

 

Remark 1. A more refined upper bound for δ is given by the following. Let r denote the quantity aL /aR , i.e., one minus the rate of the code. Suppose that aR > 1/r—an assumption that is automatically satisfied if the average left degree is larger than one—and consider the function f(x) = x−r(1−(1−x)aR ). We have f(0) = 0, f(1) = 1 − r > 0, f  (0) < 0, and f  (x) has exactly one root. Hence, f has exactly one nonzero root, denoted by δˆaR ,r and this root is between 0 and 1. The previous theorem asserts that the maximum δ satisfying the inequality (1) is smaller than δˆaR ,aL /aR . The following definition is inspired by Corollary 1. Definition 1. A sequence (λm , ρm ) of degree distributions giving rise to codes of rate R ≥ R0 for a fixed R0 > 0 is called asymptotically quasi-optimal if there exists a constant µ (possibly depending on R) and for each m there exists a positive δm with δm λm (1 − ρm (1 − x)) ≤ x for x ∈ (0, δm ] such that the average right degree aR satisfies aR ≤ µ log(1 − δ/(1 − R)).

New Sequences of Erasure Codes

71

The significance of quasi-optimal is the following: we want to construct codes that can recover as many erasures as possible in as little as possible time. The running time of the erasure recovery algorithm is proportional to the blocklength of the code times aR , the average right degree. Hence, we are looking for a trade-off between aR and δ. Corollary 1 shows that we cannot make aR too small. In fact, it implies that aR ≥ log(1 − δ/(1 − R))/ log R. However, we might hope that we can make aR smaller than log(1 − δ/(1 − R)) times a (negative) constant µ. In this way, we have maintained a qualitative optimality of the code sequence in the sense that the running time increases only logarithmically with the relative increase of the erasure correction capability. The heavy tail/Poisson sequence (2) is asymptotically quasi-optimal as is shown in [8]. Starting in the next section, we will introduce another quasi-optimal sequence which has certain advantages over the heavy tail/Poisson sequence. We close this section by stating another useful upper bound on the maximum possible value of δ. Lemma 2. Suppose that λ and ρ are polynomials and δ is such that δλ(1−ρ(1− x)) < x for x ∈ (0, δ]. Then δ ≤ ρ (1)/λ (0). Proof. Let f(x) = δλ(1 − ρ(1 − x)) − x. By assumption, f(x) < 0 for x ∈ (0, δ]. This implies that f  (0) ≤ 0, where f  (x) is the derivative of f with respect to x. But f  (0) = δλ (0)ρ (1) − 1, which gives the assertion.  

4

Fractional Binomial Coefficients

As can be seen from their description (2), the heavy tail/Poisson sequence is closely related to the Taylor series of − ln(1 − x). The new sequence that we will describe in the next section will be related to the Taylor expansion of (1 − x)α where 1/α is an integer. The coefficients of this expansion are fractional binomial coefficients. For this reason, we will recall some well-known facts about these numbers. For real α and a positive integer N we define
 α α(α − 1) · · · (α − N + 1) := N N!
 α α

α 1− ··· 1 − (1 − α) . = (−1)N −1 N N −1 2  For convenience, we also define α0 := 1. We have the Taylor expansion ∞
  α α 1 − (1 − x) = (4) (−1)k+1 xk . k k=1

Note that for 0 < α < 1 the coefficients of the right hand power series above are all positive. Furthermore, we have
 N −1
  α N α (−1)N +1 , (−1)k+1 = 1 − (5) α N k k=1

72

and

M. Amin Shokrollahi N −1
 k=1

α  (−1)N +1 α− N α (−1)k+1 = , k+1 α+1 k

(6)

as can easily be proved by induction. In the following, we will derive some estimates on the size of the binomial coefficients. Proposition 1. Let α be a positive real number less than 1/2 and let N ≥ 2 be an integer. There is a constant c independent of α and N such that
 α α cα ≤ (−1)N +1 ≤ α+1 . α+1 N N N α N +1 Proof. First note that N (−1)  i is positive. Taking its logarithm and expanding the series − ln(1 − x) = i x /i, we obtain

  N N  1 α2  1 α N +1 (−1) ln = ln(α/N ) − α − ···. − k 2 k2 N k=1

(7)

k=1

(Note that the series involved are absolutely convergent, so that we can rearrange the orders of the summations.) For an upper bound on this sum we replace N s k=1 1/k by 1 for s ≥ 2 and use the left hand side of the inequality ln(N ) + γ
1 in the nonlinear congruential method, we can overcome the restriction in first-order recurrences that the period length cannot exceed the modulus. For these reasons, it is of interest to study nonlinear congruential generators of higher orders. When m = 1, for sequences of the largest possible period t = p, a number of results about the distribution of the fractions un/p in the interval [0, 1) and, more generally, about the distribution of the points
 un un+s−1 ,..., (2) p p in the s-dimensional unit cube [0, 1)s have been obtained in the case where n runs through the full period, n = 0, . . . , p − 1. Many of these results are essentially best possible. We refer to [4,5,8,9,10,13,14,15,16,17,18,19] for more details and precise references to original papers. The case of periods t < p is of interest as well. Quite recently, in the series of papers [8,16,17,18,19] a new method has been introduced and successfully applied to the case m = 1. In the present paper we show that the same method works for nonlinear generators of arbitrary order m ≥ 1. In particular, we obtain rather weak but nontrivial bounds on the discrepancy of the points (2) when n runs over a part of the full period. In the very special case where f(X) = X e , that is, for the power generator , an alternative approach has been proposed in [6]. This approach, although it has produced quite strong results for the power generator, cannot be extended to other nonlinear generators. Moreover, a combination of the approach of [8,16,17,18,19] and the present work with some additional considerations has been used in [7] to improve the results of [6].

2

Definitions and Auxiliary Results

For a sequence of N points N

Γ = (γ1,n , . . . , γs,n )n=1 of the half-open interval [0, 1)s , denote by ∆Γ its discrepancy, that is,    TΓ (B)  ∆Γ = sup  − |B| , N B⊆[0,1)s where TΓ (B) is the number of points of the sequence Γ which hit the box B = [α1 , β1 ) × . . . × [αs , βs ) ⊆ [0, 1)s

(3)

Distribution of Nonlinear Recursive Congruential Pseudorandom Numbers

89

and the supremum is taken over all such boxes. For an integer vector a = (a1 , . . . , as ) ∈ ZZ s we put |a| = max |ai |,

r(a) =

i=1,...,s

s 

max{|ai |, 1}.

(4)

i=1

We need the Erd¨ os–Tur´ an–Koksma inequality (see Theorem 1.21 of [3]) for the discrepancy of a sequence of points of the s-dimensional unit cube, which we present in the following form. Lemma 1. There exists a constant Cs > 0 depending only on the dimension s such that, for any integer L ≥ 1, for the discrepancy of a sequence of points (3) the bound       s    N  1 1 1  ∆ Γ < Cs  + exp 2πi aj γj,n   L N r(a) n=1  j=1 0 0, and it would be very important to extend the range of N for which Ds (N ) = o(1). We believe that Theorems 1 and 2 hold for much more general polynomials, not necessarily only polynomials with a dominating term. Obtaining such an extension would be very important. We note that for univariate polynomials this condition is automatically satisfied, thus our results indeed generalise those of [17]. It would be very interesting to extend the results of this paper to the case of nonlinear generators with rational functions f(X1 , . . . , Xm ) ∈ IFp (X1 , . . . , Xm ).

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Frances Griffin, Harald Niederreiter, and Igor E. Shparlinski

At the beginning of this paper we have mentioned two special generators in the case m = 1, namely the inversive generator and the power generator, for which essentially stronger results than in the general case are known, see [8,16,19] and [6,7], respectively. It is desirable to understand what are the analogues of these special generators in the case m ≥ 2 and extend the results of [6,7,8,16,19] to these analogues. Finally we remark that our method works for generators modulo a composite number as well. But one should expect weaker results because instead of the very powerful Weil bound one will have to use bounds on exponential sums with composite denominator which are essentially weaker, see [20].

References 1. T. Cochrane, ‘Trigonometric approximation and uniform distribution modulo 1’, Proc. Amer. Math. Soc., 103 (1988), 695–702. 2. P. Deligne, ‘Applications de la formule des traces aux sommes trigonom´etriques’, Lect. Notes in Mathematics, Springer-Verlag, Berlin, 569 (1977), 168–232. 3. M. Drmota and R.F. Tichy, Sequences, discrepancies and applications, SpringerVerlag, Berlin, 1997. 4. J. Eichenauer-Herrmann, E. Herrmann and S. Wegenkittl, ‘A survey of quadratic and inversive congruential pseudorandom numbers’, Lect. Notes in Statistics, Springer-Verlag, Berlin, 127 (1998), 66–97. 5. M. Flahive and H. Niederreiter, ‘On inversive congruential generators for pseudorandom numbers’, Finite Fields, Coding Theory, and Advances in Communications and Computing (G.L. Mullen and P.J.-S. Shiue, eds.), Marcel Dekker, New York, 1993, 75–80. 6. J. B. Friedlander, D. Lieman and I. E. Shparlinski, ‘On the distribution of the RSA generator’, Proc. SETA ’98 (C.S. Ding, T. Helleseth and H. Niederreiter, eds.), Springer-Verlag, Singapore, (to appear). 7. J. B. Friedlander and I. E. Shparlinski, ‘On the distribution of the power generator’, Preprint, 1999. 8. J. Gutierrez, H. Niederreiter and I. E. Shparlinski, ‘On the multidimensional distribution of inversive congruential pseudorandom numbers in parts of the period’, Monatsh. Math., (to appear). 9. D. Knuth, The art of computer programming, Vol. 2, 3rd ed., Addison-Wesley, Reading, MA, 1998. 10. R. Lidl and H. Niederreiter, ‘Finite fields and their applications’, Handbook of Algebra (M. Hazewinkel, ed.), Vol. 1, Elsevier, Amsterdam, 1996, 321–363. 11. R. Lidl and H. Niederreiter, Finite fields, Cambridge University Press, Cambridge, 1997. 12. G. Marsaglia, ‘The structure of linear congruential sequences’, Applications of Number Theory to Numerical Analysis (S.K. Zaremba, ed.), Academic Press, New York, 1972, 249–285. 13. H. Niederreiter, Random number generation and quasi–Monte Carlo methods, SIAM, Philadelphia, 1992. 14. H. Niederreiter, ‘Finite fields, pseudorandom numbers, and quasirandom points’, Finite Fields, Coding Theory, and Advances in Communications and Computing (G.L. Mullen and P.J.-S. Shiue, eds.), Marcel Dekker, New York, 1993, 375–394.

Distribution of Nonlinear Recursive Congruential Pseudorandom Numbers

93

15. H. Niederreiter, ‘New developments in uniform pseudorandom number and vector generation’, Lect. Notes in Statistics, Springer-Verlag, Berlin, 106 (1995), 87–120. 16. H. Niederreiter and I. E. Shparlinski, ‘On the distribution of inversive congruential pseudorandom numbers in parts of the period’, Preprint, 1998. 17. H. Niederreiter and I. E. Shparlinski, ‘On the distribution and lattice structure of nonlinear congruential pseudorandom numbers’, Finite Fields and Their Appl., 5 (1999), 246–253. 18. H. Niederreiter and I. E. Shparlinski, ‘On the distribution of pseudorandom numbers and vectors generated by inversive methods’, Preprint, 1999. 19. H. Niederreiter and I. E. Shparlinski, ‘Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus’, Acta Arith., (to appear). 20. S. B. Steˇckin, ‘An estimate of a complete rational exponential sum’, Trudy Mat. Inst. Steklov., 143 (1977), 188–207 (in Russian).

A New Representation of Boolean Functions Claude Carlet1 and Philippe Guillot2
1

INRIA Projet CODES, BP 105, 78153 Le Chesnay Cedex, France and GREYC, Universit´e de Caen, France 2 Thomson-CSF Communication, 66 rue du Foss´e blanc 92231 Gennevilliers Cedex, France

Abstract. We study a representation of Boolean functions (and more generally of integer-valued / complex-valued functions), not used until now in coding and cryptography, which yields more information than the currently known representations, on the combinatorial, spectral and cryptographic properties of the functions. Keywords: Boolean function, Fourier spectrum, cryptography.

1

Introduction

Let n be any positive integer. From a cryptographic and coding theoretic point of view, we are interested in Boolean (i.e. {0, 1}-valued) functions defined on the set F2 n of all binary words of length n. This set is viewed as an F2 -vector space of dimension n (the addition in this vector-space is mod 2 and will be denoted by ⊕). Since {0, 1} can be viewed either as a subset of Z or as the field F2 , we need to distinguish between the addition
in Z (denoted by +, the sum of r several terms b1 , · · · , br will then be denoted by i=1 bi ) and the addition r in F2 (denoted by ⊕, the sum of several terms b1 , · · · , br will be denoted by i=1 bi ). The basic Boolean functions are the affine functions: f(x1 , · · · , xn ) = a1 x1 ⊕ · · · ⊕ an xn ⊕ a0 = a · x ⊕ a0 where a = (a1 , · · · , an ) ∈ F2 n and a0 ∈ F2 . The expression a · x = a1 x1 ⊕ · · · ⊕ an xn ⊕ a0 is the usual inner product in F2 n . The set of all the affine functions is the Reed-Muller code of length 2n and order 1 (a Boolean function is identified with the 2n -long binary word of its values, assuming that some order on F2n is chosen). Important parameters of Boolean functions are: – the Hamming weight: w(f) is the size of the support of f, i.e. of the set {a ∈ F2 n | f(a) = 1}; the Hamming distance between two functions f and g is d(f, g) = w(f ⊕ g); – the Fourier
spectrum, i.e. the data of all the values of the Fourier transform of f: f(a) = x∈F2 n f(x) (−1)a·x . The related Fourier spectrum of the function χf = (−1)f = 1 − 2f is equal to χ f =  1 − 2f = 2n δ0 − 2f, where δ0 is the Dirac symbol at the all-zero word (recall that, in general, δa (x) equals 1 if x = a and 0 otherwise).


AMS classification numbers: 06E30, 11T23, 94A60.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 94–103, 1999. c Springer-Verlag Berlin Heidelberg 1999 

A New Representation of Boolean Functions

95

– the nonlinearity Nf of f, i.e. the Hamming distance between f and the set of affine functions. We have Nf = 2n−1 − 12 maxa∈F2 n | χf (a)|; The Boolean functions whose nonlinearity is maximum are called bent functions (cf. [9]). They are interesting from a coding point of view, since they correspond to the words of length 2n whose distance to the Reed-Muller code of order 1 is equal to the covering radius of this code. The bent functions on F2 n with n even have extra properties which make them also interesting from a cryptographic point of view. They are called perfect nonlinear [6] and characterized by the fact that, for any nonzero word a, the Boolean function x → f(x) ⊕ f(x ⊕ a) is balanced (i.e. takes the values 0 and 1 equally often). We describe now the representations of Boolean functions currently used in cryptography and in coding theory. The truth-table of a Boolean function f is the one-dimensional table, indexed by the elements of F2 n (some order being chosen), whose entry at a ∈ F2 n is f(a). The advantage of this representation is its simplicity
together with the fact that the weight of f is directly computed from w(f) = a∈F2 n f(a). But it has important drawbacks: – it does not give any information on the algebraic degree of the function and on the number of terms in its algebraic normal form (see below); – the affine functions have the same complexity as any other function; – it does not directly characterize perfect nonlinear functions. The Algebraic Normal Form A.N.F. (cf. [7]): n    ui f(x1 , · · · , xn ) = au xi ; au ∈ F2 u∈F2 n

(1)

i=1

is the sum, modulo 2, of monomials in which each of the n binary variables n has degree at most 1. For simplicity, we shall write xu instead of i=1 xi ui . The algebraic degree of f is the global degree of its A.N.F. The A.N.F. can be computed from the truth table with a complexity O(n 2n ) by a butterfly algorithm (cf. [4]); conversely, the truth table  can be computed from the A.N.F. with the same complexity. We have au = xu f(x), where x = (x1 , . . . , xn )

u = (u1 , . . . , un ) ⇐⇒ ∀i ∈ {1, . . . , n} xi ≤ ui . The qualities of the A.N.F. are the following: – it leads to an important class of linear codes: the Reed-Muller codes; – the complexity of the A.N.F. of a function is coherent with its algebraic complexity, i.e. its implementation with and/xor gates; but it has also some drawbacks: – there exists no simple formula which computes the weight of a function, given its A.N.F.; the only known formula, cf. [10], page 124 is complex; – a fortiori, there exists no simple formula which computes the Fourier spectrum or the nonlinearity of a function, given its A.N.F.; – it does not directly characterize perfect nonlinear functions.

96

Claude Carlet and Philippe Guillot

The Fourier spectrum defined above can be also considered as a representation. It can be computed from the truth table by a butterfly algorithm with complexity O(n 2n ) (cf. [4]). Its qualities are the following: – the weight of f is directly given by w(f) = f(0); – the affine functions are characterized by the fact that one of the values of the Fourier spectrum of χf is equal to ±2n and all the others are null; – perfect nonlinear functions are directly characterized by means of the Fourier spectrum of χf : f is perfect nonlinear if and only if, for every word a ∈ F2 n , n the value at a of the Fourier transform of χf is equal to ±2 2 . But this representation has at least two important drawbacks: – there is no simple characterization of the fact that a list (µa )a∈F2 n of integers corresponds to the Fourier spectrum of a Boolean function; – the algebraic degree cannot be directly deduced from the Fourier spectrum.

2

The Numerical Normal Form

This representation of Boolean functions (and more generally of complex-valued functions) is similar to the Algebraic Normal Form, but with numerical coefficients. It has been used in the context of circuit complexity. We study this representation in a more systematic way which allows direct formulae for the weight and the Fourier spectrum. It leads to a characterization of perfect nonlinear functions and to divisibility properties of the weight. Boolean functions are characterized with a single formula by means of this representation. The overcost of such an improvement is the storage of 2n integers in the range {−2n , · · · , 2n } instead of 2n bits. Definition 1. Let f be a complex-valued function on F2 n . We call Numerical Normal Form (N.N.F.) of f, the following expression of f as a polynomial : n  



ui f(x1 , · · · , xn ) = λu xi λu xu , λu ∈ C. (2) = u∈F2 n

i=1

u∈F2 n

Proposition 1. Any complex-valued function admits a unique Numerical Normal Form. Proof. The set of all complex-valued functions on F2 n , and the set of all polynomials over C in n variables such that every variable has degree at most 1 are vector-spaces of dimension 2n over C. The mapping which maps any such polynomial P (x1 , · · · , xn ) to the complex-valued function f : a ∈ F2 n → P (a1 , · · · , an ) is linear. It is enough to show that this linear mapping is surjective. For every x ∈ F2 n , the polynomial expression of the Dirac function δa (x) is equal to the product, when i ranges over {1, · · · , n} of xi if ai = 1, and of 1 − xi if ai = 0. By expanding this product, we deduce the existence of a polynomial representing δa . A polynomial representing
any complex-valued function is then obtained by using the equality f(x) = a∈F2 n f(a) δa (x).

A New Representation of Boolean Functions

3 3.1

97

Relations between N.N.F. and other Representations N.N.F. and Truth Table

According to the definition of the N.N.F.,
the truth table of f can be recovered n from its N.N.F. by the formula f(a) = ua λu , ∀a ∈ F2 . Conversely, it is possible to derive an explicit formula giving the N.N.F. of the function by means of its truth table. Proposition 2. Let f be any complex-valued function on F2 n . For every u ∈ F2 n , the coefficient λu of the monomial xu in the N.N.F. of f is:

λu = (−1)w(u) (−1)w(a)f(a) a∈F2 n | au

where w(u) denotes the Hamming weight of the word u (i.e. the number of nonzero coordinates).
Proof. Since we have f(x) = a∈F2 n f(a) δa (x), it is enough to prove that the coefficient of xu in the N.N.F. of the Dirac symbol δa is (−1)w(u)−w(a) if a u and 0 otherwise. by supp(a) Denoting  the set {i = 1, · · · , n | ai = 1}, the value of δa (x) is equal to i∈supp(a) xi i∈ / supp(a) (1 − xi ) . Let Ia be the complement of supp(a)
w(v) vi in {1, · · · , n}, we have i∈Ia (1 − xi ) = v∈F2 Ia (−1) i∈Ia xi , and the result holds, denoting by u the word whose support equals the union of those of a and v. We give now a butterfly algorithm for computing the coefficients λu . The truth table of a function f, in n variables, is the concatenation of the truth tables of the two functions, in (n − 1) variables: f0 : (x2 , . . . , xn ) → f(0, x2 , . . . , xn ) f1 : (x2 , . . . , xn ) → f(1, x2 , . . . , xn ).

Let f0 (x) = u∈F n−1 µu xu and f1 (x) = u∈F n−1 νuxu be respectively the 2 2 N.N.F.’s of f0 and f1 . It is easy to deduce from proposition 2 that λ(0,u2,... ,un) = µ(u2,... ,un) and λ(1,u2,... ,un ) = ν(u2,... ,un ) − µ(u2,... ,un ) . This leads to the F.F.T.-like algorithm which computes the N.N.F. of any function with a complexity of n2n−1 elementary substractions: procedure NNF(f) for i ← 0 to n − 1 do b ← 0; repeat for x ← b to b + 2i − 1 do f[x + 2i ] ← f[x + 2i ] − f[x]; end for; b ← b + 2i+1 ; until b = 2n ; end for; end procedure

98

Claude Carlet and Philippe Guillot

In this algorithm, the function f is given by an array of 2n numbers (integer, real or complex), and the
value of f at vector u = (u1 , . . . , un ) is the k-th entry n of this array, where k = i=1 ui 2n−i−1 = u1 2n−1 + · · · + un . 3.2

N.N.F. and A.N.F.

Given the N.N.F. of a Boolean function f, the A.N.F. of f is simply equal to its N.N.F. mod 2. Conversely, the coefficients of the N.N.F. can be recovered from those of the A.N.F. This uses the well-known Poincar´e’s formula which computes the xor of a set of binary values by using usual arithmetic operations. Lemma 1 ((Poincar´ e formula)). Let a1 , . . . , am be elements of the set {0, 1}. The following formula holds: m 

ai =

i=1

m

(−2)k−1

k=1



ai1 · · · aik .

1≤i1 d. Thus, for k < d , i.e. for   there

k≤

w(u) d

− 1, every term of sum (3) has a null factor auj with w(uj ) > d.

Moreover, if w(u1 ∨ · · · ∨ uk ) > rd + (k − r)(d − 1) = k(d − 1) + r, then w(u)−r there exists an index j such that w(uj ) > d. Thus, for k < d−1 , i.e. for   w(u)−r k≤ − 1, every term of sum (3) has a null factor auj with w(uj ) > d. d−1 Corollary 1. Under the same hypothesis as in Proposition 6, the coefficient λu w(u) w(u)−r of xu in its N.N.F. is a multiple of 2 d −1 and of 2 d−1 −1 . Mc Eliece theorem on cyclic codes and Ax Theorem imply the following result on Boolean functions (cf. [7], page 447): Proposition 7. Let f be a Boolean function of degree d, then the weight of f n is a multiple of 2 d −1 . Corollary 1 and relation (5) give a new proof of this result. Indeed, according to Corollary 1 each term of the  sum in relation (5) is multiple of w(u) w(u) n−w(u)+ d −1 − 1 is minimum for w(u) = n. 2 . And n − w(u) + d

We know that the bound given by Proposition 7 is tight. However, Corollary 1 improves upon it for the functions which have few monomials of highest degree in their A.N.F.: Proposition 8. Let f be a Boolean function of degree d. Assume that the numn ber of monomials of degree d in its A.N.F. is r < nd . Then n−r d−1 > d and the n−r weight of f is a multiple of 2 d−1 −1 . Proof. According to Corollary 1, each term of the sum in relation (5)is a multiple  w(u) w(u)−r w(u) of 2n−w(u)+ d −1 and of 2n−w(u)+ d−1 −1 . We have n−w(u)+ d −1 ≥   w(u)−r n − w(u) + − 1 if and only if w(u) ≤ rd and it is a simple matter to d−1     w(u) w(u)−r check that max(n − w(u) + d − 1, n − w(u) + d−1 − 1) is minimum for w(u) = n and the result holds.

A New Representation of Boolean Functions

6

103

Conclusion

Since integer-valued functions are characterized by an integer-valued N.N.F. and thanks to Proposition 4, the N.N.F. representation allows more easily to construct a general Boolean function with prescribed combinatorial properties such as bentness or high nonlinearity. It was not possible before as Boolean functions are not easily characterized with the Fourier transform, and combinatorial properties are not directly ensured from the truth table or A.N.F. representation.

References 1. C. Carlet. Generalized Partial Spreads, IEEE Transactions on Information Theory vol 41 (1995) 1482-1487 2. C. Carlet and P. Guillot. A characterization of binary bent functions, Journal of Combinatorial Theory, Series A, Vol. 76, No. 2 (1996) 328-335 3. J. F. Dillon. Elementary Hadamard Difference sets, Ph. D. Thesis, Univ. of Maryland (1974). 4. C.J.A. Jansen. Investigation on Nonlinear Streamcipher Systems: Construction and Evaluation Methods Philips (1989). 5. J.P.S. Kung. Source Book in Matro¨ıd Theory, Birkha¨ user (1986). 6. Meier, W. and O. Staffelbach. Nonlinearity Criteria for Cryptographic Functions, Advances in Cryptology, EUROCRYPT’ 89, Lecture Notes in Computer Science 434, 549-562, Springer Verlag (1990). 7. F. J. Mac Williams and N. J. Sloane. The theory of error-correcting codes, Amsterdam, North Holland 1977. 8. Gian-Carlo Rota. On the foundations of Combinatorial Theory; Springer Verlag (1964) ; reprint in [5]. 9. O. S. Rothaus. On bent functions, J. Comb. Theory, 20A (1976) 300- 305. 10. J. H. Van Lint. Coding Theory, Springer Verlag 201 (1971).

An Algorithm to Compute a Nearest Point in the Lattice A∗n I. Vaughan and L. Clarkson
Department of Electrical and Electronic Engineering The University of Melbourne, Parkville, Victoria 3052, Australia [email protected]

Abstract. The lattice A∗n is an important lattice because of its covering properties in low dimensions. Conway and Sloane [3] appear to have been the first to consider the problem of computing the nearest lattice point in A∗n . They developed and later improved
 [4] an algorithm which is able to compute a nearest point in O n2 arithmetic steps. In this paper, a new algorithm is developed which is able to compute a nearest point in O(n log n) steps.

1

Introduction

The study of point lattices is of great importance in several areas of number theory, particularly the studies of quadratic forms, the geometry of numbers and simultaneous Diophantine approximation, and also to the practical engineering problems of quantisation and channel coding. They are also important in studying the sphere packing problem and the kissing number problem [5]. Let us now define what is meant by the term ‘point lattice’. Definition 1. Consider a set B = {b1 , b2 , . . . , bn } of linearly independent points in IRm , m
n. The set Λ = {a1 b1 + a2 b2 + · · · + an bn | a1 , a2 , . . . , an ∈ ZZ} is a (point) lattice of rank n in IRm and B is a basis of Ω. The lattice that is studied in this article, known as ‘A∗n ’ following the notation of Conway and Sloane or sometimes known as ‘Voronoi’s principal lattice of the first type’, is remarkable because of its covering properties in low dimensions [5]. The author’s interest in the lattice arises from his work in the engineering problem of pulse train deinterleaving [1,2]. The lattice A∗n reduces to the hexagonal lattice when n = 2 and to the body-centred cubic lattice when n = 3. This is illustrated in Fig. 1. The computational problem of finding a nearest lattice point to a given point is the particular problem of interest here. van Emde Boas [7] showed that 

This work was supported by the Australian Research Council under Grant S499721.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 104–120, 1999. c Springer-Verlag Berlin Heidelberg 1999 

An Algorithm to Compute a Nearest Point in the Lattice A∗n

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Fig. 1. Examples of the lattice A∗n for n = 2 (the hexagonal lattice) and n = 3 (the body-centred cubic lattice).

the problem is NP -complete under certain conditions when the lattice itself, or rather a basis thereof, is considered as an additional input parameter. For specific lattices, the problem is considerably easier. The problem of computing the nearest lattice point in A∗n was first studied by Conway and Sloane [3]. By decomposing the lattice A∗n into the finite superposition of translations of its dual lattice An , the authors discovered for computing the nearest
an algorithm  lattice point to a given point in O n2 log n arithmetic operations. [4],
Later  they were able to improve the execution time of the algorithm to O n2 steps. In this paper, a new algorithm is presented which is able to compute a nearest lattice point in A∗n to a given point in O(n log n) arithmetic steps. After discussing some mathematical preliminaries in Sect. 2, we investigate in Sect. 3 the so-called Voronoi regions and relevant sets of vectors in lattices and, particularly, in A∗n . A Voronoi region surrounds each lattice point, and any point within this region is closer to the lattice point at the region’s centre than to any other lattice point. The relevant vectors are those lattice vectors which can be used to define the boundary of the Voronoi region. Having found a relevant set of vectors for A∗n , we find it convenient in Sect. 4 to define three notions of proximity of a given point to a lattice point. These notions allow us to decompose the algorithm into three smaller algorithms, discussed in turn in Sects. 5–7, each of which calculates a successively closer lattice point to the given point until at last a nearest lattice point has been found.

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Mathematical Preliminaries

In this section, we will briefly discuss the notation and terminology used throughout the sequel. Firstly, we note that we will, on occasion and without notice, use elements of IRm as if they were column vectors (elements of IRm×1 ). We will use the notation IPm to refer to the set of (elementwise) permutations of the point (1, 2, . . . , m) ∈ ZZ m . We also define the parameterised set σ(·) as follows. Definition 2. We define the parameterised set σ(·) such that if x ∈ IRm and s ∈ σ(x) ⊆ IPm then xs1  xs2  . . .  xsm . Finally, it is necessary to define the lattice A∗n . Definition 3. The lattice A∗n is the lattice of rank n in IRn+1 whose basis vectors are any n columns of the matrix   n −1 · · · −1 −1 n · · · −1 1   (1) 11T =  . . . B=I− .  . n+1  .. .. . . ..  −1 −1 · · · n Remark 1. The matrix B is a symmetric projection matrix, which is to say that B = B T and B 2 = B. Remark 2. Notice that bi · bj =

 n  n + 1   −1 n+1

if i = j, (2) otherwise,

and, more generally, that x · bj = xj

(3)

for any x ∈ BIRn+1. Remark 3. The lattice A∗n is constituted of vectors in a vector space whose dimension, n + 1, exceeds the rank of the lattice, n. Remark 4. In Figs. 1 and 2, A∗n is represented with respect to the hyperplane BIRn+1 on which the lattice vectors lie. That is, representative coordinates of the lattice points in IRn are generated by premultiplication by QT where Q ∈ IR(n+1)×n is a matrix whose columns form an orthonormal basis of BIRn+1 . In the case of A∗3 , this is followed by an orthographic projection into IR2 .

An Algorithm to Compute a Nearest Point in the Lattice A∗n

3

107

Relevant Sets of Lattice Vectors

Consider a lattice Λ in IRm and a norm · . Around the origin, there is a region V consisting of points which are closer to the origin than to any other lattice point. That is, we define V as the set V = {x ∈ IRn ; x  x − v ∀v ∈ Λ} . Such a region is called the Voronoi cell of the origin. The whole of IRm can be tessellated by translating these cells by the lattice vectors. Therefore, an algorithm to find the nearest lattice point to a given point can be interpreted as an algorithm to find the particular translation of the Voronoi cell to which the given point belongs. Consider the Voronoi cell of the origin. It is a convex polytope with faces that lie in the hyperplanes at the midway points along the lines connecting the origin to nearby lattice points. The set of vectors which define the faces are the (Voronoi-)relevant vectors of the lattice. Definition 4. For a lattice Λ in IRm , a set of non-zero lattice vectors R is relevant if, for every x ∈ IRm which satisfies x 

1 2

x − r

(4)

for every r ∈ R, this same inequality is also satisfied for every r ∈ Λ. In other words, a relevant set contains lattice vectors such that, for any point which is closer to the origin than to any of the vectors in the set, that point is closer to the origin than to any other lattice point. A straightforward implication of the definition is that a lattice point v ∈ Λ is closest to some point u ∈ IRm if (4) is satisfied with x = u − v for each lattice point r in a relevant set. For the Euclidean norm, we note that (4) is equivalent to x · x  (x − r) · (x − r) = x · x − 2x · r + r · r and, after cancellation and rearrangement of terms, it is equivalent to x·r 

1 2

r

2

.

(5)

Let us now turn our attention to the lattice A∗n with the Euclidean norm. Henceforth, we will use the notation · exclusively to denote the Euclidean norm of its argument. Consider the set

m   n+1 R= bsi ; s ∈ IP ,1  m  n . (6) i=1

We will show that this set is relevant to A∗n .

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Fig. 2. Examples of the Voronoi regions of the lattice A∗n for n = 2 (a hexagon) and n = 3 (a truncated octahedron). Lemma 1. Every element v ∈ A∗n can be expressed in the form v=

j n  

c j bs i ,

(7)

j=1 i=1

where the cj ∈ ZZ, cj
0, j = 1, 2, . . . , n and s ∈ IPn+1 . Proof. We can express any v with respect to the basis vectors b1 , b2 , . . . , bn in the obvious way, which is to say that v=

n 

aj bj ,

j=1

where the aj ∈ ZZ, j = 1, 2, . . . , n. Clearly, we can extend the summation to n + 1 terms, so that v=

n+1 

aj bj

j=1

with the coefficient an+1 = 0. Now, choose s ∈ σ(−a), so that as1
as2
. . .
asn+1 . Then, using the fact that bsn+1 = −bs1 − bs2 − · · · − bsn ,

An Algorithm to Compute a Nearest Point in the Lattice A∗n

109

we find that n 


v=

 asj − asn+1 bsj

j=1 j n  


=

 asj − asj+1 bsi .

j=1 i=1

Therefore, with cj = asj − asj+1 , we satisfy (7).



Lemma 2. If v, w ∈ A∗n can be expressed as v=

p 

bs i ,

w=

i=1

q 

bs i ,

i=1

where s ∈ IPn+1 , 1  p  n and 1  q  n then v · w > 0. Proof. Suppose p  q. Then, v·w = =

q p  

bs i · bs j

i=1 j=1 p  

bs i · bs i +

i=1

=

q 

 bs i · bs j

j=1 j=i q 

 p   1 n − n + 1 j=1 n + 1 i=1 j=i

p(n + 1 − q) = >0 . n+1 Since we can use the labels v and w arbitrarily, we conclude that v · w > 0, regardless of whether p  q.

 Theorem 1. The set R as defined in (6) is relevant to A∗n . Proof. We prove the theorem statement by working directly from the definition of a relevant set of lattice vectors in Definition 4. Consider some x ∈ IRn+1 which satisfies (5) for every r ∈ R. Consider also some v ∈ A∗n . From Lemma 1, we can write v=

n  j=1

cj w j

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where each cj
0, j = 1, 2, . . . , n and wj =

j 

bs i

i=1

where s ∈ IPn+1. Thus, w j ∈ R. Now, our assumption that (5) is satisfied implies that x·v = 

n 

cj x · w j j=1 n  2 1 cj w j 2 j=1

.

(8)

Furthermore, 2

v =

n  n 

ci cj w i · w j

i=1 j=1

=


n  j=1 n 

2

c2j wj + 2

n  i−1 

ci cj w i · w j

i=2 j=1

c2j wj

2

(9)

j=1

because w i · w j > 0 from Lemma 2. From (8) and (9) and bearing in mind that cj
0, we have x·v 

1 2

v

2

, 

as required.

4

Degrees of Proximity of a Lattice Point

In order to explain the workings of subsequent algorithms, we find it convenient to define degrees of proximity of a lattice point in A∗n with respect to another point in BIRn+1. Firstly, we note the following fact. Theorem 2. Consider some x ∈ IRn+1. If v ∈ A∗n is a closest point to Bx with respect to the Euclidean norm, where B is defined in (1), then it is also a closest point to x. Proof. The proof follows from a simple decomposition of the vector x into orthogonal components. Consider a lattice point w ∈ A∗n . We have 2

2

x − w = (x − Bx) + (Bx − w)

.

An Algorithm to Compute a Nearest Point in the Lattice A∗n

111

Now, w can be expressed as w = Bz, where z ∈ ZZ n+1 . Thus, 2

x − w = (x − Bx) + B(x − z)

2

.

Consider the inner product (x − Bx) · (Bx − w). We have (x − Bx) · (Bx − w) = xT B(x − z) − xT B T B(x − z) = 0 , since B T B = B. Clearly, then, 2

2

x − w = x − Bx + Bx − w 2

2

2


x − Bx + Bx − v = x − v

2

, 

as required.

From this theorem, we see that it is sufficient to consider only the nearest lattice points to points on the plane BIRn+1. We now define three degrees of proximity to a lattice point in A∗n with respect to points in BIRn+1 . Definition 5. Consider a lattice point v ∈ A∗n and a point y ∈ BIRn+1 . Let δ = y − v. The lattice point v is α-close to y if |δi − δj |  1

(10)

for all i, j = 1, 2, . . . , n + 1. The lattice point v is β-close to y if |δi | 

1 2

(11)

for all i = 1, 2, . . . , n + 1 and it is γ-close if   m  m(n + 1 − m)    δsi      2(n + 1)

(12)

i=1

for all m = 1, 2, . . . , n and s ∈ IPn+1. Theorem 3. If v ∈ A∗n is γ-close to a point y ∈ BIRn+1 then v is a nearest lattice point to y. Proof. From (3), we find that, with δ = y − v, m 

δsi = δ · w = −δ · w 

i=1

where s ∈ IPn+1, w=

m  i=1

bs i

and

w = −w =

n+1  i=m+1

bs i .

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Clearly, both w and w  are elements of R as defined in (6). Furthermore, it is easily confirmed that w = w = 2

2

m(n + 1 − m) . n+1

Therefore, satisfaction of the inequality (12) for any particular value of s is equivalent to simultaneous satisfaction of δ·w 

1 2

w

2

and

δ · w 

1 2

w 

2

(13)

for the corresponding vector w. Satisfaction of (12) for all values of s ∈ IPn+1 is then equivalent to satisfaction of (13) for all w ∈ R. However, from Theorem 1, we know that R is relevant to A∗n and so the origin is the closest lattice point to δ. By extension, v is the closest lattice point to y when v is γ-close to y. 

We now describe three algorithms. The first algorithm takes as input a point x ∈ IRn+1 and, in O(n) arithmetic steps, outputs a point z ∈ ZZ n+1 such that Bz is α-close to Bx. The second algorithm and third algorithms both take a point x ∈ IRn+1 and a point z ∈ ZZ n+1 as inputs and, after O(n log n) operations, output a new value of z. For the second algorithm, the inputs z and x are assumed to be such that Bz and Bx are α-close and, on output, they are β-close. For the third algorithm, the input z and x are assumed to be such that Bz and Bx are β-close and, on output, they are γ-close. From Theorem 3, we see that the application of these algorithms in series results in an algorithm which finds a nearest lattice point in A∗n to an input point in O(n log n) steps.

5

The First Algorithm

We will now set out to prove that the algorithm described below, which takes as its input some x ∈ IRn+1 , outputs a vector z ∈ ZZ n+1 such that the lattice point Bz is α-close to Bx. In the following, we will use the notation · to denote a function which returns a nearest integer to its real argument. Algorithm 1. 1 2 3 4 5 6

begin for i := 1 to n + 1 do zi := xi ; od; output(z); end.

Proposition 1. If x ∈ IRn+1 is input to Algorithm 1 then a point z ∈ ZZ n+1 is output such that Bz is α-close to Bx.

An Algorithm to Compute a Nearest Point in the Lattice A∗n

113

Proof. If we write y = Bx and v = Bz then we have yj − vj = (xj − xj ) − Now, − 12  xi − xi 
n+1 − 12 do zsm := zsm + 1; m := m − 1; od; output(z); end.

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Lemma 3. Let v be a lattice point of A∗n and let y be a point in BIRn+1. If v is α-close to y then, with δ = y − v, there exists a permutation s ∈ IPn+1 and an integer 0  m  n + 1 such that  δs2  . . .  δsm  ηm ηm − 1  δs1  δsm+1  δsm+2  . . .  δsn+1  ηm + 1

(14)

where m 1 − . (15) n+1 2 Proof. We will prove the lemma by contradiction. We begin by choosing a value of s ∈ σ(δ), for if we did not, we could not possibly satisfy (14). However, suppose there is no value of m which will satisfy (14), even with s ∈ σ(δ). With the value m = 0 or m = n + 1, this implies that there is some index i such that |δi | > 12 . Note that, because v is α-close to y, there does not exist a pair of indices i and j such that ηm =

δi < − 12

δj >

and

1 2

because this would contradict (10) of Definition 5. So assume, without loss of generality, that there exists some index i such that δi < − 12 . This implies that δs1 < − 12 and that δsn+1 < 12 . Now, we will show that, even though (14) is not satisfied for m = 0 or m = n + 1, there must exist some value of m with 1  m  n such that the slightly weaker condition δs1  δs2  . . .  δsm  ηm  δsm+1  δsm+2  . . .  δsn+1

(16)

is satisfied. If this were not the case then, by applying an inductive argument from our assumption that δs1 < η0 , we would have δsi < ηi−1 for each i = 1, 2, . . . , n. However, this would imply that n+1  i=1

δsi = δsn+1 +

n 

δsi
12 . That is, either δs1 < − 12 or δsn+1 > 12 . These inequalities cannot both be satisfied at once because v is α-close to y. This implies that exactly one of the while loops on lines 7–10 and 12–15 will be entered. Suppose the while loop which is entered is the first one, on lines 7–10. This is equivalent to the supposition that δs1 < − 12 and δsn+1 < 12 . The while loop continues while δsm+1 < ηm , incrementing m at the end of each loop. Lemma 3 guarantees us that the loop will terminate with 1  m  n, since (14) is not satisfied for m = 0 or m = n + 1. Thus, the loop requires O(n) steps to complete. When it terminates, (16) will be satisfied with 1  m  n. As discussed in the proof of Lemma 3, the fact that δs1  ηm  δsn+1 implies that (14) is satisfied for that value of m. Furthermore, by the end of the loop, the new value of z — and let us denote this new value as z — differs from the input value by −1 in a given element whenever its index is in the set {s1 , s2 , . . . , sm }. That is, Bz  = Bz −

m 

bs i .

(19)

i=1

From Theorem 4, Bz  is then β-close to Bx. Finally, suppose instead that it is the second while loop on lines 12–15 that is entered, which implies that δsn+1 > 12 and δs1 > − 12 . The argument follows along similar lines to those employed when the first while loop is entered. We are assured that the loop will terminate with 1  m  n, thus requiring

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O(n) operations to complete. In turn, this implies that (16) will be satisfied with 1  m  n and that, as a result, (14) will also be satisfied. When the loop terminates, the new value of z which has been computed, say z , differs from the input value by 1 in a given element whenever its index is in the set {sm+1 , sm+2 , . . . , sn+1 }. That is, n+1 

Bz = Bz +

bs i .

i=m+1

However, substituting the identity n+1 

bs i = −

i=m+1

n 

bs i ,

i=1

we find we again have (19) and so, from Theorem 4, Bz  is β-close to Bx.

7



The Third Algorithm

Having presented algorithms which compute an α-close lattice point in A∗n to an input point in BIRn+1 and, given an α-close point, produce one which is β-close, we now present an algorithm that can produce a γ-close point from one which is β-close. Algorithm 3. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

begin y := project(x); v := project(z); δ := y − v; s := sortindices(δ); m := 0; t := 0; τ := 0; for i := 1 to n do i−1 n t := t + δsi + n+1 − 2(n+1) ; if t < τ then τ := t; m := i; fi od; for i := 1 to m do zsi := zsi − 1; od; output(z); end.

Lemma 4. A point v ∈ A∗n is γ-close (a nearest lattice point) to y ∈ BIRn+1 if and only if m  i=1

(δsi − pi )
0

(20)

An Algorithm to Compute a Nearest Point in the Lattice A∗n

117

for all m = 1, 2, . . . , n + 1, where δ = y − v, s ∈ σ(δ) and p=

1 (−n, −n + 2, . . . , n − 2, n) . 2(n + 1)

(21)

Proof. First of all, let us prove that (20) is a necessary condition for v to be γ-close to y. Noting that m 

pi = −

i=1

m(n + 1 − m) , 2(n + 1)

we see that if (12) of Definition 5 is satisfied then (20) must be satisfied also. Having proved necessity, let us now prove sufficiency. For any r ∈ IPn+1 and 1  m  n + 1, we use the fact that the sum of any m elements of δ must be greater than or equal to the sum of the m smallest elements to obtain m 

δri


i=1

m 

δsi


m 

i=1

pi = −

i=1

m(n + 1 − m) . 2(n + 1)

Similarly, using the fact that the sum of any m elements of δ must be less than or equal to the sum of the m largest elements, we find that m  i=1

δri 

n+1 

δsi = −

n+1−m 

i=n+2−m

δsi  −

n+1−m 

i=1

i=1

pi =

m(n + 1 − m) . 2(n + 1)

Thus, we find that if (20) is satisfied then, for any r ∈ IPn+1 and 1  m  n,   m   m(n + 1 − m)   δri      2(n + 1) i=1

and, since this is simply (12) of Definition 5, v is γ-close to y.



Lemma 5. Consider a lattice point v ∈ A∗n and a point y ∈ BIRn+1 . Let δ = y − v and s ∈ σ(δ). Suppose v is β-close to y. If v  = v − w where w=

m 

bs i

(22)

i=1

for some 1  m  n then, with δ  = y − v  , there exists some s ∈ σ(δ  ) for which
   s1 , s2 , . . . , sn+1 = (sm+1 , sm+2 , . . . , sn+1 , s1 , s2 , . . . , sm ) . (23) That is, there exists some s ∈ σ(δ  ) which is the elementwise left rotation of s by m positions.

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Proof. Since the form of w in (22) is identical to that in (17), it follows from (18) that δs i  δs j whenever 1  i  j  m and whenever m + 1  i  j  n + 1. Furthermore, since v is β-close to y, we know that δs1
− 12 and δsn+1  12 . Hence, δs 1 = δs1 + 1 −

m 1 m
− n+1 2 n+1

δs n+1 = δsn+1 −

m 1 m  − . n+1 2 n+1

and

This implies that δs m+1  δs m+2  . . .  δs n+1  δs 1  δs 2  . . .  δs m 

which in turn implies (23), as required.

Corollary 1. Consider a lattice point v ∈ A∗n and a point y ∈ BIRn+1. Let δ = y − v, s ∈ σ(δ) and  = (δs1 − p1 , δs2 − p2 , . . . , δsn+1 − pn+1)

(24)

where p is defined in (21). Suppose v is β-close to y. If v  = v − w where w is given in (22) then, with δ  = y − v  , there exists some s ∈ σ(δ  ) such that, when  = (δs
1 − p1 , δs
2 − p2 , . . . , δs
n+1 − pn+1) ,  is the elementwise left rotation of  by m positions. Proof. From Lemma 5, we have, when 1  i  n + 1 − m, m δs
= δs i+m = δsi+m − i n+1 and so !i = δs
i − pi = δsi+m −

2(m + i − 1) − n = δsi+m − pi+m = !i+m . 2(n + 1)

Similarly, when n + 1 − m < i  n + 1, we have δs
= δs i+m−n−1 = δsi+m−n−1 + 1 − i

m n+1

which implies that !i = δs
i − pi = δsi+m−n−1 −

2(i + m − n − 2) − n = δsi+m−n−1 − pi+m−n−1 2(n + 1)

= !i+m−n−1 and hence  is the elementwise left rotation of  by m positions.



An Algorithm to Compute a Nearest Point in the Lattice A∗n

Lemma 6. Consider a point  ∈ BIRn+1. If m = arg

min

j=1,2,...,n+1

j   !i

119

(25)

i=1

and  is the elementwise left rotation of  by m positions then j 

!i
0

(26)

i=1

for all 1  j  n + 1. Proof. If 1  j  n + 1 − m then j  i=1

!i

=

m+j   m    !i = !i − !i
0 .

m+j  i=m+1

i=1

i=1

On the other hand, if n + 1 − m < j  n + 1 then we find that j+m−n−1   n+1  j+m−n−1   m  j       !i = !i + !i = !i − !i
0 i=1

i=1

i=m+1

and so (26) is true for any 1  j  n + 1.

i=1

i=1



Theorem 5. Consider a lattice point v ∈ A∗n and a point y ∈ BIRn+1. Let δ = y − v, s ∈ σ(δ) and let  be as defined in (24). Suppose v is β-close to y. If m is defined according to (25) then v  = v − w is γ-close to y, where w is defined according to (22). Proof. The proof follows directly from application of Lemma 6, Corollary 1 and Lemma 4. 

Proposition 3. If x ∈ IRn+1 and z ∈ ZZ n+1 are input to Algorithm 3 and Bz is β-close to Bx then, after O(n log n) arithmetic operations, a new value of z is output such that Bz is γ-close to Bx. Proof. The proposition is proved merely by observing that Algorithm 3 expresses in a programmatic way the construction of a γ-close point given in Theorem 5. Specifically, the first for loop on lines 7–10 calculates the value of m as defined in (25), albeit that the value of m is restricted to lie in the range 0  m  n rather than 1  m  n + 1 — a variation that has no mathematical implications. The second for loop on lines 11–13 constructs the new value of z, say z  , such that v  = Bz  = v − w, where w is defined as in (22). In terms of the amount of calculation required, both for loops require only O(n) arithmetic steps to complete and so the total execution time is dominated by the execution of the sortindices procedure, which requires O(n log n) operations. 

120

I. Vaughan and L. Clarkson

References 1. I. Vaughan L. Clarkson. Approximation of Linear Forms by Lattice Points with Applications to Signal Processing. PhD thesis, The Australian National University, 1997. 2. I. Vaughan L. Clarkson, Stephen D. Howard, and Iven M. Y. Mareels. Parameter estimation and association from pulse time-of-arrival data. Submitted to Signal Process., 1997. 3. J. H. Conway and N. J. A. Sloane. Fast quantizing and decoding algorithms for lattice quantizers and codes. IEEE Trans. Inform. Theory, IT–28(2):227–232, March 1982. 4. J. H. Conway and N. J. A. Sloane. Soft decoding techniques for codes and lattices, including the Golay code and the Leech lattice. IEEE Trans. Inform. Theory, IT– 32(1):41–50, January 1986. 5. J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups. SpringerVerlag, Berlin, 1988. 6. Donald E. Knuth. The Art of Computer Programming, volume 3 (Sorting and Searching). Addison-Wesley, Reading, Ma., 1973. 7. P. van Emde Boas. Another N P -complete partition problem and the complexity of computing short vectors in a lattice. Technical Report 81–04, Mathematisch Instituut, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands, April 1981.

Sequences from Cocycles Kathy J. Horadam
RMIT University, Melbourne, VIC 3001, Australia [email protected]

Abstract. Binary and quaternary sequences with perfect periodic autocorrelation, and perfect nonlinear pm -ary sequences, are both shown to equate to orthogonal coboundaries — the simplest class of orthogonal cocycles. We consider doubly-indexed sequences defined by cocycles. We give a new construction — a generalised multiplication — of orthogonal cocycles and show it gives perfect nonlinear sequences for parameters where 1-dimensional PN sequences cannot exist.

1

Introduction

Sequences with desirable correlation or distribution properties are much sought for use in signal transmission, optical imaging and encryption. Much effort has been expended on constructing and classifying sequences which are optimal or nearly optimal with respect to some measure of merit. The sequence is frequently regarded as a mapping from an index set G (such as the modular group Zv or the finite field Fpm ) to a sequence set C (typically taking binary, complex or pm -ary values). The measure of merit can be off-peak correlation, or uniform distribution amongst sequence values, or nonlinearity. Here we observe that sequences with perfect periodic autocorrelation and perfect nonlinear sequences both equate to orthogonal coboundaries — the simplest class of a set of functions called orthogonal cocycles which are 2-dimensional on G. This allows us to generalise from 1-dimensional to 2-dimensional mappings and hence consider sequences defined by cocycles, which have improved performance over the optimal 1-dimensional functions against these figures of merit. Cocycles are mappings ψ : G × G → C, where G and C are finite groups with C abelian, which satisfy a particular quasi-associative equation (1). They arise naturally in the topology of surfaces, in quantum dynamics, in projective representation theory, and in combinatorial design theory, as well as in the cohomology theory of groups. Increasingly, links have been found between the optimal 1-dimensional sequences and difference sets [2,3,15]. In the 2-dimensional case, there is a precise link: we know each orthogonal cocycle is equivalent to a semiregular central relative difference set, and conversely [17]. Semiregular relative difference sets are of interest precisely because of their good Hamming correlation properties for FHMA. For example, perfect binary 

This work was supported by Australian Research Council Grant A49701206.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 121–130, 1999. c Springer-Verlag Berlin Heidelberg 1999


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arrays are equivalent to splitting abelian relative (4u2 , 2, 4u2 , 2u2)-difference sets, and Jedwab’s generalised perfect binary arrays are equivalent to abelian relative (4t, 2, 4t, 2t)-difference sets [13, Theorem 3.2]. Kumar’s ideal matrices for FHMA communications systems are the two-dimensional characteristic functions of relative (v, v, v, 1)-difference sets in Z2v [16], and hence are rare. In this paper we firstly give a new construction (Theorem 2) of orthogonal cocycles. Secondly, we show that orthogonal coboundaries are the same as 1-dimensional sequences with perfect periodic autocorrelation and with perfect nonlinearity. We generalise these measures to 2-dimensional sequences of cocycle values and prove that the distribution of values of a cocyclic sequence is invariant (Theorem 3) under the relevant equivalence operations on the cocycle. We prove that there are many 2-dimensional sequences with perfect nonlinearity (Example 5) for parameters where 1-dimensional PN functions cannot exist, and give constructions for Almost-PN and ∆-nonlinear cocyclic sequences (Corollary 3). Earlier, the author and Perera [10] introduced a very general description of cocyclic codes in order to demonstrate the previously unrecognised (and wellhidden) presence of cocycles in several code construction techniques. Category I of these codes comprised those constructed from a cocyclic generalised Hadamard matrix. Many standard constructions of (generalised) Hadamard matrices are in fact cocyclic, and in [9] many well-known codes are shown to be Category I. Since (generalised) Hadamard matrices determine nonlinear codes which meet the Plotkin bound, and orthogonal cocycles are equivalent to cocyclic generalised Hadamard matrices [17], the new orthogonal cocycles identified here also determine optimal nonlinear codes.

2

Cocycles and Generalised Hadamard Matrices

Throughout, G will be a finite group of order v and C will be a finite abelian group of order w. A (2-dimensional) cocycle is a mapping ψ : G × G → C satisfying the cocycle equation ψ(g, h) ψ(gh, k) = ψ(g, hk) ψ(h, k), ∀g, h, k ∈ G.

(1)

This implies ψ(g, 1) = ψ(1, h) = ψ(1, 1), ∀g, h ∈ G, so we follow standard usage and consider only normalised cocycles, for which ψ(1, 1) = 1. Each cocycle ψ determines a central extension 1 → C → Eψ → G → 1 of C by G, in which the group Eψ of order vw consists of the set of ordered pairs {(a, g) : a ∈ C, g ∈ G} with multiplication (a, g)(b, h) = (abψ(g, h), gh),

(2)

and the image C × {1} of C lies in the centre of Eψ . A cocycle is naturally displayed as a G-cocyclic matrix; that is, a square matrix Mψ whose rows and columns are indexed by the elements of G under some fixed ordering, and whose entry in position (g, h) is ψ(g, h). If ψ is symmetric (ψ(g, h) = ψ(h, g) always), Mψ is a symmetric matrix. We write Mψ = [ψ(g, h)]g,h∈G .

(3)

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123

Definition 1. A cocycle is a coboundary ∂φ if it is derived from a set mapping φ : G → C having φ(1) = 1 by the formula ∂φ(g, h) = φ(g)−1 φ(h)−1 φ(gh). Two cocycles ψ and ψ  are cohomologous if there exists a coboundary ∂φ such that ψ  = ψ · ∂φ. Example 1. If G = g : g v = 1 ∼ = Zv and a is an element of order n in C, then ψ(g i , g j ) = aij , for all 0 ≤ i, j ≤ v − 1, is a symmetric cocycle and Mψ is a Vandermonde matrix. If n = v, Mψ is the matrix of the Discrete Fourier Transform (or the Mattson-Solomon polynomial). Example 2. If G = Zn2 and C = {±1} ∼ = Z2 then ψ(u, v) = (−1)u·v , for all u, v ∈ G, is a symmetric cocycle and Mψ is the Sylvester Hadamard matrix of order 2n . Example 3. If V is a finite-dimensional vector space over a field F and ψ is a bilinear form on V then ψ : (V, +) × (V, +) → F is a cocycle. We will term an additively-written abelian group “distributive” if it also carries a distributive multiplication. That is, a distributive group is an abelian group G = (G, +) (with identity 0) having a second binary operation · such that ∀g, h, k ∈ G, g · (h + k) = g · h + g · k and (h + k) · g = h · g + k · g. (We write g · h = gh when the context is clear.) Such distributive groups include finite rings and presemifields (where (G, ·) is a quasigroup; that is, for any nonzero g, h ∈ G there are unique solutions in G to gx = h and to yg = h.) In particular, Galois fields, Galois rings and semifields (such as Dickson commutative semifields) are distributive groups. If a distributive group is a finite commutative semifield but not a field, the only field property which does not hold is associativity of multiplication, and (see [1, VI.8.4], [14, p.269]), G is an elementary abelian group of order pa ≥ 16, with a ≥ 3. Example 4. Let G be a distributive group. The mapping µ : G × G → G defined by µ(g, h) = gh, ∀g, h ∈ G, is a normalised cocycle, the multiplication cocycle. Since G is abelian, µ is symmetric ⇔ Eµ is abelian ⇔ (G, ·) is commutative. The last four examples are instances of a general class of cocycles: any mapping f : G × G → C which is homomorphic in each coordinate (that is, f(gh, k) = f(g, k)f(h, k) and f(g, hk) = f(g, h)f(g, k) always) is a normalised cocycle. In particular we can generalise the multiplication cocycle. Example 5. Let C be a distributive group. Let λ, ρ : G → C be mappings, and define µλ,ρ : G × G → C by µλ,ρ(g, h) = λ(g)ρ(h), ∀g, h ∈ G. If λ and ρ are homomorphisms then µλ,ρ is a normalised cocycle. This link is emphasised by the following partial converse.

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Corollary 1. Let C be a distributive group with a multiplicative unity 1, and let λ, ρ : G → C be mappings. If µλ,ρ is a normalised cocycle, 1 ∈ Imλ (resp. Imρ) and λ (resp. ρ) is a homomorphism then ρ (resp. λ) is a homomorphism. Proof. Since µλ,ρ is a cocycle, λ(g)ρ(h) + λ(gh)ρ(k) = λ(g)ρ(hk) + λ(h)ρ(k). If λ is a homomorphism, the result follows on setting λ(g) = 1.   To link cocycles with generalised Hadamard matrices and perfect sequences we will be interested in the frequency with which a cocycle takes each value in the target abelian group C. Definition 2. Let ψ : G × G → C be a cocycle and for each g ∈ G, a ∈ C define N (g, a) = |{h ∈ G : ψ(g, h) = a}|. The set of frequencies {N (g, a) : g ∈ G, a ∈ C} is the distribution of ψ. A generalised Hadamard matrix GH(w, v/w) over W is a v × v matrix H with entries from the group W of order w|v such that the list of quotients hij h−1 kj , , the 1 ≤ j ≤ v, contains each element of W exactly v/w times. With h∗ij = h−1 ji defining matrix equation over ZW is
HH ∗ = vIv + (v/w)( u)(Jv − Iv ). (4) u∈W

Any GH(w, v/w) is equivalent to a normalised GH(w, v/w) with its first row and column consisting entirely of the unit element of W . A Hadamard matrix is a GH(2, v/2). All known examples of GH(w, v/w) require w to be a prime power. When the GH(w, v/w) H is also cocyclic; that is, W = C is abelian, w|v and H = Mψ for some cocycle ψ, this is equivalent to imposing a combinatorial condition we call orthogonality, on the cocycle. Definition 3. The cocycle ψ : G × G → C is orthogonal when w|v if the noninitial rows of Mψ are uniformly distributed over the elements of C; that is if, for each g = 1 ∈ G, N (g, a) = v/w, ∀a ∈ C (5)   or equivalently, if in ZC, for each g = 1 ∈ G, h∈G ψ(g, h) = v/w ( a∈C a). For instance, in Example 1 if n|v then ψ is orthogonal (with entries in C  = a ) when n = p and v = pm , for p a prime. The cocycles of Example 2 are orthogonal. The multiplication cocycle for any finite presemifield is orthogonal. The designs corresponding to normalised GH(w, v/w) are divisible designs. When the GH matrix is also cocyclic, these designs have been completely characterised. Furthermore, they correspond to relative difference sets. A relative (v, w, k, λ)-difference set [4] in a finite group E of order vw relative to a normal subgroup N of order w, is a k-element subset R of E such that the multiset of quotients d1 d−1 2 of distinct elements d1 , d2 of R contains each element of E\N exactly λ times, and contains no elements of N . (The ordinary (v, k, λ)difference sets correspond to the case N = {1}.) There is always a short exact

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125

sequence 1 → N → E → E/N → 1 . We will be concerned only with relative difference sets having k = v and therefore also k = wλ. A relative difference set with the latter property is termed semiregular. Theorem 1. (Equivalence Theorem) [17, Lemma 2.7, Theorem 4.3] Let w|v and let Mψ be a G-cocyclic matrix with entries in C. Then Mψ is a GH(w, v/w) if and only if the design Dψ developed from {(1, g), g ∈ G} ⊂ Eψ is a divisible (v, w, v, v/w)design, class regular with respect to C, if and only if Rψ = {(1, g), g ∈ G} ⊂ Eψ is a relative (v, w, v, v/w)-difference set relative to the central subgroup C × {1}, if and only if the cocycle ψ is orthogonal.

3

New Classes of Orthogonal Cocycles

In this section we give a new construction of orthogonal cocycles, and show how to obtain new orthogonal cocycles from old. In [8] generalisations of the multiplication cocycle for a finite field were given, together with conditions under which they were orthogonal. Example 6. [8, Lemma 3.11] If F is a finite field and G = (F, +), then the power i cocycles µi : G×G → G defined by µi (g, h) = g p h, 0 ≤ i ≤ a−1, are orthogonal. (Here µ0 = µ.) Udaya [19] has pointed out that the Frobenius automorphism in this example can be replaced by an arbitrary linearised permutation polynomial. This prompts the following generalisation, giving a very large class of orthogonal cocycles which have not previously been recognised. Lemma 1. Let G be a distributive group which is a presemifield and let λ, ρ : G → G be homomorphisms. If λ (resp. ρ) is an automorphism, then µλ,ρ : G×G → G is an orthogonal cocycle if and only if ρ (resp. λ) is an automorphism. Proof. Here v = w, so v/w = 1. From Example 5, µλ,ρ is a cocycle. By symmetry we may assume λ is an automorphism, and we only need to show that µλ,ρ is orthogonal if and only if ρ is one-to-one. This is clear because, given any λ(g) = 0 and k ∈ G, there is a unique solution h ∈ G to the equation λ(g)ρ(h) = k if and only if |{h ∈ G : µλ,ρ(g, h) = k}| = 1.   The composition of an (orthogonal) cocycle G×G → C with an epimorphism C → C  of abelian groups is an (orthogonal) cocycle. For example the field multiplication cocycle projects via the relative trace function. Example 7. Let F = Fqm and K = Fq be finite fields and let T = T rF/K be the relative trace function. Then T ◦ µ : (F, +)2 → (K, +) is an orthogonal cocycle. A more general version of this example is this consequence of Lemma 1.

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Kathy J. Horadam

Theorem 2. If G is a distributive group which is a presemifield, λ, ρ : G → G are automorphisms and γ : G → C is an epimorphism then γ ◦ µλ,ρ : G × G → C is an orthogonal cocycle. The composition of an automorphism of G acting diagonally on G × G with an (orthogonal) cocycle G × G → C is an (orthogonal) cocycle. For example, if ρ is an automorphism then µλ,ρ ◦ (ρ−1 × ρ−1) = µλ◦ρ−1 ,1 , so that µλ,ρ and µλ◦ρ−1 ,1 lie in the same orbit under diagonal Aut(G)-action. When further composed with an automorphism of C, we obtain an (Aut(C) × Aut(G))-action on the abelian group of cocycles which partitions it into (Aut(C) × Aut(G))-orbits which either consist entirely of orthogonal cocycles or contain no orthogonal cocycles. Denote by O(ψ) the orbit of ψ under the action ψ (γ,θ) (g, h) = γ(ψ(θ(g), θ(h))), γ ∈ Aut(C), θ ∈ Aut(G).

(6)

In [7] it was shown that these orbits may be collected into bundles under a further G-action termed shift equivalence. Definition 4. Cocycles ψ, ϕ : G × G → C are shift-equivalent, written ψ ∼s ϕ via k, if there exists k ∈ G such that ψ = ϕ · ∂ϕk , where ϕk (g) = ϕ(k, g), g ∈ G, or, equivalently, if there exists k ∈ Gsuch that ψ(g, h) = ϕ(kg, h) ϕ(k, h)−1 . The bundle B(ψ) of ψ is B(ψ) = ϕ∼s ψ O(ϕ). Shift equivalence preserves orthogonality, so these bundles consist wholly of orthogonal cocycles or of non-orthogonal cocycles. In fact, each orthogonal bundle corresponds uniquely to an equivalence class of relative difference sets, and vice versa (the Bundle Isomorphism Theorem [7]). Finally we show that the distribution of a cocycle is an invariant of the bundle containing that cocycle. It is easy to show it is an invariant of the (Aut(C) ×  Aut(G)) orbit of the cocycle. Write h∈G ψ(g, h) = a∈C N (g, a)a. If k ∈ G then (by (1))

ψ(kg, h)ψ(k, h)−1 = ψ(kgk −1 .k, h)ψ(k, h)−1 = (7)
h∈G

h∈G

ψ(kgk

−1

h∈G −1

, k)

ψ(kgk

−1

, kh) =




N (kgk −1 , a)(ad), d = ψ(kgk −1, k)−1 .

a∈C

Hence the distribution is invariant under shift equivalence. Theorem 3. The distribution D(B(ψ)) of a cocycle ψ is an invariant of its bundle B(ψ).

4

Sequences from Cocycles

For binary sequences φ : G → {±1} indexed by an abelian group G, the (periodic) autocorrelation function A : G → Z is defined [15] to be A(g) =  φ(g + h)φ(h), and the sequence is a perfect binary array if A(g) = 0 h∈G

Sequences from Cocycles

127

for every g = 0. It is well-known that a perfect binary array corresponds to a Menon-Hadamard difference set in G and to a splitting relative (4u2 , 2, 4u2 , 2u2 )difference set in Z2 × G, and vice versa. Perfect binary arrays are investigated as one response to the lack of examples of perfect binary sequences (for which G = Zv in the definition above). A recent survey of progress in the search for generalisations of perfect binary sequences, and their links to difference sets and divisible difference sets, appears in [15]. More generally, the autocorrelation function for a sequence φ : G → C where G and C are multiplicatively written  finite groups with C abelian and φ(1) = 1, is A : G → ZC where A(g) = h∈G φ(gh)φ(h)−1 , and φ is a perfect array if A(g) = 0 for every g = 1. We see that for g = 1,

φ(gh)φ(h)−1 = 0 ⇔ φ(gh)φ(h)−1 φ(g)−1 = 0 (8) h∈G

h∈G

⇔




∂φ(g, h) = 0,

(9)

h∈G

so that a perfect binary array is also equivalent to an orthogonal coboundary with C = {±1}. A perfect quaternary array has C = {±1, ±i} and in this case the orthogonal coboundaries determine the subclass of balanced perfect quaternary arrays: those for which each value of C is taken equally often. Clearly the coboundary in (9) may be replaced by a cocycle. Definition 5. A cocyclic perfect array is a cocycle ψ : G × G → C, where G is a finite group, C is a finite abelian group, such that in ZC,
∀ g = 1, ψ(g, h) = 0. (10) h∈G

It is balanced if each element of C appears equally often in the summation.  Whenever a∈C a = 0 in ZC, such as in the cases of interest mentioned above, balanced cocyclic perfect arrays are identical to orthogonal cocycles, and vice versa. Hughes [12] has investigated the relationships between autocorrelation functions and orthogonal cocycles in detail. He has, in a sense, reversed the process above, deriving from each orthogonal cocycle ψ a 1-dimensional sequence Eψ → C with uniformly distributed autocorrelation function for values h of Eψ away from a specified forbidden subgroup. This sequence is the characteristic function of a central relative (v, w, v, v/w)-difference set. The autocorrelation is one measure of the behaviour of a sequence φ relative to its translates. Another approach is to look at the spread, as h ranges over G, of the values φ(g + h)φ(h)−1 , and to take the flatness of the distribution as the measure of merit. Sequences for which this distribution is as flat as possible are good potential S-box functions because of their resistance to differential cryptanalysis. What is sought is sequences φ : Fpm → Fpm such that ∆ = max |{x ∈ Fpm : φ(a + x) − φ(x) = b}| a =0,b

(11)

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Kathy J. Horadam

is as small as possible. A function φ with ∆ = 1 is perfect nonlinear (PN) and with ∆ = 2, almost perfect nonlinear (APN). It is known that for p = 2, ∆ = 2 is the best possible, while for p ≥ 3, ∆ = 1 is achievable. For p = 2, extensive current research focuses on the identification and classification of APN functions and their relationship to maximally nonlinear sequences (eg. [3]). It is also clear that further links to cyclic difference sets are being uncovered [2,3]. In this section, we propose a cocyclic construction of nonlinear functions with desirable distribution properties. Note that in (11), ∆ = 1 if and only if ∀a = 0, b ∈ Fpm , |{x ∈ Fpm : φ(a + x) − φ(x) = b}| = 1 if and only if ∀a = 0, c ∈ Fpm , |{x ∈ Fpm : φ(a + x) − φ(x) − φ(a) = c}| = 1 if and only if ∂φ : G × G → G, G = (Fpm , +) is an orthogonal coboundary. Corollary 2. φ : Fpm → Fpm is a PN function if and only if ∂φ is an orthogonal coboundary if and only if {(φ(g), g) : g ∈ Fpm } is a splitting relative (pm , pm , pm , 1)-difference set in Fpm × Fpm relative to Fpm × {0}. In [8] the author shows that the constructions of abelian splitting relative (pm , pm , pm , 1)-difference sets known to her [14,18] for odd p are isomorphic to those defined by the multiplication cocycle in some commutative semifield. In these cases, µ is (necessarily) a coboundary [8, Theorem 3.8]. By Ganley’s result [5], no abelian splitting relative (2m , 2m , 2m , 1)-difference set exists, so ∆ = 2 is indeed the best possible for p = 2, in contrast to the situation for odd primes. However in the same article, the author shows that the field multiplication cocycle defines the known [6,18] abelian (non-splitting) relative (2m , 2m , 2m , 1)difference sets, and thus, if we relax the requirement that a nonlinear function be 1-dimensional, a wealth of constructions of sequences with excellent distribution properties exists. Definition 6. Let ψ : G×G → C be a cocycle and define ∆ = maxg =1,c N (g, c). We say ψ is a ∆-nonlinear cocycle (briefly: ψ is perfect nonlinear (PN) if ∆ = 1 and almost perfect nonlinear (APN) if ∆ = 2). Combining this definition with the results of the previous section we obtain: Corollary 3. If ψ : G × G → C is an orthogonal cocycle it is v/w-nonlinear, and if γ : C → C  is any monomorphism, γ ◦ ψ is v/w-nonlinear. Theorem 4. If ψ is ∆-nonlinear, so is every cocycle in its bundle B(ψ). We illustrate the above ideas in full detail for the small case G = C = Z22 . Example 8. If G = C = Z22 , each cocycle ψ is identified by a quadruple x, y, z, t , where x, y, z, t ∈ C. With indexing {0 = 00, 10, 01, 11}, the cocyclic matrix Mψ is   0 0 0 0 0 x z x+z   . 0 z+t y y + z + t 0 x+z +t y+z x+y +t

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129

There are 16 bundles of cocycles [7] and on computing their distributions from a representative cocycle ψ we obtain the following table, in which bundles are grouped according to the cohomology class of ψ. The first group (of 2 bundles) consists of the coboundaries. The first three groups consist of symmetric cocycles (and Eψ is abelian) and the second three groups consist of non-symmetric cocycles (and Eψ is nonabelian). The distributions are read off in order from the rows of the corresponding Mψ and abbreviated, so that for example, the third row of M a,0,c,0 is 0, c, 0, c, which is represented as 22 and the fourth is 0, a + c, c, a, which is represented as 14 . (Here distinct variables refer to distinct nonzero elements of C.) ψ 1.1 0, 0, 0, 0 1.2 0, 0, c, 0 2.1 a, 0, 0, 0 2.2 a, a, 0, 0 2.3 a, 0, c, 0 3.1 a, b, 0, 0 3.2 a, b, c, 0 4.1 0, 0, 0, d 4.2 0, 0, c, d 5.1 0, b, 0, d 5.2 0, b, c, c 5.3 0, b, b, d 6.1 a, a, 0, a 6.2 a, a, 0, d 6.3 a, a, a, d 6.4 a, a, c, a

D(B(ψ))

Eψ

{4, 4, 4, 4} Z42 2 2 2 {4, 2 , 2 , 2 } Z42 {4, 22 , 4, 22 } Z22 × Z4 {4, 22 , 22 , 22 } Z22 × Z4 {4, 14 , 22 , 14 } Z22 × Z4 {4, 22 , 22 , 14 } Z24 {4, 14 , 14 , 14 } Z24 {4, 4, 22 , 22 } Z2 × D4 {4, 22 , 22 , 14 } Z2 × D4 {4, 4, 14 , 14 } E5 [7] {4, 22 , 22 , 22 } E5 {4, 22 , 14 , 22 } E5 {4, 22 , 22 , 22 } Z4  Z4 {4, 22 , 14 , 14 } Z4  Z4 {4, 22 , 14 , 22 } Z4  Z4 {4, 14 , 14 , 14 } Z4  Z4

∆ Comments 4 2 4 2 2 2 1 4 2 4 2 2 2 2 2 1

Trivial cocycle APN ≡ best 1 − dim. function APN APN, µ for F2 + uF2 APN, µ for F2 × F2 PN, µ for F4 APN APN APN APN APN APN PN, power cocycle µ1 for F4

The 2-dimensional APN functions in this table are all new sequences with distribution as good as or flatter than the best 1-dimensional APN functions (which must equate to coboundaries, ie those in bundle 1.2 above). The 2-dimensional balanced APN functions involving a single variable (bundles 1.2, 2.2 and 6.1 above) can also be derived as the monomorphic images of orthogonal (hence APN) cocycles G × G → Z2 using Corollary 3. The 2-dimensional PN functions in this table (bundles 3.2 and 6.4 above) are new sequences whose distributions achieve the theoretical limit not attainable by any 1-dimensional functions. Theorem 5. Let G be the additive group of a presemifield and let λ, ρ : G → G be any automorphisms. Then µλ,ρ has distribution {N (0, 0) = v, N(0, c) = 0, c = 0, N (g, c) = 1, g = 0}, so that ∆ = 1 and µλ,ρ is a PN cocycle. There are many presemifield structures with underlying additive group G ∼ = (for example any finite field or Dickson commutative semifield). In this case Zm p

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there are (pm − 1)(pm − p) . . . (pm − pm−1 ) automorphisms of G, each of which can be expressed as a unique linearised permutation polynomial of the finite field structure on G. We hereby obtain new sequences whose distributions achieve the theoretical limit.

References 1. Colbourn, C. J. and Dinitz, J. H. (Eds), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, 1996. 2. Dillon, J. F., Multiplicative difference sets via additive characters, preprint August 1998. 3. Dobbertin, H., Kasami power functions, permutation polynomials and cyclic difference sets, NATO-ASI Workshop “Difference sets, sequences and their correlation properties”, August 1998. 4. Elliott, J. E. H. and Butson, A. T., Relative difference sets, Illinois J. Math. 10 (1966), 517–531. 5. Ganley, M. J., On a paper of Dembowski and Ostrom, Arch. Math. 26 (1976), 93–98. 6. Hammons, A. R. et al, The Z4 -linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. IT 40 (1994), 301–319. 7. Horadam, K. J., Equivalence classes of central semiregular relative difference sets, preprint April 1999. 8. Horadam, K. J., Multiplication cocycles and central relative (pa , pa , pa , 1)difference sets, Research Report 7, Mathematics Department, RMIT, May 1998. 9. Horadam, K. J., Cocyclic Hadamard codes, preprint 1998, abstract in Proc. ISIT 1998, IEEE, p. 246. 10. Horadam, K. J. and Perera, A. A. I., Codes from cocycles, in AAECC-12, T. Mora and H. Mattson, eds, LNCS 1255, Springer, Berlin 1997. 11. Hughes, D. R. and Piper, F. C., Projective planes, GTM 6, Springer, New York, 1973. 12. Hughes, G., Characteristic functions of relative difference sets, correlated sequences and Hadamard matrices, in AAECC-13, LNCS, Springer, 1999, to appear. 13. Jedwab, J., Generalised perfect arrays and Menon difference sets, Des., Codes Cryptogr. 2 (1992) 19–68. 14. Jungnickel, D., On automorphism groups of divisible designs, Canad. J. Math. 24 (1982), 257–297. 15. Jungnickel, D. and Pott, A., Perfect and almost perfect sequences, Des., Codes Cryptogr., to appear. 16. Kumar, P. V., On the existence of square dot-matrix patterns having a specific three-valued periodic-correlation function, IEEE Trans. IT 34 (1988) 271–277. 17. Perera, A. A. I. and Horadam, K. J., Cocyclic generalised Hadamard matrices and central relative difference sets, Des., Codes Cryptogr. 15 (1998) 187–200. 18. Pott, A., Finite Geometry and Character Theory, LNM 1601, Springer, Berlin, 1995. 19. Udaya, P., Cocyclic generalised Hadamard matrices over GF (pn ) and their related codes, Proc. AAECC-13, Honolulu, Hawaii, November 1999.

On the Second Greedy Weight for Binary Linear Codes Wende Chen1 and Torleiv Kløve2 1

2

System and Control Lab., Institute of Systems Science Academia Sinica, Beijing 100080, China Department of Informatics, University of Bergen, N-5020 Bergen, Norway

Abstract. The difference g2 − d2 for a binary linear [n, k, d] code C is studied. Here d2 is the smallest size of the support of a 2-dimensional subcode of C and g2 is the smallest size of the support of a 2-dimensional subcode of C which contains a codeword of weight d. For codes of dimension 4, the maximal value of g2 −d2 is determined. For general dimensions, upper and lower bounds on the maximal value are given.

1

Introduction

Ozarow and Wyner [6] suggested one application of linear codes to cryptology, namely to the wire-tap channel of type II. For this channel, an adversary is assumed to be able to tap s bits (of his choice) of n bits transmitted. The goal for the sender is to encode k bits of information into n transmitted bits in such a way that the adversary gets as little information as possible. One of their schemes was to use the dual of an [n, k] binary linear code C. The code has 2k cosets, each representing a binary k-tuple. If the sender wants to transmit k bits of information to the receiver, he selects a random vector in the corresponding coset. The channel is assumed to be noiseless, so the receiver can determine the corresponding coset of the received vector. It is assumed the adversary has full knowledge of the code, but not of the random selection of a vector in a coset. In his studies of this scheme, Wei [7] introduced a set of parameters of a binary code which he called the generalized Hamming weights. The same parameters had also been studied previously in another context [4] and has since proved important also in other contexts. For any code D, let χ(D), the support of D, be the number of positions where not all the codewords of D are zero. Further, the support weight of D is |χ(D)|. For an [n, k] code C and any r, where 1 ≤ r ≤ k, Wei defined dr (C) = min{|χ(D)| | D is an (n, r) subcode of C}. In particular, the minimum distance of C is d1 (C). The weight hierarchy of C is the set {dr (C) | 1 ≤ r ≤ k}. For the Ozarow-Wyner scheme, it was shown by Wei [7] that the adversary can obtain r bits of information if and only if s ≥ dr (C). Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 131–141, 1999. c Springer-Verlag Berlin Heidelberg 1999


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Cohen et al. [2], [3] considered the following variation of the problem. The adversary is greedy. He first reads d = d1 positions to obtain one symbol of information as soon as possible. He then reads a minimal number of further positions to get one additional symbol of information and so on. Let gr denote the minimal number of symbols he has to read to get r symbols of information in this way. Note that g1 = d1 and gk = dk . We call the sequence (g1 , g2 , g3 , . . . , gk ) the greedy weight hierarchy. In particular, g2 is the smallest support of a 2dimensional subcode of C which contains a codeword of weight d. The cost to the adversary (in extra positions read) to get two symbols of information using this algorithm is g2 − d2 . We consider here how large g2 − d2 can be for given n, k, and d. In general, we consider three main cases: Case I: There exist two codewords c1 , c2 ∈ C, such that w(c1 ) = d and |χ(c1 ) ∪ χ(c2 )| = d2 . Here and in the following w() denotes the Hamming weight. I this case g2 = d2 by definition and there is nothing more to be said. Case II: We do not have case I, but there exist three codewords c1 , c2 , c3 ∈ C, such that w(c1 ) = d, χ(c2 ) ∪ χ(c3 )| = d2 , and |χ(c1 ) ∪ χ(c3 )| = g2 . Case III: We do not have case I or II. Then we consider in detail the 4-dimensional subspace D generated by four codewords c1 , c2 , c3 , c4 ∈ C such that two of them generate a subspace of support d2 and the other two a subspace of support g2 which contains a codeword of weight d.

2

Case II Codes

We first consider briefly [n, 3, d] codes. Let m3 = m3 (n, d) be defined by
 m3 (n, d) = max g2 (D) − d2 (D) | D is an [n, 3, d] code without zero-positions . The exact value of m3 (n, d) was determined in [1]:   

  − n−d if n+3 ≤ d ≤ 2n−3 ,  d 3 4 5    

 4n−15 2n−3d−3 n−d 2n−3 m3 = − 3 if 1/3, we only get an upper bound on the maximum).

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Acknowledgments This work was supported by The National Science Foundation of China and The Norwegian Research Council.

References 1. W. Chen and T. Kløve, ”On the second greedy weight for linear codes of dimension 3”, to appear in Discrete Mathematics. 2. G. D. Cohen, S. B. Encheva and G. Z´emor, ”Antichain codes”, Proc. IEEE Intern. Symp. Inform. Theory, Cambridge, Mass, Aug. 16-20, 1998, p. 232. 3. G. D. Cohen, S. B. Encheva and G. Z´emor, ”Antichain codes”, Designs, Codes and Cryptography, to appear. 4. T. Helleseth, T. Kløve, J. Mykkeltveit, ”The weight distribution of irreducible cyclic codes with block lengths n1 ((q l − 1)/N )”, Discrete Math., vol. 18, pp. 179211, 1977. 5. T. Helleseth, T. Kløve, Ø. Ytrehus, “Generalized Hamming weights of linear codes”, IEEE Trans. Inform. Theory, vol. 38, pp. 1133-1140, 1992. 6. L. H. Ozarow and A. D. Wyner, ”Wire-tap channel II”, AT& T Bell Labs Technical Journal, vol. 63, pp. 2135-2157, 1984. 7. V. K. Wei, ”Generalized Hamming Weights for Linear Codes”, IEEE Trans. Inform. Theory, vol. 37, pp. 1412-1418, 1991.

On the Size of Identifying Codes Uri Blass1 , Iiro Honkala2 , and Simon Litsyn1 1

2

Department of Electrical Engineering – Systems Tel-Aviv University, Ramat-Aviv 69978, Israel blass [email protected], [email protected] Department of Mathematics, University of Turku, 20014 Turku, Finland [email protected]

Abstract. A code is called t-identifying if the sets Bt (x) ∩ C are all nonempty and different. Constructions of 1-identifying codes and lower bounds on the minimum cardinality of a 1-identifying code of length n are given.

1

Introduction and Basics

Denote by IF2 the binary alphabet {0, 1}. The (Hamming) distance d(x, y) between any two vectors x, y ∈ IFn2 is the number of coordinates in which they differ from each other. The integer d(x, 0), where 0 denotes the all-zero vector (0, 0, . . . , 0), is called the weight of x. If x, y ∈ IFn2 and d(x, y) ≤ t, then we say that x t-covers y (and vice versa). Any nonempty subset C of of IFn2 is called a binary code of length n. The covering radius of a code is the smallest integer R with the property that every vector in IFn2 is within distance R from at least one codeword. As usual, we denote for x ∈ IFn2 , Bt (x) = {y ∈ IFn2 | d(y, x) ≤ t} and

St (x) = {y ∈ IFn2 | d(y, x) = t}.

The purpose of this paper is to study codes that can be used for identification in the following situation: Assume that 2n processors are arranged in the nodes of an n-dimensional hypercube. A processor can check itself and all the neighbours in the hypercube within Hamming distance t, and reports YES if everything is all right and NO if there is something wrong in the processor itself or one of these neighbours. We wish to find a code C consisting of some of the nodes in the hypercube such that if the checking is done in the corresponding processors, then based on the answers we receive, we can tell where the problem lies — assuming there is a problem in at most one of the processors. Such a code is called t-identifying. This problem has been studied in Karpovsky, Chakrabarty and Levitin [4]. More formally, a t-identifying code is defined as follows. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 142–147, 1999. c Springer-Verlag Berlin Heidelberg 1999


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Definition 1. A binary code C of length n is called t-identifying (where t < n) if the sets Bt (x) ∩ C, x ∈ IFn2 are nonempty and different. The fact that the sets Bt (x) are all nonempty implies that C has covering radius at most t. Moreover, for all x, y ∈ IFn2 , the condition Bt (x)∩C = Bt (y)∩C implies that x = y. The results of this paper are from [1], where more detailed proofs of our results can be found. The minimum cardinality of a t-identifying code of length n is denoted by Mt (n). Theorem 1. Assume that C is 1-identifying. The direct sum {0, 1} ⊕ C is 1identifying if and only if d(c, C \ {c}) ≤ 1 for all c ∈ C. Proof. Denote the code {0, 1} ⊕ C by D. If there is a codeword c ∈ C such that d(c, C \ {c}) > 1, then clearly for both z = (0, c) and z = (1, c) we have B1 (z) ∩ D = {(0, c), (1, c)} and therefore D is not 1-identifying. Assume therefore that the condition d(c, C \ {c}) ≤ 1 always holds, and let z = (a, b) ∈ IFn+1 , where a ∈ IF2 and b ∈ IFn2 , be arbitrary. If b ∈ / C, then 2 there are no codewords not beginning with a in B1 (z) ∩ D; if b ∈ C, then there is a unique codeword (a + 1, b) not beginning with a in B1 (z) ∩ D, and by the condition at least two codewords in B1 (z) ∩ D that begin with a. In both cases we can immediately identify z.

We do not know if M1 (n) ≤ 2M1 (n−1) holds — or more generally, Mt1 +t2 (n1 + n2 ) ≤ Mt1 (n1 )Mt2 (n2 ) — but we can prove the following slightly weaker result. Theorem 2. If C is 1-identifying then so is C ⊕ {00, 01, 10, 11}. Proof. Assume that C ⊆ IFn2 and let z = (x, y) ∈ IFn+2 , where x ∈ IFn2 and 2 2 y ∈ IF2 , be arbitrary. Without loss of generality, assume y = 00. If x ∈ C, then B1 (z) contains the words (x, 00), (x, 01), (x, 10), and (x, 00) is the only vector covered by all of them. If x ∈ / C, then all the codewords in B1 (z) end in y and we can identify z because C is 1-identifying.



2

Constructions

Theorem 3. M1 (4) = 7. Proof. The seven codewords 0000, 0001, 0010, 0101, 0110, 1011, 1101 form a 1-identifying code of length four. For the proof of optimality, we refer to [1].

In the following theorems we mention the best previous upper bounds from [4] in parenthesis. Theorem 4. M1 (6) ≤ 19 (20).

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Proof. Take as codewords i) the four words 010000, 001000, 000010, 000001, ii) all the words of weight two except 110000 and its cyclic shifts and iii) all the words of weight five. It is not difficult to check that this code is indeed 1identifying.

Theorem 5. M1 (7) ≤ 32 (40), M1 (8) ≤ 64 (80), M1 (9) ≤ 128. Proof. In this proof all pairs (x, y) denote binary vectors with x ∈ IF52 , y ∈ IF22 . Denote C = {(x, y) | w(x) = 1} \ {(00001, 00), (00010, 01), (00100, 10), (01000, 11)}. We claim that the sets B1 (u) ∩ C where u = (a, b) and w(a) ≤ 2 are nonempty and different. Then it is clear that the codewords of C together with their complements form a 1-identifying code with 32 words. This code has the property of Theorem 1, which gives the other two upper bounds. Let u = (a, b) be fixed. If w(a) = 0, then B1 (u) ∩ C consists of four or five codewords (x, b) ∈ C. If w(a) = 1, then all the codewords B1 (u) ∩ C are of the form (a, y), and as we shall see, there are at least two different choices for y. If a = 10000, then y has the following choices: b 00 01 10 11 00 00 00 01 y 01 01 10 10 10 11 11 11 If a = 01000, the choice y = 11 is not available (because (01000, 11) was removed), but the sets of choices for y still remain different and all contain at least two elements. The same is true in the three remaining cases. If w(a) = 2, the (one or two) codewords in B1 (u) ∩ C are of the form (x, b). If there are two codewords in B1 (u) ∩ C, we immediately know u; if there is only one, we know that the vector ending in b among the removed four vectors also has to cover u, and we again know u.

Theorem 6. For all even m ≥ 4, M1 (2m − 1) ≤ (2m + 2m−1 − 2)22

m

−2m

.

∗ Proof. (Sketch) Denote by Pm the punctured Preparata code for all even m ≥ 4. We take as codewords all the vectors x ∈ IFn2 such that d(x, c) = 1 for some ∗ ∗ c ∈ Pm together with all the vectors in Hm \ Pm we obtain a 1-identifying code. For a detailed proof, see [1]. m ∗ The cardinality of Pm is 22 −2m and therefore the cardinality of our code m equals (n + (2m−1 − 1))22 −2m .



Corollary 1. M1 (15) ≤ 5632 (5760).



On the Size of Identifying Codes

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145

Lower Bounds

The following theorem has been proved in [4]. We prove it using a short counting argument, which also leads to further results. In the proof we use the concept of excess, cf. e.g. [3, Section 6.3]: Assume that C ⊆ IFn2 has covering radius 1. If a vector x ∈ IFn2 is 1-covered by exactly i + 1 codewords of C, we say that the excess E(x) (by C) on x is i. In general, the excess E(V ) on a subset V ⊆ IFn2 is defined by
E(V ) = E(x). x∈V Theorem 7. M1 (n) ≥

n2n . V (n, 2)

Proof. Assume that C is a 1-identifying code with K = M1 (n) codewords. There may be some at most K points in the space IFn2 covered by a unique codeword; for all the other points x we have E(x) ≥ 1. Every point x with E(x) = 1 is called a son; every point x with E(x) > 1 is called a father. Every son has a unique father defined as follows. A son x is covered by exactly two codewords, who necessarily have distance 1 or 2 from each other. Consequently, they both 1-cover exactly one other point, which is called the father of x. Since C is 1-identifying, the father must be covered by at least one more codeword and therefore automatically has excess at least 2. We now divide the space into families: all the sons that have a common father, form a family with their father. The families are disjoint; the families together with the uniquely covered points partition the whole space. There may be fathers with no sons. If the father of a family is covered by exactly i codewords, then it can have  at most 2i sons. Indeed, each son is covered by exactly two codewords that also cover the father. The average excess on the points in a family whose father is covered by exactly i ≥ 3 codewords is therefore at least     f(i) := ( 2i + i − 1)/( 2i + 1). This is a decreasing function on i ≥ 4; and f(3) = f(6). Assume that n ≥ 6. Then f(i) ≥ f(n) whenever 3 ≤ i ≤ n. If there were a codeword c ∈ C such that B1 (c) ⊆ C then we could remove c from the code without violating the identification property. Since K = M1 (n), this is not the case. In other words, there are no families with i = n + 1. The total excess E(IFn2 ) trivially equals K(n + 1) − 2n . We now estimate it in a different way. The uniquely covered points contribute nothing. The number of such points is at most K. Apart from the uniquely covered points, the remaining points form families, each of which has average excess at least f(n). Therefore K(n + 1) − 2n ≥ (2n − K)f(n),

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The claimed lower bound on K now follows. For n = 5, the argument gives 6K − 25 ≥ (25 − K)f(3) and hence K ≥ 10, so the formula of the theorem also holds when n = 5. The formula also holds for n = 4 by Theorem 3, and it is easy to see that it is also valid for n = 2 and n = 3.

A code C is called e-error-correcting if d(a, b) ≥ 2e + 1 whenever a, b ∈ C, a = b. An e-error-correcting code is called perfect if it has covering radius e. Theorem 8. Assume that n > 2. Equality holds in the previous theorem if and only if there exists a perfect 2-error-correcting code of length n. Proof. (Sketch) Assume that n ≥ 7. It has been proved in [4] that if there exists a perfect 2-error-correcting code of length n, then equality holds in Theorem 7. From the previous proof we see that if equality holds, there are no families with i =  n + 1 or i < n, i.e., all the families have i = n and, moreover, consist of n 2 + 1 points. Assume then that a code C attains the bound with equality. If the distance between any two fathers is less than five, it is not difficult to show that at least one of them does not have enough sons. To prove that D is perfect, we show that the covering radius of D is at most two. Let z ∈ IFn2 be arbitrary. We claim that d(z, D) ≤ 2. If z itself is a father or a son, there is nothing to prove. Assume that z is 1-covered by a unique codeword c of C. Because B1 (z) ∩ C = B1 (c) ∩ C, we know that c is covered by another codeword c ∈ C. However, since B1 (c) ∩ C and B1 (c ) ∩ C are different and both contain c and c , at least one of them contains another codeword c ∈ C, i.e., either c or c is a father.

It is well known that there is no perfect 2-error-correcting code of length 90, and therefore 90 · 290 M1 (90) > = 90 · 278 V (90, 2) The following optimality result gives a case where the argument of the previous theorem can be sharpened. Theorem 9. M1 (7) = 32. Proof. (Sketch) By Theorem 5 it suffices to prove that M1 (7) ≥ 32. Step 1: By Theorem 7, M1 (7) ≥ 31. Assume that C is a 1-identifying code of length seven with K = 31 codewords. From the proof of Theorem 7 we know that if there are less than K points x ∈ IFn2 with E(x) = 0, then 120 = K(n + 1) − 2n ≥ (2n − K + 1)f(7) > 120, which is a contradiction. Hence for every c ∈ C there is a vector x ∈ B1 (c) such that E(x) = 0. Step 2: The 97 points with excess at least one are divided into families, say F1 , F2 , . . . , Fk with fathers y1 , y2 , . . . , yk , respectively. A careful analysis similar to the one used in the proof of Theorem 7 in this specific shows that we

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may assume that F1 , F2 , F3 are families with E(y1 ) = E(y2 ) = E(y3 ) = 6 and all have at least 21 elements. Step 3: Moreover, since n = 7, two of these three fathers are at most distance four apart, say d(y 1 , y2 ) ≤ 4. It is not difficult to show that this leads to a contradiction with the fact that the families are so large or with the observation made in Step 1.

The following table presents the currently known bounds on M1 (n) when n ≤ 9. For results on M2 (n), we refer to [4] and [2]. n M1 (n) 3 a4a 4 b7b 5 c 10 a 6 c 18–19 e 7 d 32 f 8 c 56–64 f 9 c 101–128 f Key a Karpovsky et al. [4] b Theorem 3 c Theorem 7 (which is from [4]) d Theorem 9 e Theorem 4 f Theorem 5

Acknowledgement The second author would like to thank the Finnish Academy for financial support.

References 1. U. Blass, I. Honkala, S. Litsyn: Bounds on identifying codes. Discrete Mathematics, to appear 2. U. Blass, I. Honkala, S. Litsyn: On binary codes for identification. Journal of Combinatorial Designs, submitted 3. G. Cohen, I. Honkala, S. Litsyn, A. Lobstein: Covering Codes. Elsevier, Amsterdam (1997) 4. M. G. Karpovsky, K. Chakrabarty, L. B. Levitin: On a new class of codes for identifying vertices in graphs. IEEE Trans. Inform. Theory 44 (1998) 599–611

Fast Quantum Fourier Transforms for a Class of Non-abelian Groups Markus P¨ uschel1, Martin R¨ otteler2 , and Thomas Beth2 1

2

Dept. of Mathematics and Computer Science, Drexel University 3141 Chestnut Street, Philadelphia, PA 19104 [email protected] Institut f¨ ur Algorithmen und Kognitive Systeme, Universit¨ at Karlsruhe Am Fasanengarten 5, D-76128 Karlsruhe, Germany {roettele,EISS Office}@ira.uka.de

Abstract. An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvable groups on a quantum computer. The inherent structure of the Hilbert space imposed by the qubit architecture suggests to consider groups of order 2n first (where n is the number of qubits). As an example, fast quantum Fourier transforms for all 4 classes of nonabelian 2-groups with cyclic normal subgroup of index 2 are explicitly constructed in terms of quantum circuits. The (quantum) complexity of the Fourier transform for these groups of size 2n is O(n2 ) in all cases.

1

Introduction

Quantum algorithms are a recent subject and possibly of central importance in physics and computer science. It has been shown that there are problems on which a putative quantum computer could outperform every classical computer. A striking example is Shor’s factoring algorithm (see [27]). Here we address a problem used as a subroutine in almost all known quantum algorithms: The quantum Fourier transform (QFT) and its generalization to arbitrary finite groups. In classical computation there exist elaborate methods for the construction of Fourier transforms (e. g., [3], [4], [5], [6], [10], [19]), therefore it is highly interesting to adapt and modify these methods to get a quantum algorithm for the Fourier transform with a much better performance (with respect to the quantum complexity model, see Section 3) than any classical algorithm. First attempts in this direction have been proposed by Beals [2] and Høyer [16]. In this paper we present an algebraic approach using representation theory which can be seen as a first step towards the realization of a large class of generalized Fourier transforms on a quantum computer. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 148–159, 1999. c Springer-Verlag Berlin Heidelberg 1999


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2

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Representation Theory and Fourier Transforms

Fourier transforms for finite groups are an interesting and well studied topic for classical computers. We refer to [3], [6], [19], [24] as representatives for a vast number of publications. The reader not familiar with the standard notations concerning group representations should refer to these publications or to standard references like [9] or [26]. For the convenience of the reader we will briefly present the terms and notations from representation theory which we are going to use and recall the definition of Fourier transforms. A representation of a finite group G of degree deg(φ) = n is a homomorphism φ : G → GLn (K) from G into the group of invertible (n × n)-matrices over a field K. We denote by 1G : g → 1 the trivial representation of G (of degree 1). If A ∈ GLn (K), then φA : g → A−1 · φ(g) · A is the conjugate of φ by A. φ and ψ are called equivalent, if φ = ψ A . If φ,
ψ are representations of G, then the  φ(g) 0 representation φ ⊕ ψ : g → φ(g) ⊕ ψ(g) = is called the direct sum of 0 ψ(g) φ and ψ. φ is called irreducible, if it cannot be conjugated to be a direct sum. In this paper, we will deal only with ordinary representations, i. e. the characteristic of K does not divide the group order |G| (Maschke condition). In this case, every representation φ can be conjugated, by a suitable matrix A, to ak direct sum of irreducible representations ρi (Maschke’s theorem), i. e. φA = i=1 ρi , which is called a decomposition of φ and A is referred to as a decomposition matrix for φ. Let φ be a representation of H ≤ G, and φ a representation of G which is equal to φ when restricted to H (φ ↓ H = φ). Then φ is called an extension of φ to G and φ is called extensible (to G). Note, that an extension does not exist in general. If φ is a representation of H ≤ G and t ∈ G then φt : h → φ(tht−1 ) is a representation of H t , called the inner conjugate of φ by t. If H ≤ G is a subgroup with transversal (i. e. a system of representatives of the right cosets of ˙ i gt−1 ) | i, j = 1 . . . n], where H in G) T = (t1 , . . . , tk ), then (φ ↑T G)(g) = [φ(t j ˙ φ(x) = φ(x) for x ∈ H and the all-zero matrix else, is called the induction of φ to G with transversal T . A regular representation is an induction of the form φ = (1E ↑T G) where E denotes the trivial subgroup of G. Let φ be a regular representation of G. A Fourier transform for G is any decomposition matrix A of φ with the additional property that equivalent irreducibles in the corresponding decomposition are even equal. Note, that the definition says nothing about the transversal fixing φ, nor the choice of the irreducible representations. As an example let G = Zn = x | xn = 1 be the cyclic group of order n with regular representation φ = 1E ↑T G, T = (x0 , x1 , . . . , xn−1 ), n−1 and ωn a primitive nth root of unity. Then φA = i=0 ρi , where ρi : x → ωni and A = DFTn = √1n [ωnij | i, j = 0 . . . n − 1] is the (unitary) discrete Fourier transform well-known from signal processing. If A is a Fourier transform for the group G, then we will refer to a fast Fourier transform as any fast algorithm for the multiplication with A. Of course, the term “fast” depends on the chosen complexity model. Since we are primarily interested in the realization of a fast Fourier transform on a quantum computer (QFT) we first have to define the measure of complexity on this architecture.

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The Complexity Model Used in Quantum Computing

Quantum computing is a topic of recent interest which emerged after the discovery of polynomial algorithms for integer factoring and discrete logarithms by P. Shor (see [27]). The state of a quantum computer is given by a normalized vector in a Hilbert space Hn of dimension 2n , which is given the natural tensor structure Hn = C2 ⊗ . . . ⊗ C2 (n factors). The restriction of the computational space to Hilbert spaces of this particular form is motivated by the idea of a quantum register consisting of n quantum bits. A quantum bit, also called qubit, is a state corresponding to one tensor component of Hn and has the form |ϕ = α|0 + β|1 ,

|α|2 + |β|2 = 1,

α, β ∈ C.

The possible operations this computer can carry out are the elements of the unitary group U(2n ). To study the complexity of performing unitary operations on n-qubit quantum systems we introduce the following two types of computational primitives (see also [15], this volume): Local unitary operations on a qubit i are matrices of the form U (i) := 12i−1 ⊗ U ⊗ 12n−i , where U is an element of the unitary group U(2) of 2 × 2-matrices. Furthermore we need operations which affect two qubits at a time, the standard choice for which is the so-called controlled NOT gate (also called measurement gate) between the qubits i (control) and j (target) defined by  1 1 CNOT(i,j) := 1 1

when restricted to the tensor component of the Hilbert space spanned by the qubits i and j. We assume that these so-called elementary quantum gates can be performed with cost O(1). In the graphical notation using quantum wires (for details see [1]) these transforms are written as shown in Figure 1. The lines correspond to the qubits, unaffected qubits are omitted, and the dot • sitting on a wire denotes the control bit. U (i) =

U

i

CNOT(2,1) =

•

2

❢

1

Fig. 1. Elementary quantum gates. These two types of gates suffice to generate all unitary transformations, which is the content of the following theorem from [1]. Theorem 1. The set G = {U (i) , CNOT(i,j) | U ∈ U(2), i, j = 1 . . . n, i = j} is a generating set for the unitary group U(2n ). This means that for each U ∈ U(2n ) there is a word w1 w2 . . . wk (where wi ∈ G for i = 1 . . . k is an elementary gate) such that U factorizes as U = w1 w2 . . . wk . In general only exponential upper bounds for the minimal length occurring in factorizations have been proved (see [1]) but there are many interesting classes

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151

of unitary matrices in U(2n ) affording only polylogarithmic word length, which means that the minimal length is asymptotically O(p(n)) where p is a polynomial. In the following, we give examples of some particular unitary transforms admitting short factorizations which will be useful in the rest of the paper. – The symmetric group Sn is embedded in U(2n ) by the natural operation of Sn on the tensor components (qubits). Let τ ∈ Sn and Πτ the corresponding permutation matrix on 2n points. Then Πτ has a O(n) factorization as shown in [22]. As an example in Figure 2 the permutation (1, 3, 2) of the qubits (which corresponds to the permutation (2, 5, 3)(4, 6, 7) on the register) is factored according to (1, 3, 2) = (1, 2)(2, 3). ❅✁✁ ❅ ❅✁ ✁❅

=

•

❣

•

❣

•

❣

•

❣

•

❣

•

❣

Fig. 2. Factorization (1, 3, 2) = (1, 2)(2, 3). – Following the notation of [1] we denote a k-times controlled U by Λk (U). As an example for the graphical notation we give in Figure 3 a Λ1 (U) gate for arbitrary U ∈ U(2n ) with normal, and a gate with inverted control bit including the represented matrix. Lemma 7.2 and Lemma 7.5 in [1] show that for U ∈ U(2) the gate Λk (U) can be realized with gate complexity O(n), for k < n − 1, and Λn−1 (U) in O(n2 ). r Λ1 (U ) = 12n ⊕ U =

.. .

U

❜

n+1 .. .

n .. .

U ⊕ 12n =

.. .

U

n+1 .. .

1

n .. . 1

Fig. 3. Controlled gates with a) normal and b) inverted control bit. – The Fourier transform DFT2n can be performed in O(n2 ) elementary operations on a quantum computer (see [27], [8]). – Let Pn ∈ S2n be the cyclic shift acting on the states of the quantum register as x → x + 1 mod 2n . The corresponding permutation matrix is the 2n cycle (1, 2, . . . , 2n ). Pn can be realized using O(n2 ) basic operations (see [13], Section 4.4). Let U ∈ U(2n ). The cost for a (single) controlled U is settled by the following Lemma 1. If U ∈ U(2n ) can be realized in O(p(n)) elementary operations then Λ1 (U) ∈ U(2n+1 ) can also be realized in O(p(n)) basic operations. Proof: First we assume without loss of generality that U is written in elementary gates. Therefore we have to show that a double controlled NOT and a single controlled U ∈ U(2) can be realized with a constant increase of length. This follows from [1].  

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Creating Fast Fourier Transforms

In Section 2 we have explained that a Fourier transform for a group G is a decomposition matrix B for a regular representation φ of G with the additional property that equivalent irreducible summands are equal, i. e. φB = ρ1 ⊕ . . . ⊕ ρk

fulfilling ρi ∼ = ρj ⇒ ρi = ρj .

A “fast” Fourier transform (on a classical computer) is given by a factorization of B into a product of sparse matrices. (see [3], [6], [19], [25]). For a solvable group G, this factorization can be obtained recursively using the following idea. First, a normal subgroup of prime index (G : N ) = p is chosen. Using transitivity of induction, φ = 1E ↑ G is written as (1E ↑ N ) ↑ G (note, that we have the freedom to choose the transversals appropriately). Then 1E ↑ N , which again is a regular representation, is decomposed (by recursion) yielding a Fourier transform A for N . In the last step, B is derived from A using a recursion formula. In the following, we will explain this procedure in more detail by first presenting the two essential theorems (without proof) and then stating the actual algorithm for deriving fast Fourier transforms for solvable groups. The special tensor structure of the recursion formula mentioned above will allow us to use the very same algorithm as a starting point to also obtain fast quantum Fourier transforms in the case G being a 2-group (i. e. |G| is a power of 2). The statements in this section are all taken from the first chapter of [24] where decomposition matrices for arbitrary monomial representations in general are investigated. The first thing we need is Clifford’s Theorem which explains the relationship between the irreducible representations of N and those of G. Theorem 2 (Clifford). Let N ✂ G be a normal subgroup of prime index p with (cyclic) transversal T = (t0 , t1 , . . . , t(p−1)) and denote by λi : t → ωpi , i = 0 . . . p − 1, the p irreducible representations of G arising from G/N . Assume ρ is an irreducible representation of N . Then exactly one of the two following cases applies: 1. ρ ∼ = ρt and ρ has p pairwise inequivalent extensions to G. If ρ is one of them, then all are given by λi · ρ, i = 0 . . . p − 1. p−1 i 2. ρ ∼  ρt and ρ ↑T G is irreducible. Furthermore, (ρ ↑T G) ↓ N = i=0 ρt = and (λi · (ρ ↑T G))

D⊗1d

= ρ ↑T G,

D = diag(1, ωp , . . . , ωp(p−1))i .

The following theorem provides the recursion formula which had already been used in [3] to obtain fast Fourier transforms. Theorem 3. Let N ✂ G be a normal subgroup of prime index p with transversal T = (t0 , t1 , . . . , t(p−1)) and φ a representation of degree d of N . Suppose that A is matrix decomposing φ into irreducibles, i. e. φA = ρ = ρ1 ⊕ . . . ⊕ ρk and that ρ is an extension of ρ to G. Then (φ ↑T G)B =

p−1  i=0

λi · ρ,

Fast Quantum Fourier Transforms for a Class of Non-abelian Groups

153

where λi : t → ωpi , i = 0 . . . p − 1, are the p irreducible representations of G arising from the factor group G/N , B = (1p ⊗ A) · D · (DFTp ⊗ 1d ),

and

D=

p−1 

ρ(t)i .

i=0

If, in particular, ρ is a direct sum of irreducibles, then B is a decomposition matrix of φ ↑T G. In case of an cyclic group G the formula yields exactly the well-known CooleyTukey decomposition, [7], in which D is usually called the Twiddle matrix. Assume that N ✂ G is a normal subgroup of prime index p with Fourier m transform A and decomposition φA = ρ = i=1 ρi . We can reorder the ρi , such that the first, say k, ρi have an extension ρi to G and the other ρi occur (p−1) of inner conjugates (cf. Theorem 2, note as sequences ρi ⊕ ρti ⊕ . . . ⊕ ρti j that irreducibles ρi , ρti have the same multiplicity since φ is regular). In the first case the extension may be calculated by Minkwitz’ formula, [21], in the latter case each sequence can be extended by ρi ↑T G (Theorem 2, Case 2). We do not state Minkwitz’ formula here, since we will not need it in the special cases treated in Section 5. Altogether we obtain an extension ρ of ρ and can apply p Theorem 3. The remaining task is to assure, that equivalent irreducibles in i=1 λi · ρ are equal. For summands of ρ of the form ρi we have that λj · ρi and ρi are inequivalent and hence there is nothing to do. For summands of ρ of the form ρi ↑T G, we conjugate λj · (ρi ↑T G) onto ρi ↑T G using Theorem 2, Case 2. Now we are ready to formulate the recursive algorithm for constructing a fast Fourier transform for a solvable group G. Algorithm 1. Let N ✂ G a normal subgroup of prime index p with transversal T = (t0 , t1 , . . . , t(p−1) ). Suppose that φ is a regular representation of N with (fast) Fourier transform A, i. e. φA = ρ1 ⊕ . . . ⊕ ρk , fulfilling ρi ∼ = ρj ⇒ ρi = ρj . A Fourier transform B of G with respect to the regular representation φ ↑T G can be obtained as follows. 1. Determine a permutation matrix P rearranging the ρi , i = 1 . . . k, such that the extensible ρi (i. e. those satisfying ρi = ρti ) come first followed by the oth(p−1) . (Note: ers ordered into sequences of length p equivalent to ρi , ρti , . . . , ρti (p−1) which is established in These sequences need to be equal to ρi , ρti , . . . , ρti the next step). 2. Calculate a matrix M which is the identity on the extensibles and conjugates (p−1) the sequences of length p to make them equal to ρi , ρti , . . . , ρti . 3. Note that A · P · M is a decomposition matrix for φ, too, and let ρ = φA·P ·M . Extend ρ to G summandwise. For the extensible summands use Minkwitz’ (p−1) formula, the sequences ρi , ρti , . . . , ρti can be extended by ρi ↑T G. p−1  ρ(t)i . 4. Evaluate ρ at t and build D = i=0

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5.  Construct a blockdiagonal matrix C with Theorem 2, Case 2, conjugating p−1 i=0 λi · ρ such that equivalent irreducibles are equal. C is the identity on the extended summands. Result: B = (1p ⊗ A · P · M) · D · (DFTp ⊗ 1|N |) · C

(1)

is a fast Fourier transform for G.

 

It is obviously possible to construct fast Fourier transforms on a classical computer for any solvable group by recursive use of this algorithm. Since we restrict ourselves to the case of a quantum computer consisting of qubits, i. e. two-level systems, we apply Algorithm 1 to obtain QFTs for 2-groups (|G| = 2n , p = 2). In this case the two tensor products occurring in (1) fit very well to yield a coarse factorization as shown in Figure 4. The lines correspond to the qubits like in Section 3 and a box ranging over more than one line denotes a matrix admitting no a priori factorization into a tensor product. The remaining problem is the realization of the matrices A, P, M, D, C in terms of elementary building blocks as presented in Section 3. At present, however, this realization remains a creative process which might be performed by hand if a certain class of groups is given. In Section 5 we will apply Algorithm 1 to a class of non-abelian 2-groups. n

DFT2 .. .

A

.. .

P

.. .

M

D

.. .

.. .

C

.. .

n–1 .. . 1

Fig. 4. Coarse quantum circuit visualizing Algorithm 1 for 2-groups.

5

Generating QFTs for a Class of 2-Groups

In the case of G being an abelian 2-group the realization of a fast quantum Fourier transform has been settled by [18]. This case is also covered by the method presented here. In this section we will apply Algorithm 1 to the class of non-abelian 2-groups containing a cyclic normal subgroup of index 2. Fast quantum Fourier transforms for these groups have already been constructed by Høyer in [16]. According to [17], p. 90/91, there are for n ≥ 3 exactly four isomorphism types of non-abelian groups of order 2n+1 affording a cyclic normal subgroup of order 2n : 1. 2. 3. 4.

The The The The

n

dihedral group D2n+1 = x, y | x2 = y 2 = 1, xy = x−1 . n quaternion group Q2n+1 = x, y | x2 = y 4 = 1, xy = x−1 . n n−1 group QP2n+1 = x, y | x2 = y 2 = 1, xy = x2 +1 . n n−1 quasi-dihedral group QD2n+1 = x, y | x2 = y 2 = 1, xy = x2 −1 .

Fast Quantum Fourier Transforms for a Class of Non-abelian Groups

155

Observe that the extensions 1, 3, and 4 of the cyclic subgroup Z2n = x split, i. e. the groups have the structure of a semidirect product of Z2n by Z2 . The three isomorphism types correspond to the three different embeddings of Z2 = y into (Z2n )× ∼ = Z2 × Z2n−2 . 5.1

QFT for the Dihedral Groups D2n+1

In this section we construct a QFT for the dihedral groups D2n+1 step by step according to Algorithm 1 and explicitly state the occurring quantum circuits. n Let G = D2n+1 = x, y | x2 = y 2 = 1, xy = x−1 with normal subgroup N = x ✂ G of index 2 and transversal T = (1, y). We consider the regular n representation φ = (1E ↑S N ) ↑T G of G with S = (1, x, . . . , x2 −1 ). Obviously the regular representation (1E ↑S N ) of N is decomposed by A = DFT2n into ρ0 ⊕ . . . ⊕ ρ2n −1 where ρi : x → ω2i n . Now we are ready to apply Algorithm 1 to obtain a decomposition matrix  B for φ. For convenience we denote ω2n simply as ω and H = DFT2 = √12 · 11 −11 . 1. Since ρyi (x) = ρi (yxy −1 ) = ρi (x−1 ) = ρ2n −i (x) we see that there are exactly two extensible ρi namely for i = 0, 2n−1 . The sequences of inner conjugates are given by ρi , ρ2n −i , i = 0, 2n−1 . Thus we need a permutation P reordering the ρi as ρ , ρ n−1 , ρ , ρ n , . . . , ρi , ρ2n −i , . . . , ρ2n−1 −1 , ρ2n−1 +1 . 0 2 1 2 −1

 extensibles

pairs of inner conjugates

This can be accomplished by the circuit given in Figure 5 since the n-cycle on the qubits which is performed first yields a decimation by two on the indices, i. e. the indices 0, . . . , 2n−1 − 1 have found their correct position. The only thing which remains to do is to perform the operation x → −x on the odd positions. This can be done by an inversion of all (odd) bits followed by a x → x + 1 shift Pn−1 on the odd states of the register. ❉ ...

❉

❉ ❉.

..

❉ ❉ ❉

.. .

x → −x

•

.. .

=

...

❢ ❢

❉ ❉.

..

❉ ❉

❢ ❢ ❉

•

Pn−1

.. .

•

Fig. 5. Ordering the irreducibles of Z2n ✂ D2n+1 . 2. M can be omitted since all the ρi are of degree 1. 3. Let φA·P = ρ. We extend ρ summandwise to ρ: – ρ0 = 1N can be extended by 1G . – ρ2n−1 can be extended through ρ2n−1 (y) = 1. – The sequences ρi ⊕ ρ2n −i , i = 0, 2n−1 can be extended by ρi ↑T G and (ρi ↑T G)(y) = ( 01 10 ).

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4. Evaluation of ρ at the transversal T yields the Twiddle matrix D = ρ(1) ⊕ ρ(y) = 12n ⊕ 12 ⊕ ( 01 10 ) ⊕ . . . ⊕ ( 01 10 ),

 2n−1 −1

which is realized by the quantum circuit given in Figure 6. r .. .

❢

r ❜

r

❜ ❜ ❢

.. .

r ❜

.. .

.. .

H

Fig. 6. Twiddle matrix for D2n+1 .

❢

❜ ❜ ❢ H

Fig. 7. Equalizing inductions.

5. According to Theorem 2, Case 2, the matrix C has the following diagonal form: C = 12n ⊕ diag(1, 1, 1, −1, . . . , 1, −1),

 2n−1 −1 pairs

which is realized by the quantum circuit given in Figure 7. We obtain that B = (1p ⊗ A · P · M) · D · (DFTp ⊗ 1|N |) · C is a decomposition matrix for φ and a fast quantum Fourier transform for G. The whole circuit is shown in Figure 8. r

...

DFT2n

...

❢ ❢

❉ ❉ ❉.

..

❉ ❉

❢ ❢ ❉

•

Pn−1

•

.. .

❢

r ❜ ❜ ❜ ❜ ❢

r

H

r ❜ ❜ .. .

H

❢

❜ ❜ ❢ H

Fig. 8. Complete QFT circuit for the dihedral group D2n+1 . 5.2

QFT for the Groups Q2n+1 , QP2n+1 , and QD2n+1

In the following we give the circuits for the groups Q2n+1 , QP2n+1 , and QD2n+1 . In all cases we have x = N ✂ G so that Algorithm 1 has to be performed only once for the last step. For the sake of brevity we will state only those parts of the circuit which differ from the dihedral group. We use the same notation as in the last section. Q2n+1 : The irreducibles ρi extend or induce in the same way as in the dihedral case. Hence the QFT only differs in the Twiddle matrix D since for a not

Fast Quantum Fourier Transforms for a Class of Non-abelian Groups

 . Thus D is given by    0 1 ⊕ 12 ⊕ −1 0 ⊕ . . . ⊕ −10 10

 

extensible ρi we have (ρi ↑T G)(y) = D = 1 2n

157

01 −1 0

2n−1 −1

and can be realized by the circuit in Figure 9. r

r ❜

.. .

.. .

❜ ❜ 0 1 -1 0

❈ ✄ ❈ ✄ ❈✄ . .. .. ✄❈ . ✄❈ ✄ ❈ ✄ ❈

0 1 -1 0

Fig. 9. Twiddle matrix for Q2n+1 .

Fig. 10. Permutation for QP2n+1 .

QP2n+1 : To determine which ρi are extensible we observe ρyi (x) = ρi (yxy −1 ) = n−1 n−1 n−1 = 1 ⇔ 2 | i and there ρi (x2 +1 ). Hence ρi = ρyi ⇔ ω i = ω i·(2 +1) ⇔ ω i·2 are exactly 2n−1 extensible ρi . The reordering permutation P has the easy form shown in Figure 10, and the matrix D is given by D = 12n ⊕ 12n−1 ⊕ ( 01 10 ) ⊕ . . . ⊕ ( 01 10 ),

 2n−2

which is simply a double controlled NOT as visualized in Figure 11. The matrix C then is given by Figure 12. r r ...

r r ...

...

❢

H

Fig. 11. Twiddle matrix for QP2n+1 . QD2n+1 : Here we have ρyi (x) = ρi (x2 ρi = ρyi ⇔ ω i = ω i·(2

n−1

−1)

n−1

...

❢

H

Fig. 12. Equalizing for QP2n+1 . −1

) and

⇔ ω i·(2

n−1

−2)

= 1 ⇔ i = 0, 2n−1 .

Thus everything is the same as in the dihedral case beside the ordering permutation P which takes the more complicated form shown in Figure 13. Concerning the complexity of these QFTs we have the following theorem. Theorem 4. The Fourier transforms for the groups G = D2n , Q2n , QP2n , and QD2n can be performed on a quantum computer in O(log2 |G|) elementary operations.

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❇ .. .

❇ ❇

❇❇

❈ ✄ ❈ ✄ .. ❈✄ .. . ✄❈ . ✄ ❈ ✄ ❈ ❜

❢ ❢ ❢ r

Pn−2

. ..

r

r ❜

...

r ❜

Fig. 13. The permutation for the QD2n+1 . Proof: We can treat the four series uniformly, since the Fourier transforms all have the same decomposition pattern. First, in all cases a Fourier transform for the normal subgroup Z2n−1 is performed with cost of O(n2 ) basic operations. The reordering permutation P , the Twiddle matrix D, and the equalizing matrix C cost O(n2 ) in case of D2n , Q2n , and QD2n due to Lemma 1 and the examples in Section 3. For QP2n we need only O(1) operations for P , D, and C.   All presented Fourier transforms have been implemented by the authors in the language GAP [14] using the share package AREP [12].

6

Conclusions and Outlook

A constructive algorithm has been presented allowing to attack the problem of constructing fast Fourier transforms for 2-groups G on a quantum computer built up from qubits. For a certain class of non-abelian 2-groups the algorithm has been successfully applied. All the QFTs created are of computational complexity O(log2 |G|) like in the case of the cyclic group Z2n . The main problem imposed by the implementation of certain permutation and block diagonal matrices has been solved efficiently. Using the recursion formula from Theorem 3 it should be possible to construct QFTs for other classes of groups as well as to realize certain signal transforms on a quantum computer by means of symmetry-based decomposition (see [24], [11], [20]).

Acknowledgments The authors are indebted to Markus Grassl for helpful comments and fruitful discussions. Part of this work was presented and completed during the 1998 Elsag-Bailey – I.S.I. Foundation research meeting on quantum computation. M. R. is supported by Deutsche Forschungsgemeinschaft, Graduiertenkolleg Beherrschbarkeit Komplexer Systeme under Contract No. DFG-GRK 209/3-98.

References 1. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter. Elementary gates for quantum computation. Physical Review A, 52(5):3457–3467, November 1995. 2. R. Beals. Quantum computation of Fourier transforms over the symmetric groups. In Proc. STOC 97, El Paso, Texas, 1997. 3. Th. Beth. Verfahren der Schnellen Fouriertransformation. Teubner, 1984.

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4. Th. Beth. On the computational complexity of the general discrete Fourier transform. Theoretical Computer Science, 51:331–339, 1987. 5. M. Clausen. Fast generalized Fourier transforms. Theoretical Computer Science, 67:55–63, 1989. 6. M. Clausen and U. Baum. Fast Fourier Transforms. BI-Verlag, 1993. 7. James W. Cooley and John W. Tukey. An Algorithm for the Machine Calculation of Complex Fourier Series. Mathematics of Computation, 19:297–301, 1965. 8. D. Coppersmith. An Approximate Fourier Transform Useful for Quantum Factoring. Technical Report RC 19642, IBM Research Division, 1994. 9. W.C. Curtis and I. Reiner. Methods of Representation Theory, volume 1. Interscience, 1981. 10. P. Diaconis and D. Rockmore. Efficient computation of the Fourier transform on finite groups . Amer. Math. Soc., 3(2):297–332, 1990. 11. S. Egner. Zur Algorithmischen Zerlegungstheorie Linearer Transformationen mit Symmetrie. PhD thesis, Universit¨ at Karlsruhe, Informatik, 1997. 12. S. Egner and M. P¨ uschel. AREP – A Package for Constructive Representation Theory and Fast Signal Transforms. GAP share package, 1998. http://avalon.ira.uka.de/home/pueschel/arep/arep.html. 13. A. Fijany and C. P. Williams. Quantum Wavelet Transfoms: Fast Algorithms and Complete Circuits. In Proc. NASA conference QCQC 98, LNCS vol. 1509, 1998. 14. The GAP Team, Lehrstuhl D f¨ ur Mathematik, RWTH Aachen, Germany and School of Mathematical and Computational Sciences, U. St. Andrews, Scotland. GAP – Groups, Algorithms, and Programming, Version 4, 1997. 15. M. Grassl, W. Geiselmann, and Th. Beth. Quantum Reed–Solomon Codes. In Proc. of the AAECC–13 (this volume), 1999. 16. P. Høyer. Efficient Quantum Transforms. LANL preprint quant–ph/9702028, February 1997. 17. B. Huppert. Endliche Gruppen, volume I. Springer, 1983. 18. A. Yu. Kitaev. Quantum Measurements and the Abelian Stabilizer Problem. LANL preprint quant–ph/9511026, November 1995. 19. D. Maslen and D. Rockmore. Generalized FFTs – A survey of some recent results. In Proceedings of IMACS Workshop in Groups and Computation, volume 28, pages 182–238, 1995. 20. T. Minkwitz. Algorithmensynthese f¨ ur lineare Systeme mit Symmetrie. PhD thesis, Universit¨ at Karlsruhe, Informatik, 1993. 21. T. Minkwitz. Extension of Irreducible Representations. AAECC, 7:391–399, 1996. 22. C. Moore and M. Nilsson. Some notes on parallel quantum computation. LANL preprint quant–ph/9804034, April 1998. 23. M. P¨ uschel. Konstruktive Darstellungstheorie und Algorithmengenerierung. PhD thesis, Universit¨ at Karlsruhe, Informatik, 1998. Translated in [24]. 24. M. P¨ uschel. Constructive representation theory and fast signal transforms. Technical Report Drexel-MCS-1999-1, Drexel University, Philadelphia, 1999. Translation of [23]. 25. D. Rockmore. Some applications of generalized FFT’s. In Proceedings of DIMACS Workshop in Groups and Computation, volume 28, pages 329–370, 1995. 26. J. P. Serre. Linear Representations of Finite Groups. Springer, 1977. 27. P. W. Shor. Algorithms for Quantum Computation: Discrete Logarithm and Factoring. In Proc. FOCS 94, pages 124–134. IEEE Computer Society Press, 1994.

Linear Codes and Rings of Matrices M. Greferath and S.E. Schmidt AT&T Labs-Research, 180 Park Ave, Florham Park, NJ 07932

Abstract. We investigate the role of matrix rings in coding theory. Using these rings we introduce an embedding technique which enables us to give new interpretations for some previous results about algebraic representations of some prominent linear codes.

Introduction Codes over rings have been enjoying growing importance during the recent decade. This is based on the observation by Nechaev [13] and Hammons et al. [9] that certain non-linear binary codes (Kerdock, Preparata and further Codes) of high quality are linear over ZZ 4 . Since then various papers have been published many of them dealing with codes over ZZ 4 but also involving other rings (cf. [8,3,2]). Matrix rings are a particular class of non-commutative alphabets which— apart from [2]—have not yet been involved in coding theory. In what follows we therefore first describe relations of coding theory over the latter class of rings to that over their base rings. Subsequently we consider embeddings of given rings into matrix rings and study their properties. To get prepared for the last section we consider matrix representations of algebras. Extensions of Skolem-Noether type theorems based on results by Nechaev [12] show that under suitable conditions an embedding of a K-algebra R into a matrix algebra Mm (K) is unique up to inner automorphisms of the target algebra, provided K is a commutative Artinian local ring. In the last section we apply the preparations of the preceeding ones and provide a different view on several known code representations via matrix rings. In particular our considerations shed new light on the results in Wolfmann [16] and Pasquier [14] (cf. also Goldberg [7]) and yield similar statements for extended binary Hamming Codes and the Octacode. In what follows the reader is supposed to be familiar with basic facts of ring and module theory as well as order theory. We will frequently denote the set of all linear (left) codes of length n over a ring K—i.e. the lattice of all submodules of K K n —by L(K K n ). Hence, our use of the term lattice is the order theoretical one rather than the one usually referred to in coding theory.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 160–169, 1999. c Springer-Verlag Berlin Heidelberg 1999


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1

161

Linear Codes under Morita Equivalence

Let K be a ring, m a positive integer, and let S denote the full ring of all (m × m)-matrices over K. We establish a connection between linear codes over K and those over S referring to what is known as Morita equivalence. For a module K M let S operate on the abelian group M m via: S × M m −→ M m
((aij )i,j , (mj )j ) −→ ( aij mj )i , j

where the index domain of (aij )i,j and (mj )j are thought to be understood. Hence, S operates in the same way on M m , as it operates on K m ; the only difference is that the matrix elements are multiplied with module elements rather than ring elements. Lemma 1. For every module isomorphic to S M m .

SN

there exists a module

KM

such that

SN

is

The following lemma clarifies the connection between the submodule lattices involved. For an element x of a module K M let x(m) denote the m-fold concatenation of x, i.e. the element (x1 , . . . , xm ) ∈ M m with xi = x for all i = 1, . . . , m. Proposition 1. 1. The mapping M −→ M m with x −→ x(m) is a semilinear embedding where the accompanying ring homomorphism K −→ S is given by the natural embedding of K in S with λ → λIm . 2. Its induced lattice mapping L(K M) −→ L(S M m ) is given by C −→ C m and is an isomorphism. Proof. It is easily verified that the mapping in question is additive and that (λx)(m) = (λIm )x(m) , for all λ ∈ K and x ∈ M, which is our first claim. The second claim is a direct consequence.   We are interested in how the property of being free carries over to the respective module. Lemma 2. Let S be the ring of (m × m)-matrices over the ring K, let K M and m S N be modules with S N  S M . Then S N is free of rank n if and only if K M is free of rank nm. For the coding theoretical context we now summarize what we have seen in a theorem: Theorem 1. For a ring K and its ring of all (m × m)-matrices Mm (K) there is a 1 − 1-correspondence between linear codes of length mn over K and linear codes of length n over Mm (K). This correspondence is induced by (semilinearly) embedding the former as their m-fold concatenations into the latter.

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Remark 1. There is a natural relation between the respective decoding problems: 1. Any decoding scheme A for a K-linear code C can be extended to a scheme A∗ used to decode its Mm (K)-linear image C m under the natural embedding established above. It proceeds in m steps at the cost of one application of A per step. It corrects any matrix error pattern each full row of which is!correctable by A. 2. Any decoding scheme A∗ for an Mm (K)-linear code C m can be reduced to a scheme A used to decode its K-linear preimage C under the natural embedding. It proceeds at the cost of one application of A. It corrects any error pattern e the m-fold concatenation e(m) of which is correctable by A∗ . The correspondence between codes over rings and those over their matrix rings is purely algebraic so far. It does not imply any further going equivalence since it does not involve any metrical aspect. The following considerations are devoted to this point. We establish a weight function on Mm (K) for which the semilinear embedding mentioned in Prop. 1 is an isometry. Let again S denote the ring of all (m × m)-matrices over the ring K, and let wK : K −→ IR be a weight function on K, which shall be thought to be completed additively on K m . We define a function wS : S −→ IR A −→ max wK (Ai ), i=1...m

where Ai denotes the ith row of A. This kind of row-sum norm is known to satisfy the triangle inequation and is strictly positive. We have furthermore wS (A) = wS (−A) for all A ∈ S. As usual we complete wS additively to S n (n ∈ IN). Then our definition gets justified by the following statement. Proposition 2. Let wK be a weight function on K and wS the just defined weight function on S. The semilinear embedding K K nm −→ S S n with x −→ x(m) is an isometry of (K nm , wK ) into (S n , wS ). Proof. It is easily verified that wS (x(m) ) = wK (x) for all x ∈ K nm .

 

For the set of all linear Codes of length nm over K and those of length n over S we are now able to prove the following: Theorem 2. Let K be a unital ring, wK a weight function on K and S the ring of all (m × m)-matrices over K together with the above-defined weight function wS . For every natural number n the lattice isomorphism L(K K nm ) −→ L(S S n ) induced by the isometry K nm −→ S n preserves minimum distance. Proof. Let C be a K-linear code of length nm, and let C m be its S-linear image under the lattice mapping in question. If x is a word of minimal K-weight in C, then by Prop. 2 we have wS (x(m ) = wK (x) and therefore dS (C m ) ≤ dK (C). Conversely, if y is a word of minimal S-weight in C m , then each full row yi

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(i = 1, . . . , m) of this matrix word is contained in C and satisfies by definition of wS clearly wK (yi ) ≤ wS (y). This however shows, that dK (C) ≤ dS (C m ), and therefore our claim follows.   We are now able to refine our statements in Rem. 1. Theorem 3. Let K be a unital ring, wK a weight function on K and S the ring of all (m × m)-matrices over K together with the row-sum weight function wS . Let furthermore A be a decoding scheme for a K-linear code C of length nm and A∗ its extension for the S-linear code C m . Then A is bounded distance (with respect to wK ) if and only if A∗ is bounded distance (with respect to wS ). Proof. Let first A correct error patterns of K-weight up to t, and let y be an Serror pattern of S-weight s ≤ t. As we saw in the proof of Thm. 2 the K-weight of each full row of y is at most s, and hence it can be corrected by A. But this shows that A∗ corrects S-error patterns of S-weight upt to t. Conversely, if A∗ corrects error patterns of S-weight up to t, and x is a K-error pattern of Kweight s ≤ t, then by Prop. 2 combined with wS (x(m) ) = wK (x) it is immediate that x(m) can be corrected by A∗ , and therefore x can be corrected by A.  

2

Code Correspondences Induced by Ring Embeddings

The situation of the foregoing section changes significantly when considering more general embeddings. Let ϕ : R −→ S be a unital ring embedding and let n be a natural number. The given embedding clearly induces a semilinear embedding ϕ : R Rn −→ n S S defined componentwise and the latter gives rise to a lattice mapping ϕ : L(R Rn ) −→ L(S S n) with C → S ϕ(C). This mapping need not be injective, as the natural embedding of ZZ into Q shows. It is immediate, however, that ϕ is (completely)join-preserving,  i.e. for any family (Ci )i∈I of submodules of R Rn there holds ϕ( i∈I Ci ) = i∈I ϕ(Ci ). Under appropriate assumptions on the relation between R and S an according statement for finite meets can be proved. Recall that S obtains the structure of an R-bimodule by setting r · s := ϕ(r)s and s · r := sϕ(r), and it is clear that (r · s) · r = r · (s · r ) for all r, r ∈ R and s ∈ S. Keeping this in mind we find it convenient to write ϕ(C) = S ⊗ϕ R C. Lemma 3. If SR is a flat module, then ϕ : L(R Rn ) −→ L(S S n ) is (finite) meet preserving, i.e. ϕ(C ∩ D) = ϕ(C) ∩ ϕ(D) for all C, D ∈ L(R Rn ). Proof. cf. [4, §2.6].

 

The last statement is in particular valid, if SR is a projective module. In the coding theoretical context the flatness of SR might be important under a different aspect: Let an R-linear left code C of length n possess a (k × n)generator matrix G and an (m × n)-check matrix H. We have C = Im(G), which is usually represented by the short exact sequence G

0 −→ ker(G) −→ Rk −→ C −→ 0.

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Applying ϕ to this sequence and subsequent tensoring with S SR and idS results in ϕ(C) = Im(ϕ(G)) because of ⊗ϕ R being right exact. Therefore ϕ maps generator matrices for C to such for ϕ(C). On the other hand the equality C = ker(H t ) may be represented by the sequence Ht

0 −→ C −→ Rn −→ Im(H t ) −→ 0, the exactness of which is not preserved in general under the above manipulations. If however SR is presumed to be flat then ϕ(C) = ker(ϕ(H t )), which shows that ϕ maps check matrices for C to such for ϕ(C). Next, we clarify under which conditions ϕ is injective, and hence an order embedding. It is known, direct decompositions (cf. [1,  that tensoring preserves  Lemma 19.9]), i.e. if i∈I Ci = Rn then i∈I ϕ(Ci ) = S n. For our question we therefore obtain (cf. [11, 5.2.5]): Proposition 3. The restriction of the lattice mapping ϕ : L(R Rn ) −→ L(S S n ) to the set of all direct summands of R Rn is injective. We summarize what we have seen in terms of coding theory. From [8] recall a splitting code to be a linear code having a complement in its ambient module. Theorem 4. Let ϕ : L(R Rn ) −→ L(S S n) denote the lattice mapping induced by the ring embedding ϕ : R −→ S. 1. ϕ is join-preserving. Its restriction to the set of all splitting codes is injective. 2. If the module SR is flat then ϕ is meet-preserving. 3. If R is semisimple then ϕ is a lattice embedding. Proof. The first two statements result from our initial remarks together with Prop. 3 and Lem. 3. For our last claim we just remark that in the semisimple case every code is splitting and every R-module is flat because of projectivity (cf. [1, Cor. 17.4]).  

3

Matrix Representations of Algebras

In what follows, let K be a commutative ring, and R be a K-algebra. We are interested in unital embeddings R −→ Mm (K), for suitable m ∈ IN, and first give conditions under which these always exist. Proposition 4. A unital K-algebra embedding R −→ Mm (K) exists if and only if there exists a faithful (left module) operation of R on K m . Proof. Suppose there is an embedding σ : R −→ Mm (K). Then K m naturally obtains the structure of an R-module by r · x = σ(r)x for all x ∈ K m . Obviously r · K m = 0 implies r = 0, which means that R is operating faithfully on K m . If, conversely, K m is an R-module, then left-multiplication by elements of R is a K-endomorphism of K m , which provides a homomorphism σ : R −→ Mm (K) which is injective if and only if R K m is a faithful module.  

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Observing that R R is always a faithful module it is now clear that if R is free of rank m over K, then there is a natural faithful operation of R on K m . According to Prop. 4 this gives rise to an embedding of R in Mm (K). For a particular case this embedding is easy to construct. Proposition 5. Let K be an Artinian commutative local ring and let f := m i f x ∈ K[x] be a monic irreducible polynomial. Then the mapping β : i i=0 K[x] −→ Mm (K), g(x) → g(X) induces a unital embedding of K[x]/f into Mm (K) where   0 0 · · · 0 −f0  1 0 · · · 0 −f1      X :=  0 1 · · · 0 −f2  .  .. . . . . .. ..  . . . . .  0 0 · · · 1 −fm−1 The matrix X introduced in the last proposition is known as the companion matrix of the polynomial f. As we saw, it provides a particular embedding of K[x]/f into the ring Mm (K), and there immediately arises the question for a classification of all unital embeddings of K[x]/f into the ring Mm (K). In case of K being a field, this is well-known as a consequence of the following theorem. For a short proof see [10, p. 222]. Theorem 5. Let K be a field, and S be a simple K-algebra the center of which is K. Then every algebra homomorphism from a simple subalgebra R of S into R is the restriction of an inner automorphism of S. For the context at hand, there is a need for a statement clarifying the situation where K is a local ring. Problems of this type have been treated in [12] which contains a proof of the following: Theorem 6. Let K be an Artinian commutative local ring, and let f ∈ K[x] be a monic irreducible polynomial of degree m. Then each pair of unital embeddings of K[x]/f into the matrix ring Mm (K) are conjugate. We finally give four example of unital embeddings based on the previously outlined companion matrix approach. These embeddings will be used for new code representations in the following section. Example 1. 1. Let K := F2 and R := K[ξ] with ξ 2 + ξ + 1 = 0, i.e. R is the 4-element field. The companion matrix approach yields the embedding R −→ M2 (K) generated by

 01 ξ → . 11 2. Let K := F2 and R := K[ξ] with ξ 3 + ξ + 1 = 0 be the 8-element field. We will make use of the embedding R −→ M3 (K) generated by      −1  001 010 111 010 ξ →  1 1 0  =  0 0 1   1 0 1   0 0 1  . 010 110 100 110

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3. For K := F3 and the 9-element field R := K[ξ] with ξ 2 + ξ + 2 = 0 we will use the embedding R −→ M2 (K) generated by

ξ →



−1



 12 10 01 10 = . 11 12 12 12

4. Let K := ZZ 4 and consider R := K[ξ] with ξ 2 + ξ + 1 = 0, i.e. the Galois ring GR(4, 2). We introduce the embedding R −→ M2 (K) generated by

ξ →



−1



 21 31 03 31 = 11 10 13 10

which will play its role in a new representation of the quaternary Octacode.

4

Representations for Linear Codes

In the foregoing sections we dealt with different kinds of ring embeddings and clarified the induced correspondences between respective sets of linear codes. The goal of this section is to combine these results: Given a K-algebra R, a natural number m, and a unital embedding R −→ Mm (K), we study a kind of representation of all codes of length n over R by codes of length nm over K according to the following diagram. [n, k]-Codes over Mm (K)

.....................

Sect. 1

............ .........

[nm, km]-Codes over K

...... ................ ....... .... . . . .... .... ....

....... ............. ....... .... .... .... .... ...

Sect. 2,3

[n, k]-Codes over R

This may be illustrated next: Example 2. Let H(2, 3) be the extended binary Hamming Code of length 8 and dimension 4, and let S denote the ring of (2 × 2)-matrices over F2 . Row manipulations on the usual parity check matrix for H(2, 3) produce a matrix which is especially good for a representation of C via the four-element field, namely:   10101010 0 1 0 1 0 1 0 1  H :=  0 0 1 0 0 1 1 1. 00011110

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Considering the embedding F4 = F2 [ξ] −→ M2 (F2 ) as introduced in Ex. 1(1) it is obvious that the above matrix comes from an F4 -check matrix given by

1111 0 1 ξ ξ2



It checks a selfdual [4, 2, 3] code correcting one random error which carries over to the image code over M2 (F2 ), i.e. the latter code corrects up to one full matrix error. This yields more than can be directly seen from the distance properties of the code H(2, 3) that we started with: it proves in an elementary way that H(2, 3)—in addition to its single random error correction—is able to correct certain adjacent double error patterns. As a generalization we obtain: Theorem 7. For all odd r ∈ IN the extended binary Hamming code H(2, r) is induced by an F4 -linear code.

Optimal Binary Codes of Length 24 – The Binary Golay Code Pasquier [14] and Wolfmann [16] show, how to construct the binary Golay code from the extended self-dual [8, 4, 5]8 Reed-Solomon code over the eight element field (cf. also [6, p. 130]). If α is a primitive element of F8 satisfying α3 = α + 1 then the elements of the F2 -basis B := {α6 , α3 , α5 } of F8 satisfy the relation tr(xy) = δxy . The image of a coordinate-wise binary representation of the Reed-Solomon code at hand using B is the [24, 12, 8] Golay-Code. Note that the matrix representation of the mapping F8 −→ F8 , x → αx is given by  111 1 1 0, 100 

i.e. the matrix in Ex. 1(2). Representing all other F8 -Reed-Solomon codes in this way, we obtain binary linear codes of length 24 and dimensions 21, 18, 15, 9, 6, 3, respectively. Elementary row and column manipulations prove that the resulting codes have the parameters [24, 21, 2], [24, 18, 4], [24, 15, 4], [24, 9, 8], [24, 6, 8], [24, 3, 8]. According to [15] the first 4 of them are optimal. For the [24, 12, 8] Golay code this yields a generator matrix for the Wolfmann-description [16], which we write down as an illustration.

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

 100100100100100100100100 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0   0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1   0 0 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 0 1 1 0 0 0 1   0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1   0 0 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 0 1 0 1 1 0   0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 1 0   0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 1 0 1   0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 1 0 1 0 1 0   0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 0 1 0   0 0 0 0 1 0 1 0 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 000001101110111010100011 The Ternary Golay Code Goldberg [7] gives a construction of the [12, 6, 6]-Golay code as a ternary image of a self-dual code over F9 . We give a slight modification based on the embedding in Ex. 1(3). The matrix   1 0 0 1 ξ −ξ 0 1 0 ξ 1 ξ , 0 0 1 −ξ ξ 1 is a check matrix for a kind of [6, 3, 4] (hermitian self-dual) hexacode over F9 . Its image under the given embedding produces the ternary (6 × 12)-matrix   100000101221 0 1 0 0 0 0 0 1 1 1 2 2   0 0 1 0 0 0 1 2 1 0 1 2   0 0 0 1 0 0 1 1 0 1 1 1   0 0 0 0 1 0 2 1 1 2 1 0 000001221101 which by elementary row and column manipulations can be seen to be a check matrix for the extended ternary [12, 6, 6] Golay Code. The Octacode In Ex. 1(4) we introduced a unital embedding of the Galois ring GR(4, 2) into M2 (ZZ 4 ). It induces the (2 × 4)-check matrix

 1 0 ξ ξ2 0 1 ξ 2 −ξ for a self-dual code over GR(4, 2) (a kind of tetracode) being mapped to the (4 × 8)-matrix   10002113 0 1 0 0 1 1 3 2   0 0 1 0 1 3 2 3. 00013233

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Up to equivalence the latter matrix can be seen to be a check matrix for the Octacode.

Acknowledgments The authors like to thank AT&T Labs-Research for their support. They are in particular indebted to T. Y. Lam, E. Rains and N. J. A. Sloane for helpful discussions.

References 1. F. W. Anderson and K. R. Fuller, Rings and categories of modules, Springer Verlag, New York, Berlin, Heidelberg, 1992. 2. C. Bachoc, Applications of coding theory to the construction of modular lattices, J. Combin. Theory, Series A 78 (1997), 92–119. 3. A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over F2 + uF2 , Preprint, 1997. 4. N. Bourbaki, Elements of mathematics, ch. Commutative algebra, Addison-Wesley, Reading, MA, 1972. 5. A. E. Brouwer, Bounds on the size of linear codes, Handbook of coding theory, Vol. I, II, North-Holland, Amsterdam, 1998, pp. 295–461. 6. P. J. Cameron and J. H. van Lint, Designs, graphs, codes and their links, Cambridge University Press, Cambridge, 1991. 7. D. Y. Goldberg, Reconstructing the ternary Golay code, J. Combin. Theory Ser. A 42 (1986), no. 2, 296–299. 8. M. Greferath, Cyclic codes over finite rings, Discrete Math. 177 (1997), no. 1-3, 273–277. 9. A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sol´e, The ZZ 4 -linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40 (1994), 301–319. 10. N. Jacobson, Basic Algebra II, W. H. Freeman and Company, New York, 1989. 11. J. Lambek, Lectures on rings and modules, Blaisdell Publ. Co., Waltham, Massachusetts-Toronto-London, 1966. 12. A. A. Nechaev, Similarity of matrices over commutative Artinian rings, J. of Soviet. Math. 33 (1986), 1221–1237. 13. A. A. Nechaev, Kerdock codes in a cyclic form, Discrete Math. Appl. 1 (1991), 365–384. 14. G. Pasquier, The binary GOLAY code obtained from an extended cyclic code over F8 , European J. Combin. 1 (1980), no. 4, 369–370. 15. V. S. Pless, W. C. Huffman, and R. A. Brualdi (eds.), Handbook of coding theory. Vol. I, II, North-Holland, Amsterdam, 1998. 16. J. Wolfmann, A new construction of the binary Golay code (24, 12, 8) using a group algebra over a finite field, Discrete Math. 31 (1980), no. 3, 337–338.

On ZZ 4 -Simplex Codes and Their Gray Images Mahesh C. Bhandari, Manish K. Gupta, and Arbind K. Lal Department of Mathematics, Indian Institute of Technology, Kanpur, 208 016 India {mcb,mankg,arlal}@iitk.ac.in, [email protected]

Abstract. In [7] Rains has shown that for any linear code C over ZZ 4 , dH , the minimum Hammimg distance of C and dL , the minimum Lee distance of C satisfy dH ≥  d2L . C is said to be of type α(β) if dH =  d2L  (dH >  d2L ). In this paper we define Simplex codes of type α and β, namely, Skα and Skβ , respectively over ZZ 4 . Some fundamental properties like 2-dimension, Hamming and Lee weight distributions, weight hierarchy etc. are determined for these codes. It is shown that binary images of Skα and Skβ by the Gray map give rise to some interesting binary codes.

1

Introduction

The key motivation for studying codes over ZZ 4 , the ring of integers modulo 4 is that they can be used to obtain desirable types of good binary codes. Such codes have been studied recently in connection with the construction of lattices, sequences with low correlation and in a variety of other contexts [3]-[5],[10], [11]. Many good nonlinear binary codes of high minimum distances have a simple description as a linear codes over ZZ 4 . Being a linear code decoding becomes simplified. A linear code C, of length n, over ZZ 4 is a submodule of ZZ n4 . The minimum Hamming distance dH of C is given by dH = min{wH (x − y) : x, y ∈ C, x = y}, where wH (x) is the number of nonzero components in x. It is widely used for error correction/detection capabilities. Another distance which is not that widely used is the Lee distance. Lee weight of an element a ∈ ZZ 4 , denoted wL (a) is the minimum of {a, 4 − a}. Lee weight of a vector x ∈ ZZ n4 is the sum of Lee weights of its components and the minimum Lee distance of C is dL = min{wL (x − y) : x, y ∈ C, x = y}. In [7] Rains has shown that for any linear code C over ZZ 4 , dH ≥ 

dL . 2

C is said to be a code of type α(β) if dH =  d2L (dH >  d2L ). Note that the Octacode is of type β while Reed Muller code of first order is of type α. The Gray map φ : ZZ n4 → ZZ 2n 2 is the coordinate wise extension of the function from ZZ 4 to ZZ 22 defined by 0 → (0, 0), 1 → (0, 1), 2 → (1, 1) and 3 → (1, 0). Thus φ(C), the image of a linear code C over ZZ 4 of length n by the Gray map is a binary code of length 2n. If φ(C) is linear then C is called ZZ 2 -linear. Some well known binary nonlinear codes are images by the Gray map of linear codes over Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 170–179, 1999. c Springer-Verlag Berlin Heidelberg 1999


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ZZ 4 . Dual code C ⊥ of C is defined in the natural way. In [5] Hammons et al has shown that Kerdock and Preparata codes are dual over ZZ 4 and hence explains the duality relation between these nonlinear binary families. In this correspondence we define ZZ 4 - Simplex codes of type α &β, namely, Skα and Skβ and study the images of such codes under the gray map in two ways. In [9] Satyanarayana has shown that the Lee weight of every nonzero codeword in Skα is 22k . A characterization of such constant Lee weight codes has been obtained by Carlet [2]. We determine the fundamental parameters like 2-dimension, Hamming and Lee weight distributions and weight hierarchy for these codes. It is shown that both Skα and Skβ satisfy chain conditions. We also obtain some nonlinear and linear binary codes associated with these codes. Section 2 contains some preliminaries and notations. Definitions and basic parameters of ZB4 -simplex codes of type α and β are given in section 3. Section 4 deals with binary images of Simplex codes of type α and β under the gray map and some fundamental properties.

2

Preliminaries and Notations

A linear code C over ZZ 4 has a generator matrix G ( rows of G generate C) of the form
 Ik0 A B1 + 2B2 G= (1) 0 2Ik1 2C where A, B1 , B2 and C are matrices with entries 0 or 1 and Ik is the identity matrix of order k. Two codes are said to be equivalent if one can be obtained from the other by rearrangements of columns or by multiplying one or more coordinates by a unit in ZZ 4 . For each a ∈ ZZ 4 let a ¯ be the reduction of a modulo 2 then the code C (1) = {(c¯1 , c¯2 , . . . , c¯n ) : (c1 , c2 , . . . , cn ) ∈ C} is a binary linear code called the residue code of C. Another binary linear code associated with C is the torsion code C (2) which is defined by c C (2) = { : c = (c1 , . . . , cn) ∈ C and ci ≡ 0 (mod 2) for 1 ≤ i ≤ n}. 2 If k1 = 0 then C (1) = C (2) . For details and further references see [3],[8]. A vector v is a 2-linear combination of the vectors v1 , v2 , . . . , vk if v = l1 v1 + . . . + lk vk with li ∈ ZZ 2 for 1 ≤ i ≤ k. A subset S = {v1 , v2 , ..., vk } of C is called a 2-basis for C if for each i = 1, 2, ..., k − 1, 2vi is a 2−linear combination of vi+1 , ..., vk , 2vk = 0, C is the 2-linear span of S and S is 2-linearly independent [13]. The number of elements in a 2-basis for C is called the 2-dimension of C. It is easy to verify that the rows of the matrix   Ik0 A B1 + 2B2 (2) 2B1  B =  2Ik0 2A 0 2Ik1 2C form a 2-basis for the code C generated by G given in (1). The following lemma will be needed in Section 4.

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Lemma 1. The images of the rows of B under the gray map φ are linearly independent over ZZ 2 . Proof. Applying the gray map φ to the rows of B with a suitable rearrangement of rows yields a binary matrix of the type   01  11     ..  0 .     0 0 01     0 0 11     0 0 0 11      .  0 0 0 0 ..  0 0 0 0 0 11 


01 with k0 blocks of and k1 blocks of 1 1 . It is easy to see that the rows 11 of the the above matrix are linearly independent. A linear code C over ZZ 4 ( over ZZ 2 ) of length n, 2-dimension k, minimum Hamming distance dH and minimum Lee distance dL is called an [n, k, dH , dL ] ([n, k, dH ]) or simply an [n, k] code. A necessary and sufficient condition for ZZ 2 -linearity is given by the following theorem. Theorem 1. [5] C is ZZ 2 -linear if and only if whenever c = (c1 , . . . , cn ), c = (c1 , . . . , cn ) ∈ C, 2¯ c c¯ = (2c¯1 c¯1  , . . . , 2c¯nc¯n  ) ∈ C. Thus, if C is an [n, k, dH , dL ] ZZ 2 -linear code over ZZ 4 then φ(C) is a binary linear [2n, k, dL ] code. Hence by the Griesmer bound for binary linear codes [6] we have n≥

k−1 1  dL  . 2 i=0 2i

(3)

Note that the Octacode O8 meets the bound given by (3) even though it is not ZZ 2 -linear. For 1 ≤ r ≤ k, the r-th Generalized Hamming weight of C is defined by dr (C) = min{wS (Dr ) : Dr is an [n, r] subcode of C}, where wS (D), called support size of D, is the number of coordinates in which some codeword of D has a nonzero entry. The set {d1 (C), d2 (C), . . . , dk (C)} is called the weight hierarchy of C. In [15] Yang et al has obtained a lemma connecting Lee weights to the support size of a subcode. Thus for any linear code C over ZZ 4 , dr (C) may also be defined by dr (C) =

 1 min{ wL (d) : Dr is an r − dimensional subcode of C} r 2 d∈Dr

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The two definitions are equivalent. C is said to satisfy the chain condition if there exists a chain D1 ⊆ D2 ⊆ · · · ⊆ Dk , of subcodes of C satisfying wS (Dr ) = dr (C), 1 ≤ r ≤ k. In [14] Yang and Helleseth has shown that the Goethals code over ZZ 4 of length 8 satisfies chain condition. A relation between dr (C) and dr (C ⊥ ) is given by the following theorem. Theorem 2. [1] Let C be an [n, k] linear code over ZZ 4 Then {dr (C) : 1 ≤ r ≤ k} = {1, 1, 2, 2, . . . , n, n}\{n + 1 − dr (C ⊥ ) : 1 ≤ r ≤ 2n − k}.

3

ZZ 4 -Simplex Codes of Type α and β

Let Gk be a k × 22k matrix over ZZ 4 consisting of distinct columns. Inductively, Gk may be written as
 00 · · · 0 11 · · · 1 22 · · · 2 33 · · · 3 Gk = Gk−1 Gk−1 Gk−1 Gk−1 with G1 =[0123]. The code Skα generated by Gk has been visited earlier [9,2]. In [9] Satyanarayana has shown that the Lee weight of every nonzero codeword of Skα is 22k . While Carlet, in [2] has classified all constant Lee weight codes over ZZ 4 . Clearly, the 2-dimension of Skα is 2k. The following observations are useful in determining Hamming (Lee) weight distributions of Skα. α and if i = (i, i, i, ..., i) Remark 1. If Ak−1 denotes an array of codewords in Sk−1 α then an array of all codewords of Sk is given by

 Ak−1 Ak−1 Ak−1 Ak−1  Ak−1 1 + Ak−1 2 + Ak−1 3 + Ak−1     Ak−1 2 + Ak−1 Ak−1 2 + Ak−1  Ak−1 3 + Ak−1 2 + Ak−1 1 + Ak−1 

Remark 2. If R1 , R2 , ..., Rk denote the rows of the matrix Gk then wH (Ri ) = 3 · 22k−2 , wH (2Ri ) = 22k−1 and wL (Ri ) = 22k = wL (2Ri ). It may be observed that each element of ZZ 4 occurs equally often in every row of Gk . In fact we have the following lemma. Lemma 2. Let c ∈ Skα . If one of the coordinates of c is a unit then every element of ZZ 4 occurs 4k−1 times as a coordinate of c. Otherwise wH (c) = 22k−1 .

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α Proof. By Remark 1, any x ∈ Sk−1 gives rise to the following four codewords of α Sk



y1 = x x x x ,



y2 = x 1 + x 2 + x 3 + x ,



y3 = x 2 + x x 2 + x and y4 = x 3 + x 2 + x 1 + x . Hence by induction, the assertion follows. Let G(Sk ) ( columns consist of all nonzero binary k-tuples) be a generator matrix for an [n, k] binary simplex code Sk . Then the extended binary simplex   0   code Sˆk is generated by the matrix G(Sˆk ) =  ... G(S )  . Inductively, k

0
G(Sˆk ) =

 00 . . . 0 11 . . . 1 ˆ ) G(Sk−1 ˆ ) . G(Sk−1

(4)

Lemma 3. The torsion code of Skα is equivalent to the 2k copies of Sˆk . Proof. Observe that the torsion code of Skα is the set of codewords obtained by replacing 2 by 1 in all 2-linear combination of the rows of the matrix
 00 · · · 0 22 · · · 2 00 · · · 0 22 · · · 2 2Gk = . (5) 2Gk−1 2Gk−1 2Gk−1 2Gk−1 Now the assertion follows by induction on k and by regrouping the columns in (5) according to (4). As a consequence of Lemmas 2 and 3 one obtains Hamming and Lee weight distributions of Skα . Theorem 3. The Hamming and Lee weight distribution of Skα are : 1. AH (0) = 1, AH (22k−1 ) = 2k − 1, AH (3 · 22k−2 ) = 2k (2k − 1), and 2. AL (0) = 1, AL (22k ) = 22k − 1. where AH (i) (AL (i)) denotes the number of vectors of Hamming (Lee) weight i in Skα. Proof. By Lemma 2, each nonzero codeword of Skα has Hamming weight either 3 · 4k−1 or 22k−1 and Lee weight 22k . Since dimension of the torsion code is k, there will be 2k − 1 codewords of the weight 22k−1 . Hence the number of codewords having weight 3 · 4k−1 will be 4k − 2k . Remark 3. 1. Skα is an equidistant code with respect to Lee distance whereas Sk is an equidistant binary code with respect to Hamming distance. 2. Skα is of type α.

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Let c = (c1 , c2 , . . . , cn) ∈ C and let ωj (c) = |{k : ck = j}|. Then the correlation √ of c ∈ C is defined as θ(c) = (ω0 (c) − ω2 (c)) + i(ω1 (c) − ω3 (c)); where i = −1 [11]. The  Symmetrized weight enumerator (swe) of C over ZZ 4 is given by swe(x, y, z) = c∈C xω0 (c)y ω1 (c)+ω3(c)z ω2 (c) [8]. Let S¯kα be the punctured code of Skα obtained by deleting the zero coordinate. Then the swe of S¯kα is   2k−1 swe(x, y, z) = xn(k) + (2k − 1)(xz)n(k−1) z 1 + 2k y 2 , where n(k) = 4k − 1 and correlation of any c ∈ S¯kα is given by θ(c) = −1.

(6)

The length of Skα is large as compare to its 2-dimension and increases fast with increment in its 2-dimension. But one can always drop some columns from Gk in a specific manner to yield good codes over ZZ 4 in the sense of having maximum possible Lee weights for the given length and 2-dimension. Let Gβk be the k × 2k−1 (2k − 1) matrix defined inductively by
 1111 0 2 Gβ2 = , 0123 1 1 and for k > 2 Gβk =




 11 · · · 1 00 · · · 0 22 · · · 2 , Gk−1 Gβk−1 Gβk−1

α . Note that Gβk is obtained from Gk where Gk−1 is the generator matrix of Sk−1 by deleting 2k−1 (2k + 1) columns. By induction it is easy to verify that no two columns of Gβk are multiple of each other. Let Skβ be the code generated by Gβk . Note that Skβ is a [2k−1 (2k − 1), 2k] code. To determine Hamming (Lee) weight distributions of Skβ we first make few observations. β α Remark 4. If Ak−1 (Bk−1 ) denotes an array of codewords in Sk−1 (Sk−1 ) and if β i = (i, i, . . . , i) then an array of all codewords of Sk is given by   Ak−1 Bk−1 Bk−1  1 + Ak−1 Bk−1 2 + Bk−1     2 + Ak−1 Bk−1 Bk−1  3 + Ak−1 Bk−1 2 + Bk−1

Remark 5. Each row of Gβk has Hamming weight 2k−3 [3(2k − 1) + 1] and Lee weight 2k−1 (2k − 1). Proposition 1. Each row of Gβk contains 22(k−1) units and ω0 = ω2 = 2k−2 (2k−1 − 1).

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Proof. Clearly, assertion holds for the first row. Assume that the result holds for each row of Gβk−1 . Then the number of units in each row of Gβk−1 is 22(k−2). By Lemma 2, the number of units in any row of Gk−1 is 22k−3 . Hence the total number of units in any row of Gβk will be 22k−3 + 2 · 22(k−2) = 22(k−1). A similar argument holds for the number of 0’s and 2’s. Infact, similar to Skα , we have the following lemma. Lemma 4. Let c ∈ Skβ . If one of the coordinates of c is a unit then ω1 + ω3 = 22(k−1) and ω0 = ω2 = 2k−2 (2k−1 − 1). Otherwise ω0 = 2k−1 (2k−1 − 1), ω2 = 22(k−1) and ω1 = ω3 = 0. β α and y2 ∈ Sk−1 such Proof. Let x ∈ Skβ then, by Remark 4, there exist y1 ∈ Sk−1 that x can have any of the following four forms: x = (y1 |y2 |y2 ), x = (1 + y1 |y2 |2 + y2 ) Now the assertion follows by inx = (2 + y1 |y2 |y2 ) or x = (3 + y1 |y2 |2 + y2 ) duction and Lemma 2,

The proof of the following lemma, being similar to the proof of Lemma 3, is omitted. Lemma 5. The torsion code of Skβ is equivalent to the 2k−1 copies of the binary simplex code Sk . The proof of the following theorem, being similar to the proof of Theorem 3 is omitted. Theorem 4. The Hamming and Lee weight distributions of Skβ are : 1. AH (0) = 1, AH (22k−2 ) = 2k − 1, AH (2k−3 [3(2k − 1) + 1]) = 2k (2k − 1) ,and 2. AL (0) = 1, AL (22k−1 ) = 2k − 1, AL (2k−1 (2k − 1)) = 2k (2k − 1). Remark 6. 1. Skβ is of type β. 2. The correlation of each nonzero codeword of Skβ with components 0’s or 2’s is −2k−1 . 3. The swe of Skβ is given as swe(x, y, z) = xn(k) + (2k − 1)x2n(k−1) z 4

k−1

+ 2n(k)xn(k−1) y 4

k−1

z n(k−1)

where n(k) = 2k−1 (2k − 1).

4

Gray Image Families

Let C be an [n, k, dH , dL ] linear code over ZZ 4 . Then φ(C) is a binary code having 2k codewords of length 2n, and minimum Hamming distance dL . However φ(C) need not be linear. Let B be the matrix ( given in (2)) whose rows form a 2-basis for C and let φ(B) be the matrix obtained from B by applying the gray map to each entry of B. The code Cφ generated by φ(B) is a [2n, k, ≥  d2L ] binary linear code. Note that φ(C) and Cφ have same number of codewords but they are not equal in general. The following proposition shows that both φ(S¯kα) and φ(Skβ ) are not linear.

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Proposition 2. φ(S¯kα ) and φ(Skβ ) are nonlinear for all k. Proof. Let R1 , R2 , . . . Rk be the rows of the generator matrix Gk (Gβk ). Let c = Rk (R1 ) and let c = Rk−1 (Rk ) then by (6) ( Remark 6.2 ), 2¯ c c¯ ∈ / S¯kα (Skβ ). Hence, by Theorem 1, the result follows. Remark 7. 1. φ(S¯kα ) is a binary nonlinear code of length 22k+1 −2 and minimum Hamming distance 22k . It meets the Plotkin bound [6] and n < 2dH . 2. φ(Skβ ) is a binary nonlinear code of length 2k (2k −1) and minimum Hamming distance 2k−1 (2k − 1). This is an example of a code having n = 2dH [6]. 3. Even though, both S¯kα and Skβ are not ZZ 2 −linear they meet the bound given by (3). Skβ .

The next two results are about the binary linear codes obtained from S¯kα and

Theorem 5. Let C = S¯kα. Then Cφ is an [22k+1 − 2, 2k, 22k ] binary linear code consisting of two copies of binary simplex code S2k with Hamming weight distribution same as the Lee weight distribution of S¯kα . Proof. By Lemma 1, Cφ is a binary linear code of length 22k+1 − 2 and dimension 2k. Let Gk be a generator matrix of Skα in 2-basis form. Then   0...0 1...1 2...2 3...3  0...0 2...2 0...0 2...2   Gk =   Gk−1 Gk−1 Gk−1 Gk−1  2Gk−1 2Gk−1 2Gk−1 2Gk−1 and



 0000 . . . 00 0101 . . . 01 1111 . . . 11 1010 . . . 10 φ(Gk ) =  0000 . . . 00 1111 . . . 11 0000 . . . 00 1111 . . . 11  . φ(Gk−1 ) φ(Gk−1 ) φ(Gk−1 ) φ(Gk−1 )

The proof is by induction on k. Assume that φ(Gk−1 ) yields a [22k−1 , 2(k − 1), 22k−2 ] binary code in which every nonzero codeword is of weight 22k−2 . Then the possible nonzero weight from the lower portion of the above matrix will be 4 · 22k−2 = 22k . From the structure of first two rows of φ(Gk ), it is easy to verify that any linear combination of these rows with other rows has weight 22k . Puncturing the first two columns and rearranging the columns yields the code having two copies of S2k . Theorem 6. Let C = Skβ . Then Cφ is the binary MacDonald code M2k,k : [22k − 2k , 2k, 22k−1 − 2k−1 ] with Hamming weight distribution same as the Lee weight distribution of Skβ . Proof. It follows by induction and is similar to the proof of Theorem 5.

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The weight hierarchy of Skα and Skβ are given by the following two theorems. Theorem 7. Skα satisfies the chain condition and its weight hierarchy is given by r  dr (Skα ) = 22k−i = 22k − 22k−r ; 1 ≤ r ≤ 2k. i=1

Proof. By Remark 3, Any r-dimensional subcode of Skα is of constant Lee weight. Hence by definition, dr (Skα ) =

1 r (2 − 1)22k = 22k − 22k−r . 2r

Let D1 =< 2R1 >, D2 =< 2R1 , 2R2 >, D3 =< R1 , 2R1 , 2R2 >, D4 =< R1 , 2R1 , R2 , 2R2 >, . . . , D2k =< R1 , 2R1 , . . . , Rk , 2Rk > . It is easy to verify that D1 ⊆ D2 ⊆ · · · ⊆ D2k , and wS (Dr ) = dr (Skα ) for 1 ≤ r ≤ 2k. Theorem 8. Skβ satisfies the chain condition and its weight hierarchy is given by r dr (Skβ ) = n(k) − 2k−r−1 (2k − 2 2  ) 1 ≤ r ≤ 2k. where n(k) = 2k−1 (2k − 1). Proof. The proof follows by induction on k. Clearly the result holds for k = 2. β Assume that the result holds for Sk−1 . Hence if 1 ≤ r ≤ 2k − 2 then there β exists an r-dimensional subcode of Sk−1 with minimum support size n(k − 1) − k−r−2 k−1  r2  (2 − 2 ). By Remark 4, 2 β α dr (Skβ ) = 2dr (Sk−1 ) + dr (Sk−1 )

(7)

α But all r-dimensional subcodes of Sk−1 have constant support size (22k−2 − 2k−2−r ). Thus simplifying (7) yields the result. For r = 2k − 1 and 2k the result 2 can be easily proved. Let D1 =< 2R1 >, D2 =< R1 , 2R1 >, D4 =< R1 , 2R1 , R2 , 2R2 >, ... , D3 =< R1 , 2R1 , 2R2 >, D2k =< R1 , 2R1 , . . . , Rk , 2Rk > . Then it is easy to see that

D1 ⊆ D2 ⊆ · · · ⊆ D2k is the required chain of subcodes. The dual code of Skα is a code of length 22k and 2-dimension 22k+1 − 2k, whereas the dual code of Skβ is a code of length 2k−1 (2k − 1) and 2-dimension 22k −2k −2k. The Hamming and Lee weight distributions of these dual codes can be obtained with the help of Theorems 3 and 4 and the MacWilliams Identies

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[8]. Similarly, the weight hierarchies of duals can be obtained from Theorems 2, 7 & 8. In [12] Sun and Leib have considered the dual of Skβ (k ≥ 3) in a different context. They have used combinatorial arguments to obtain a code of length n = 2r−1(2r − 1), redundancy r and minimum squared noncoherent weight N + 1 − (N − 2)2 + 9, where N = n − 1. They have further punctured these codes to get some good codes in the sense of having larger coding gains over noncoherent detection. The results proved in sections 3 and 4 are easily extendible to simplex codes over ZZ 2s by suitably modifying the definition of the Gray map.

References 1. A.Ashikhmin, “On Generalized Hamming Weights for Galois Ring Linear Codes”, Designs, Codes and Cryptography, 14, 107-126 (1998) 2. C. Carlet, “One weight ZZ 4 -Linear Codes ” Int. Conf. on Coding, Crypto and related Topics, Mexico (1998), To appear in Springer Lecture Notes in Computer Science 3. J.H.Conway and N.J.A. Sloane, “Self-Dual Codes over the Integers Modulo 4”, Jr. of Comb. Theory, Series A 62, 30-45 (1993) 4. M. Greferath and U. Vellbinger, “On the Extended Error-Correcting Capabilities of the Quaternary Preparata Codes”, IEEE Trans. Inform. Theory, vol 44, No. 5, 2018-2019 (1998) 5. A.Hammons, P.V.Kumar, A.R.Calderbank, N.J.A.Sloane and P.Sole, “The ZZ 4 Linearity of Kerdock, Preparta, Goethals, and Related Codes”, IEEE Trans. Inform. Theory, vol 40, No. 2, 301-319 (1994) 6. F.J.MacWilliams and N.J.A.Sloane, The Theory of Error-correcting codes, NorthHolland, New York (1977). 7. E.M.Rains, “Optimal Self Dual Codes over ZZ 4 ”, Discrete Math., vol 203, pp. 215-228, (1999) 8. E.M.Rains and N.J.A.Sloane, Self Dual Codes: The Handbook of Coding Theory, Eds V. Pless et al, North-Holland (1998) 9. C. Satyanarayana, “ Lee Metric Codes over Integer Residue Rings ”. IEEE Trans. Inform. Theory, vol 25, No. 2, pp. 250-254 (1979) 10. A.G.Shanbhag, P.V.Kumar and T.Helleseth, “Improved Binary Codes and Sequence Families from ZZ 4 - Linear Codes”, IEEE Trans. Inform. Theory, vol 42, No. 5, 1582-1587 (1994) 11. P.Sole, “A Quaternary Cyclic Code, and a Family of Quadriphase Sequences with Low Correlation Properties”, Lecture Notes in Computer Science, vol 388, Springer-Verlag, NY, 193-201 (1989) 12. F.W.Sun and H.Leib, “Multiple-Phase Codes for Detection Without Carrier Phase Reference”, IEEE Trans. Inform. Theory, vol 44, No. 4, 1477-1491 (1998) 13. V.V.Vazirani, H.Saran and B.Sundar Rajan, “An Efficient Algorithm for Constructing Minimal Trellises for Codes over Finite Abelian Groups”, IEEE Trans. Inform. Theory, vol 42, No. 6, 1839-1854 (1996) 14. K.Yang and Tor Helleseth, “On the Weight Hierarchy of Goethals Codes over ZZ 4 ”, IEEE Trans. Inform. Theory, vol 44, No. 1, 304-307 (1998) 15. K.Yang, T.Helleseth, P.V.Kumar and A.G.Shangbhag, “On the Weight Hierarchy of Kerdock Codes over ZZ 4 ”, IEEE Trans. Inform. Theory, vol 42, No. 5, 1587-1593 (1994)

Some Results on Generalized Concatenation of Block Codes M. Bossert1 , H. Grießer1 , J. Maucher1, and V.V. Zyablov2
1

2

University of Ulm, Dept. of Information Technology Albert-Einstein-Allee 43, 89081 Ulm, Germany [email protected] Institute for Information Transmission Problems, Russian Academy of Science B. Karetnyi per. 19, 101447, Moscow GSP-4, Russia

Abstract. We consider generalized concatenation of block codes. First we give a short introduction on the notion for concatenated and errorlocating codes. Then an estimation of the hard decision error correcting capacity of concatenated codes beyond half the minimum distance is presented.

1

Introduction

Code concatenation is a useful method for constructing long codes from shorter ones. It is possible to decode these concatenated codes via their component codes, leading to a reduced decoding complexity. Moreover, many errors with weight exceeding half the minimum distance of the code can be corrected. The concatenation of block codes (CC codes) was introduced by Forney [3]. Blokh and Zyablov enhanced this definition resulting in so called generalized concatenation (GCC) that includes Forney’s approach as a special case. Later Zinov’ev [7] modified the definition of generalized concatenation in order to include nonlinear component codes. The principle of error-locating codes (EL codes) was first described by Wolf [6,5]. In [9,1] Zyablov investigated this class of codes and generalized the construction in a similar way as it was done for concatenated schemes. This led to so called generalized error-locating codes (GEL codes). GEL codes are a subclass of the GCC code class given by the definition in [7]. A formal proof of this can be found in [10,11]. For a detailed introduction on generalized concatenation see [2]. Usually the error correcting and detecting capabilities of a block code are solely characterized by the minimum Hamming distance. This is justified by using an (algebraic) decoding algorithm that can decode any error pattern with a weight up to half the minimum distance and will usually fail if an error of higher weight occurs. However, if we use a decoding algorithm that allows the correction of error patterns with weight beyond half the minimum distance this


This work was supported by Deutsche Forschungsgemeinschaft (DFG), Germany.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 181–190, 1999. c Springer-Verlag Berlin Heidelberg 1999


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should be used for an estimation of the decoding performance. We present a modified approach for the classification of the correcting capability of concatenated schemes that leads to tighter bounds on the bit error and block error rate of the generalized concatenated code.

2

Code Definitions

A code C of length n, dimension k and minimum Hamming distance d with symbols from GF(q) will be denoted by C(n, k, d)q . For the binary case the symbol q may be omitted. Let GF(q m ) be an extension field of order q m , q is a prime power. An element a ∈ GF(q m ) in exponential representation will be denoted by a if it is given as a vector consisting of m coefficients of its polynomial representation. Consequently, a vector a ∈ GF(q m )n can be written as an m × n matrix a with entries from GF(q). 2.1

Code Partitioning

Let B be a linear code with codewords b ∈ B. A µ-way partition of B is con
structed from µ disjoint subcodes Ba , a = 1, . . . , µ, for a Ba = B. We will denote such a partition by B/B . µ is called the order of the partition and a is a label identifying a unique subset Ba . If a set is partitioned L times by repeated partitioning of the subsets we get an L-level partition chain B (1)/B (2)/ · · · /B (L). Subsets in such a partition chain are labeled by a multi-part label, e.g. the label (a(1) , a(2) , . . . , a(L) ) denotes a subset in the last partition level of an L-level partitioning. The subcode that contains the all zero vector (linear subcode) will be labeled with a ‘0’. d(B) denotes the minimum Hamming distance of a code. If a subcode B (l) consists of only one single codeword we define d(B (l)) = ∞. (l+1) A linear subcode B0 of the code B (l) induces a partition of B (l) into cosets. A coset is given by adding any element of this coset, say b, to the linear (l+1) (l+1) (l+1) (l+1) subcode B0 : Ba(l) = B0 + b, b ∈ B0 . Therefore b is called coset representative of the subcode with label a(l). The union of all cosets adds to  
(l+1) B (l): B (l) = b∈[B(l) /B(l+1) ] B0 + b, where B (l)/B (l+1) is a set of all coset representatives. There are sets of coset representatives that form a linear code and can be described by a generator matrix GB(l) /B(l+1) . GB(l) /B(l+1) describes the mapping of the label a(l) to a coset representative of level l. Hence it is (L) possible to calculate a code word of the subcode b ∈ Ba(1) ,... ,a(L) :    GB(L) /B(L+1) a(L)     .. ·  ...  , where GB(1) /B(L+1) =  . . 

b = GTB(1) /B(L+1)

a(1)

GB(1) /B(2)

GB(1) /B(L+1) is the combination of all generator matrices of coset representatives.

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As the mapping done by the generator matrices is unique, there exists an inverse mapping which produces the label of a partition at level l if a coset element is given. The matrix describing this mapping is a (partial) parity check (l+1) matrix of the linear subcode B0 and will be denoted by H B(l) /B(l+1) . The combination of all the matrices H B(l) /B(l+1) , l = 1, . . . , L, defines the linear mapping from a given coset to the label:   (L)   H B(L) /B(L+1) a     .. H B(1) /B(L+1) · b =  ...  , where H B(1) /B(L+1) =  . . H B(1) /B(2)

a(1)

A code B(nb , nb , 1)q consisting of all possible nb -tuples will be denoted by GF(q)nb and the single codeword code B(nb , 0, ∞) by {0}. Note that GB(l) /{0} = GB(l) and H GF(q)nb /B(l) = H B(l) . 2.2

Generalized Concatenated Codes

The definition of GCC codes is based on the description given by Zinov’ev in [7] and is not restricted to linear component codes. Definition 1 (GCC Code). An L-level generalized concatenated code (GCC (l) (l) code) C consists of L outer codes A(l)(na , ka , da ) ma(l) and an inner code q

(1)

(1)

B (1)(nb , kb , db )q that is partitioned into a L-level partition chain (1) (1) (L+1) (L+1) B (1)(nb , kb , db )q / · · · /B (L+1)(nb , kb = 0, db = ∞)q = b. The sub(l) codes of level L + 1 consist of a single codeword only. The symbol ai of the (l) lth outer code determines a subset of B (l)a(1) ,a(2) ,... ,a(l−1) . Thus the symbols ai , i

i

i

(1)

(L)

l = 1, . . . , L, of all outer codes together form the label (ai , . . . , ai ) of a unique codeword bi . The codeword of the GCC code consists of all the codewords bi , i = 1, . . . , na . The construction principle of GCC codes is illustrated in figure 1. The GCC

L (l) (l) code has length nc = na · nb , dimension kc = l=1 ka · ma , and a minimum (l) (l) Hamming distance dc ≥ minl=1,... ,L {da · db } (see [1,4]). We will assume that (l) (l) (l+1) (L+1) m a = kb − kb , kb = 0, for all l = 1, . . . , L. If L = 1 the GCC code reduces to an ordinary concatenated code. (l) Encoding can be done by first encoding ka information symbols into a code(l) word a for l = 1, . . . , L and then mapping the label (a(1) , . . . , a(L) ) on to the codeword b. This encoding scheme, however, will not be systematic. Example 1 (GCC Code). We construct a two-level GCC code using a partition chain of the inner binary Hamming code B (1)(7, 4, 3)/B (2)(7, 3, 4)/B (3)(7, 0, ∞) and two outer codes A(1)(7, 1, 7)2 (a repetition code) and A(2)(7, 4, 4)23 . The parameters of the generalized concatenated code construction are nc = 49, kc = (1) (1) (2) (2) ka · ma + ka · ma = 1 + 12 = 13 and dc ≥ min{7 · 3, 4 · 4} = 16.

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···

a(L)

(L)

ma

..

L outer codes

···

a(1)

(1)

ma

columnwise mapping defined by a L-level partition chain of code B (1)

···

C

nb

Fig. 1. Construction principle of generalized concatenated codes.

2.3

Generalized Error-Locating Codes

Error-locating codes can be defined by the parity check matrices of the component codes [5]. Definition 2 (GEL Code). An L-level generalized error-locating code (GEL (l) (l) code) consists of L outer codes A(l)(na , ka , da ) m(l) , l = 0, . . . , L − 1, and L a q

(l)

(l)

inner codes B (l)(nb , kb , db )q , l = 1, . . . , L. Let 

 H B(l−1) /B(l)   ..   . H B(1) be a parity check matrix of the inner code B (l) and a(l) a codeword of the lth outer code A(l). Each codeword C of a GEL code in matrix form fulfills a(l) = H B(l) /B(l+1) · C, a(l) ∈ A(l)

(1)

for all l = 0, . . . , L − 1. The defining equation (eq. 1) of a GEL code is equivalent to the calculation of the syndrome. Therefore the codevector a(l) is also known as syndrome. Notice that the columns of the codeword matrix C are in general not codewords of B (1) or one of its subcodes. (l) (l) The redundancies of outer and inner codes are given by ra = na − ka and (l) (l) (l) (l) (l+1) (0) rb = nb − kb respectively. We require that ma = kb − kb , kb = n b , for all l = 0, . . . , L − 1. The GEL code has length nc = na nb , redundancy

L−1 (l) (l)

L−1 (l) (l) (L) rc = l=0 ra ma and dimension kc = na kb + l=0 ka ma = nc − rc . For L = 1 the GEL code reduces to an ordinary EL coding scheme.

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na X (L−1) information X (0) symbols

(L−1)

kb

111111111 000000000 P 111111111 000000000 .. 000000000000 111111111111 00000000000 11111111111 000000000000 111111111111 00000000000 11111111111 111111111111111 000000000000000 P 111111111111111 000000000000000  

C

nb

(L−1)

(0)

a(L−1) S =  ...  = H B(L) · C a(0)

(L)

ka

a(L−1)

u(L−1) (1)

ka

a(0)

u(0)

r(L−1)

(L−1)

ma

.. r(0)

(0)

ma

Fig. 2. Systematic encoding of generalized error-locating codes. We consider systematic encoding. The encoders for the outer codes should be systematic. Further we assume that the matrices H B(l−1) /B(l) are given by  (L−1)    Q I 0 0 . . . 0 (L−1) b ma H B(L−1) /B(L)   H B(L−2) /B(L−1)  Q(L−2) I m(L−2) 0 . . . 0  b a       ..  ..  , .. .. .. H B(L) =  = .  . . . .      H B(1) /B(2)   Q(1) I m(1) 0   b  a (0) H B(1) I m(0) Qb a

(l)

(l)

where I m(l) is a ma × ma identity matrix. Note that any H B(L) describing the a partition chain and its labeling of the inner code, can be transformed into such a representation without destroying the nested partition. This is possible if row operations within H B(l−1) /B(l) and column permutations of H B(L) are allowed. Of course, permuting the columns of H B(L) will lead to an equivalent code, but without changing the properties of the concatenated code. The matrix H B(L) cannot be made systematic while preserving the partition structure. Therefore, there is still a nonsystematic mapping that combines outer and inner codes. Figure 2 illustrates the encoding process. After filling the white part of the codeword matrix C with information symbols the information part u(l) of every outer codeword can be calculated using equation 1. Note that this can be done without knowing any of the redundancy matrices in C denoted by P (l) . Then the redundancy part r (l) of the outer codewords, l = 0, . . . , L − 1, results from ordinary systematic encoding. In the last step the matrices P (l) will be calculated. If we denote the upper right submatrix by X (l) (see figure 2) then the following equation holds for all l = 0, . . . , L − 1  

X (l)  (l) (l) (l) (l) r = Qb I m(l) = Qb X (l) + P (l) ⇒ P (l) = r (l) − Qb X (l). (l) a P

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As P (l+1) is part of X (l) that is needed for the calculation of P (l), the recursive calculation of the matrices P (l) starts with l = L − 1. Example 2 (GEL Code). A two-level GEL code is constructed using the inner codes B (1)(7, 6, 2) and B (2)(7, 3, 4), and the outer codes A(0)(7, 1, 7)2 and A(1)(7, 4, 4)23 . A sample parity check matrix H B(2) is   1011000   1 1 1 0 1 0 0 H B(1) /B(2)  H B(2) = = 1 1 0 0 0 1 0 . H B(1) 1111111 (2)

The resulting concatenated code has length nc = 49 and dimension kc = na kb + (0) (0) (1) (1) ka ma + ka ma = 34. Description of the Decoding Principle: A codeword C of the generalized concatenated code, given in matrix form, will be corrupted by some error pattern ˜ = C + E. DecodE ∈ GF(q)nb ×na after transmitting it over a noisy channel: C ing of generalized error-locating codes can be done by means of a multi-stage decoding algorithm. Decoding starts with calculating the first syndrome part ˜ Then a ˜ (0) = H B(0) /B(1) C. ˆ (0) providing the labels of the ˜ (0) is decoded into a a cosets needed to do decoding of ˜bj , j = 1, . . . , na , in some (usually) nonlinear subspace of B (0). In the second stage of the decoding first an additional part of ˜ (1) is performed and so the syndrome (˜ a(1)) is calculated. Then decoding of a on. The information of corrupted columns of the code matrix resulting from the ˜ (1). This previous decoding step can be used to erase some of the symbols of a decoding principle gave rise to the term error-locating codes. If decoding of the outer codes fails for any step then decoding of the concatenated code will fail, too. 2.4

The Connection between GCC and GEL Codes

A formal proof of this result can be found in [10,11]. Both code concatenation descriptions are based on the same partition principle of the inner code. The main difference is the labeling of the partition. For the case of GEL codes, the parity check matrix H B(L) is specified, whereas for GCC codes the labeling is not defined. If labeling of the subcodes is done by a linear mapping and consequently can be described via a matrix GB(1) then both descriptions are identical: GTB(1) · S = C

⇐⇒ S = H B(L) · C,

provided that GTB(1) = H −1 . Since H B(L) has full rank, it is always possible to B(L) find such a GB(1) . Therefore both constructions can be used to describe the same code: A linear L-level GCC code can be described as a special case of an (L + 1)-level GEL code and an L-level GEL code is a special case of an (L + 1)-level GCC code.

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L-level GEL −→ (L+1)-level GCC: An L-level GEL code with inner component (l) (l) codes B (l)(nb , nb , db )q , l = 1 . . . , L, and outer component codes (l) (l) A(l)(na , na , da ) m(l) , l = 0 . . . , L − 1, is a special case of an (L + 1)-level GCC a q

code with the additional codes B (0)(nb , nb , 1)q and A(L) (na , na , 1)

(L)

q kb

. Note that

indexing of the component codes has only been shifted by 1 compared to the original definition of GCC codes. This description reveals the minimum distance (l) (l) of the code: dc ≥ minl=0,... ,L {da db }. Example 3. The two-level GEL code from example 2 shall be interpreted as a three-level GCC code. For the GCC description we need two more component codes: B (0)(7, 7, 1) and A(2)(7, 7, 1)23 . The minimum distance of the code is dc ≥ (0) (1) (1) (2) min{da · 1, da · db , 1 · db } = 4. L-level GCC −→ (L + 1)-level GEL: An L-level GCC code with inner and outer (l) (l) (l) (l) component codes B (l)(nb , nb , db )q and A(l)(na , na , da ) ma(l) , l = 1 . . . , L, is a q

special case of an (L + 1)-level GEL code with the additional codes B (L+1)(nb , 0, ∞)q and A(0)(na , 0, ∞) r(1) . q

b

Example 4. An interpretation of the two-level GCC code from example 1 with B (1)(7, 4, 3), B (2)(7, 3, 4), A(1)(7, 1, 7)2 , and A(2)(7, 4, 4)23 as a three-level GEL code requires the additional component codes B (3)(7, 0, ∞) and A(0)(7, 0, ∞)23 . Since an L-level GEL code is only a special case of an (L + 1)-level GCC code, decoding algorithms for GCC codes can be used for GEL codes, too. Thus an algorithm of Blokh and Zyablov (see e.g. [8]) can be used to decode all error patterns whose weight does not exceed dc2−1 . However, this algorithm can correct many error patterns of higher weight, too. In the following section we will present a method for estimating the bit error rate that will take into account these additional correcting capabilities.

3

On the Hard Decision Correcting Capability of Generalized Concatenated Codes

If an algorithm is used for decoding that can correct all error patterns with a weight up to half the minimum distance of the code and in addition many patterns of higher weight, then the minimum distance is not a sufficient description of the correcting capability of the code. This is the case for the decoding of generalized concatenated codes with the algorithm of Blokh and Zyablov [8]. Therefore we present bounds for the bit error and the block error rate that take into account this additional error correcting capacity. Transmission over a binary symmetric channel (BSC) is assumed. The bounds depend on the decoding algorithm on which the estimation of the number of errors after decoding is based.

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Estimation of Bit Error Rate: Let s =  d−1 2  be the number of errors a bounded minimum distance decoder can correct. We will state an upper and a lower bound for the bit error rate of the code bits of a generalized concatenated code that can correct at least a part of all error patterns up to the weight t > s. For a good estimate, t should be chosen such that it includes most of the additional correctable error patterns. However, since t is also the maximum number of errors that can be added to a code vector in case of wrong correction, t should not be too large. Moreover the effort for the calculation of the bounds increases very fast with increasing t. One has to find a good compromise. The codewords C in matrix form consist of na columns of length nb , resulting in a total code length of nc = na · nb . Let e = (e1 , e2 , . . . , enb ) ∈ E be the distribution of the column weights of an error pattern, i.e. ei is the number of columns i errors and E is the set of all possible error distributions.

ncontaining b |e| = i=1 i· ei denotes the total number of errors. Now it is possible to split the bit error rate in a part that is due to all the errors of weight ≤ t, and a second part that is due to the remaining errors with weight > t: Pˆbit (t) = pˆ(|e| ≤ t) + pˆ(|e| > t).

(2)

We assume that the decoding algorithm will not add more errors than it is able to correct. Therefore we can upper bound the second term (p is the crossover probability of a BSC):

pˆ(|e| > t) ≤

  nc 1  nc i 1  ˆ N (e)P (e) ≤ (i + t) p (1 − p)nc −i . nc nc i=t+1 i

(3)

e∈E |e|>t

ˆ (e) ≤ (i + t) is the maximum number of bit errors that can result, given a N  weight distribution of bit errors e. P (e) = nic pi (1 − p)nc −i is the probability of this weight distribution. The first term of equation 2 is estimated by a sum of all different weight distributions of error patterns |e| ≤ t: pˆ(|e| ≤ t) ≤

1  ˆ 1  ˆ N (e)P (e) = N (e)U(e)p|e| (1 − p)nc −|e| , nc nc e∈E |e|≤t

e∈E |e|≤t

where the number U(e) of error patterns of a weight distribution results from the number of possibilities for the selection of erroneous columns times the number of possibilities to place the errors within these columns:

U(e) =

i−1   nb  t   ei  na − j=1 ej nb . · ei i

i=1

i=1

Some Results on Generalized Concatenation of Block Codes

189

0

10

−5

Pbit

10

−10

10

−15

10

upper and lower bound according to eq. 2 and 4 upper bound of a two−error correcting code upper bound of a three−error correcting code

−20

10

−7

10

−6

10

−5

10

−4

−3

10

10

−2

10

−1

10

p

Fig. 3. A comparison of some bounds on the bit error rate of the concatenated code given in example 5.

ˆ has to be done individually for all |e| ≤ t. The correspondThe calculation of N ˇ ing lower bound Pbit (t) is given by Pˇbit (t) = p(|e| ≤ t) + p(|e| > t),   nc 1  ˇ 1  nc i pˇ(|e| > t) ≥ (i − t) N (e)P (e) ≥ p (1 − p)nc −i , nc nc i=t+1 i

(4)

e∈E |e|>t

pˇ(|e| ≤ t) ≥

1  ˇ N (e)U(e)p|e| (1 − p)nc −|e| , nc e∈E |e|≤t

ˇ (e) as the minimum number of bit errors that will result from decoding. with N For p → 0 both bounds will coincide. Example 5 (Bounds on the Bit Error Rate). We consider a concatenated code of length nc = 1024 and dimension k = 988 based on a partition chain of the inner code B (0)(32, 32, 1)/B (1)(32, 31, 2)/B (2)(32, 26, 4)/B (3)(32, 21, 6) and the outer codes A(0)(32, 16, 8), A(1)(32, 29, 4)25 and A(2)(32, 31, 2)25 . As the minimum Hamming distance of the code is 6, each error of weight 1 or 2 can be corrected using the multi-stage decoding algorithm by Blokh and Zyablov. In addition almost all of the error patterns of weight 3—except the pattern given by e = (0, 0, 1, 0, . . . , 0)—can be corrected. This error pattern, however, seldom occurs as compared to all the correctable errors and can be detected. Thus the decoding result is similar to a three-error correcting code. Figure 3 compares the bounds of eq. 2 and 4 for t = 3 with upper bounds on two- and three-error correcting codes (given by eq. 3, t = 2, 3).

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Estimation of Block Error Rate: The block error rate is upper bounded by the error rate of an ordinary t error correcting code and the error rate that is caused by error patterns of weight ≤ t (we ignore error patterns of weight > t that can be corrected): BMD Pblock (t) ≤ Pblock (t) + Pblock (|e| ≤ t),   n c  nc i BMD p (1 − p)nc −i (t) = Pblock i i=t+1  ˜ (e)U(e)p|e| (1 − p)nc −|e| . Pblock (|e| ≤ t) = N e∈E |e|≤t

˜ (e) indicates whether a given weight distribution of an error pattern oversteps N the correcting capacity of the code:  ˆ (e) > 0 1 if N ˜ N= ˆ 0 if N (e) = 0.

References 1. E. L. Blokh and V. V. Zyablov. Coding of generalized concatenated codes. Problemy Peredachi Informatsii, 10(3):45–50, July–Sept. 1974. 2. M. Bossert. Channel Coding for Telecommunications. Wiley, 1999. 3. G. D. Forney, Jr. Concatenated Codes. MIT, Cambridge, MA, 1966. 4. F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-Correcting Codes. North Holland, Amsterdam, 1996. 5. J. K. Wolf. On codes derivable from the tensor product of check matrices. IEEE Trans. Inf. Theory, IT-11:281–284, 1965. 6. J. K. Wolf and B. Elspas. Error-locating codes—a new concept in error control. IEEE Trans. Inf. Theory, IT-9:113–117, 1963. 7. V. A. Zinov’ev. Generalized cascade codes. Problemy Peredachi Informatsii, 12(1):5–15, 1976. 8. V. A. Zinov’ev and V. V. Zyablov. Decoding of nonlinear generalized cascade codes. Problemy Peredachi Informatsii, 14(2):46–52, 1978. 9. V. V. Zyablov. New interpretation of localization error codes, their error correcting capability and algorithms of decoding. In Transmission of Discrete Information over Channels with Clustered errors, pages 8–17. Nauka, Moscow, 1972. (in Russian). 10. V. V. Zyablov, J. Maucher, and M. Bossert. On the equivalence of GCC and GEL codes. In Proc. 6th Int. Workshop on Algebraic and Combinatorial Coding Theory, pages 255–259, Pskov, 1998. 11. V. V. Zyablov, J. Maucher, and M. Bossert. On the equivalence of generalized concatenated codes and generalized error location codes. To appear in IEEE Trans. Inf. Theory, 1999.

Near Optimal Decoding for TCM Using the BIVA and Trellis Shaping Qi Wang1 , Lei Wei1 , and Rodney A. Kennedy2 1

Department of Engineering, The Australian National University [email protected], [email protected] 2 Research School of Information Sciences and Engineering The Australian National University [email protected]

Abstract. In this paper a bootstrap iterative decoding technique concatenated with the Viterbi algorithm (BIVA) and trellis shaping for trellis-coded modulation (TCM) is proposed. The concept of a bootstrap decoding is introduced first and then the new metric functions which take into account the bootstrap iterative decoding algorithm for TCM systems are derived. One and two dimensional bootstrap decoding are proposed for packet transmission. Furthermore, the trellis shaping technique is also considered to combine with such TCM schemes using the BIVA. The simulation results show that the performance of 1.25 dB away from Shannon limit can be achieved by the BIVA and trellis shaping for 256-state 6 bits/T TCM scheme, with low complexity and reasonable computation.

1

Introduction

Trellis-Coded Modulation (TCM) has been widely used as a combined coding and modulation technique for digital transmission over band-limited channel. Ungerboeck has shown that significant coding gains can be achieved using trelliscoded modulation with the Viterbi decoding algorithm over uncoded modulation without sacrificing bandwidth efficiency on a bandlimited channel [1]. In the past decade, many variants on the basic TCM scheme have been developed to obtain higher coding gains. Bootstrap decoding [4]-[7] is a method which imposes algebraic constraints on streams of convolutionally encoded information sequences. Such constraints can then be made use to gather extrinsic information from other streams when one stream is decoded. In [7] Wei extent the results of [4]-[6] to near optimally bootstrap decoding using long convolutional codes. One of the simplified bootstrap algorithms, which only uses the Viterbi algorithm, was given in [7] and named as BIVA in [8]. In this paper, a system of concatenating trellis codes with the bootstrap decoding is proposed. We will focus on how to apply the BIVA to TCM, and one and two dimensional bootstrap structures are designed. In addition, we noted Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 191–200, 1999. c Springer-Verlag Berlin Heidelberg 1999


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that the bootstrap iterative decoding algorithm can be modified to accommodate trellis shaping [10] and full shaping gains can be achieved. The paper is organised as follows. In section 2, the principle of encoding system with bootstrap decoding is described. We then present the BIVA which combines the bootstrap iterative decoding with Viterbi algorithm in section 3, and the new metric functions for bootstrap TCM schemes are also derived. In section 4, one and two dimensional bootstrap structures are proposed. In section 5, concatenating shaping techniques with bootstrap TCM using the BIVA is considered. The numerical simulation results and discussion will be reported in section 6 to compare the performance of the BIVA with Viterbi algorithm. Lastly, in section 7, the conclusion is given.

2

Review the Bootstrap Structure

Bootstrap decoding is a method which utilises algebraic constraints across streams of convolutional encoded information sequences. Before introducing the BIVA and its decoding algorithm, let us look at the basic concept of the bootstrap decoding. In the bootstrap decoding [4], a medium size packet can be formed by organising (mb − 1) × l information symbols a block of (mb − 1) rows and l columns. Suppose [U]j,l = [Uj,1 , Uj,2 , · · · , Uj,l ] and [V]j,l = [Vj,1 , Vj,2 , · · · , Vj,l ] denote the information symbol vector and the encoded codeword vector of the j th packet stream, respectively. As usual, we will encode each packet of l × k binary information bits into codewords of length l through the same k/n convolutional encoder and each encoded symbol Vj,i , (i = 1, 2, · · · , l) has n information bits. th An mth digit will be the b packet is then generated in such way that the i th parity of the i digits of the (mb − 1) information packets, namely, the mth b packet is a modulo 2 position-by-position sum of the above (mb − 1) information packets, i.e., Umb ,i = U1,i ⊕ U2,i ⊕ · · · ⊕ Umb −1,i ,

i = 1, 2, . . . , l.

(1)

where ⊕ denotes a modulo 2 sum. The mth b packet is therefore called the paritycheck-constraint (PCC) packet. It was shown in [7] that the free distance of such block code is doubled. The mth b PCC packet is also encoded by the same convolutional encoder. Fig. 1 shows the bootstrap structure of one block of mb rows and l columns. Because of the linearity of convolutional encoding1 , the PCC packet corresponds to a path in the coding tree whose information digits are the mod 2 sum of the information digits underlying the information packets; i.e., Vmb ,i = V1,i ⊕ V2,i ⊕ · · · ⊕ Vmb −1,i , 1

i = 1, 2, . . . , l.

(2)

All the convolutional encoders shown in this paper are assumed to be linear which guarantees that the PCC condition among the information packets still exists for coded symbols after the coding process.

Near Optimal Decoding for TCM Using the BIVA and Trellis Shaping packet 1

V1,1

V1,2

···

V1,l

packet 2

V2,1

V2,2

···

V2,l

···

Vmb −1,l

···

Vmb ,l

193

. . . packet mb − 1 packet mb

Vmb −1,1 Vmb −1,2 Vmb ,1

Vmb ,2

Fig. 1. The bootstrap structure of one block code.

Hence, all mb packets are in principle decodable. Suppose next that the mb encoded packets are sent through the channel disturbed by AWGN, and that the corresponding received signal vectors [R]j,i (j = 1, · · · , mb ; i = 1, · · · , l) are arranged by the decoder into a mb by l array. We can find that if the j-th received packet is to be decoded, the received digits of all other packets should also be taken into account, since these contain information about the transmitted digits of the j-th packet (the transmitted digits are related by the parity-checkconstraint condition).

3

Bootstrap Iterative Viterbi Algorithm for TCM

In this section, we will modify the BIVA [7][8] for TCM schemes. In [7] Wei made (p) a significant simplification on the computation of the parity metric λj,i for convolutional codes. The modified BIVA is an appropriate to decode the bootstrap TCM scheme with low complexity. Now let us consider how to construct bootstrap TCM scheme and how to derive its parity metric. In trellis-coded modulation, the key concept is the principle of mapping by set partitioning [1]. A general TCM system is shown in Fig. 2. In the trellis encoder, k the inputed information bits are divided into two parts. One part (Ul1 , · · · , Ul
) is encoded by a convolutional encoder whose output is used to select one subset
from the whole constellation; the other part (Ulk +1 , · · · , Ulk ) comprises uncoded bits which are used to determine one signal point within that selected subset. The TCM system can be combined with the fore-mentioned bootstrap structure. Next we describe how to concatenate the bootstrap iterative decoding algorithm with VA for such TCM systems. The key idea of bootstrap decoding is to produce one sequence which follows the PCC condition using (mb −1) information sequences. But for TCM systems, the question arises whether to parity check or protect each bit in one symbol? We know that the bit error probability (BER) performance of TCM is mainly determined by the minimum squared Euclidean distance d2f ree which is the minimum of parallel transition’s squared

194

Qi Wang, Lei Wei, and Rodney A. Kennedy Ul1

. ˜ . Ulk .

Encoder

.. .


k/
n

˜ Ulk+1

Ulk

Vl1

Convolutional

.. .

.. Vln˜ .

Subset Selector

˜ Vln+1 .. Vln .

Signal Selector

S

Fig. 2. Schematic representation of a typical trellis code.

distance d2parallel and coded minimum squared Euclidean distance d2f ree,c , i.e., d2f ree = min(d2parallel, d2f ree,c ). If d2f ree,c ≤ d2parallel, we can say that the bit error caused by parallel path error can be ignored at a high signal-to-noise rate (SNR), and the BER is dominated by d2f ree,c . Therefore, the performance should k are parity checked not be dramatically affected if only the coded bits U 1 , · · · , U
i

i

k +1 or protected and uncoded bits Ui
, · · · , Uik are intact or unprotected instead of 1 protecting the whole symbol Ui , · · · , Uik . The benefit of such operation is obvious — the information transmission rate is increased. However, for the TCM systems in which d2f ree,c > d2parallel, if only coded bits are protected, the performance will be much worse than the one in which the whole symbol is protected. Now let us derive the new metric function which take into account the PCC condition for TCM. In this paper, we just focus on protecting the coded bits in one symbol. Let [R]j,l = [Rj,1 , Rj,2 , · · · , Rj,l ] denote the received signal vector n+1 n of the j th packet, and (V p ; V np ) = (V 1 , · · · , V
;V
, · · · , V n ) denote the ith j,i

j,i

j,i

j,i

j,i

j,i

p encoded codeword of the j th packet, where Vj,i is the output from the convonp lutional encoder and Vj,i is the uncoded part of the input symbol. If only the p coded bits are protected, then only Vj,i satisfies the PCC condition in the encoded codeword. Suppose that packets j1 , j2 , · · · , jmb −2 have been successfully decoded using the normal VA and now we are going to decode packet jmb −1 . The original PCC condition on the ith symbols in one block (mb packets) is p p ⊕ V2,i ⊕ · · · ⊕ Vmp b ,i = 0. Wmb ,i = V1,i

(3)

p ⊕· · ·⊕Vmp b −2,i . The receiver replaces the j th (j = 1, 2, . . . , mb − Let Wmb −2,i = V1,i 2) received packets by the estimated j th transmitted packets. So we can get the estimated PCC condition

m −2,i = V p ⊕ · · · ⊕ V p W b 1,i mb −2,i .

(4)

p p is the decision of Vj,i . Now the new likehood function for decoding where Vj,i th the (mb − 1) packet is

λmb −1,i = − log[P (R1,i , · · · , Rmb −1,i , Rmb ,i |Vmp b −1,i ; Vmnpb −1,i )].

(5)

Near Optimal Decoding for TCM Using the BIVA and Trellis Shaping

195

where Vmp b −1,i and Vmnpb −1,i are coded part and uncoded part of codeword Vmb −1,i , respectively. Assuming R1,i ,· · ·,Rmb −1,i and Rmb ,i are independent of each other2 , we then have λmb −1,i = − log[P (Rmb −1,i |Vmp b −1,i ; Vmnpb −1,i )]

− log[P (R1,i , · · · , Rmb −2,i , Rmb ,i |Vmp b −1,i ; Vmnpb −1,i )], (v)

(p)

= λmb −1,i + λmb −1,i .

(6)

(v)

where λmb −1,i denotes the branch metric value which can be obtained using (p)

the VA, and λmb −1,i denotes the extrinsic metric value introduced by the PCC condition from the other packets. m −2,i = Wm −2,i , then we have If W b b p mb −2,i ⊕ V p W mb −1,i ⊕ Vmb ,i = 0.

(7)

Therefore, Vmp b ,i , the coded part of ith codeword in mth b packet, can be obtained through the PCC condition and (mb − 2) decoded received packets as well. np However, there is no PCC relation among the uncoded part Vj,i , so the question is how to determine the uncoded part Vmnpb ,i of ith codeword for mth b packet. We know that the coded part Vmp b ,i decides which subset will be selected in the constellation, and the parallel transitions error in the subset can be ignored at high SNR’s. Therefore, after Vmp b ,i is determined, Vmnpb ,i can also be decided by selecting one point (Vmp b ,i ,Vmnpb ,i ) which is the closest to the received signal Rmb ,i in this subset. Now we have λmb −1,i = − log[P (R1,i , · · · , Rmb −2,i , Rmb ,i |Vmp b −1,i ; Vmnpb −1,i )], (p)

m −2,i ; V np )]. = − log[P (R1,i , · · · , Rmb −2,i , Rmb ,i |Vmp b ,i ⊕ W b mb ,i

(8)

Exact calculation of (8) is very computation expensive and almost practically impossible when mb is large. Therefore, we approximate (8) as (p) m −2,i ; V np )]. λmb −1,i ≈ − log[P (Rmb ,i |Vmp b ,i ⊕ W b mb ,i

(9)

(p)

It is worth mentioning that λmb −1,i will not be the correct metric and the error mb −2,i = Wmb −2,i . The effect of error propagation propagation can result if W (p) can be reduced by scaling down the value of λmb −1,i with a scale factor α. So far, we have got the new metric functions for TCM systems combined with bootstrap structure. The corresponding bootstrap iterative decoding using Viterbi algorithm (BIVA) for TCM can be summarised as follows. (a) (mb − 1) × l information symbols are arranged as a block of (mb − 1) rows and l columns, where each symbol includes
k coded bits and (k − k
) uncoded 2

Actually, R1,i , · · · , Rmb ,i are weakly dependent because of the PCC condition among them.

196

(b)

(c) (d) (e) (f)

4

Qi Wang, Lei Wei, and Rodney A. Kennedy

bits. Encode each row of l information symbols into codewords of length n using a
k/
n convolutional encoder of memory length ν. All rows are encoded through the same convolutional encoder. Generate the mth b parity check row using the PCC condition. If it is only necessary to protect the coded bits, the coded bits of the ith symbol will be the parity of the coded bits of the ith symbols of the previous (mb − 1) information packets, and its uncoded bits will continue to be information bits. In the receiver, decode the first (mb −2) packets based on the VA and update the PCC condition through decoded packets. Decode the next packet based on the BIVA, using the new metric which takes into account the extrinsic information from other packets. Update the PCC condition based on the newest decoded values. Repeat step (d) and (e) for several iterations until the stop criterion is satisfied.

Variation of BIVA for Packet Transmission

In this section, we will study the BIVA decoding in packet transmission format. In previous sections, several dummy symbols in each packet are needed to make the decoding of the last information symbols reliable. Such dummy symbols can be eliminated using tail-biting [9]. In tail-biting, the encoder is first initialised by inputing the last ν information bits into the encoder and ignoring the output. Therefore, the start and end encoder states are constrained to be identical; that is, a trellis codeword starts from the state at which it will eventually end. 4.1

One Dimensional (1-D) Bootstrap Structure

We now can use the tail-biting technique in each packet to encode the whole packet information symbols. But it is noted that the much computation will be taken if each packet applies tail-biting. In this subsection, we will present a modification of the BIVA to cut down such computation without effect on its error performance. The key concept of 1-D bootstrap structure is to connect all the row packets into one big packet. In other words, instead of terminating each row packet the information symbols of all row packets will continue to feed into the encoder until the last symbol.  in the 1-D BIVA. The Next we consider how to update the PC condition W Viterbi decoder progresses over the trellis for a certain depth (i.e., truncation length), it then produces its decision result. Ideally, updating for one packet  should should be based on the decoding history of the other packets, namely, W be updated based on the newest decoded values of the other packets. In the 1-D BIVA, however, the decision results of any packet can not be determined until  is updated only once the whole super packet is finished. Thus, for all packets, W  is updated has little in every iteration. Simulation results show that the way W effect on the error performance and iteration number.

Near Optimal Decoding for TCM Using the BIVA and Trellis Shaping

4.2

197

Two Dimensional (2-D) Bootstrap Structure

We can re-apply the bootstrap structure on the structure in the above subsection 4.1 to build up a two dimensional bootstrap structure. The main motivation is to build up a block code with a large minimum free distance, but decodable by an iterative BIVA with a very low complexity. The 2-D bootstrap structure is illustrated in Fig. 3. In the 2-D structure, there are two types of parity-checks

super

packet 11

packet 21

packet 12

packet 22

.. . packet 1B

.. . packet 2B

packet 1 super

packet 2

super parity

packet B

··· ···

···

packet mb,1 packet mb,2 .. . packet mb,B

Fig. 3. 2-D bootstrap structure.

which can produce two sorts of parity metrics3 . Type (a) constraint provides the parity metric given in (9) for each super row packet. Type (b) constraint provides the parity metric for each column in the structure given in Fig. 3. Again, both parity metrics will be used after the first iteration. Updating the parity metrics for type (a) and (b) constraints is exactly the same as the procedure given in the above subsection.

5

Combining Shaping Techniques with Bootstrap TCM System

It has come to be recognised that shaping and coding are two separable and complementary components of the TCM systems. In [10], it has been shown that shaping gain can be achieved by using nonuniform, Gaussian-like signalling. One of the approaches, called trellis shaping, was proposed by Forney [10]. It was shown that a simple 4-state shaping code can achieve about 1.0 dB shaping gain, which is about 2/3 of the full 1.53 dB ultimate shaping gain. In this work, we concentrate on trellis shaping with TCM systems using the bootstrap iterative decoding algorithm. A TCM system using BIVA with shaping techniques is shown in Fig. 4. A 2n -point 2-D constellation is used to transmit k = kc + nu + rs 3

A third type of parity check constraint can be contemplated for the 2-D structure which is a combination of row and column check sums. However, it is found that this posed computational and technical difficulties which did not significantly contribute to the performance.

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Qi Wang, Lei Wei, and Rodney A. Kennedy

bits

Channel Encoder Gc

yl nc

M

bits

A Shaping zl Encoder ns

decoder

P sl rs bits

a

channel

wl nu bits

channel

xl kc

y ˆl nc

bits

x ˆl

G−1 c kc

bits

Gc

y ˆl kc

bits

w ˆl nu bits

Shaping sˆl zˆl ns bits Decoder rc bits

bits

Fig. 4. Coded modulation system with trellis shaping.

information bits/T, where kc , nu and rs are the number of channel coded bits xl , uncoded bits wl and shaping coded bits sl , respectively. The 2-D constellation is first partitioned into 2nc subsets using mapping by set partitioning [1]. The nc coded bits yl produced by the channel encoder Gc at time l specify one of the 2nc subsets. The 2-D constellation is also divided into 2ns subregions. The ns shaping bits zl produced by the shaping encoder at time l specify one of the 2ns subregions. The nu uncoded bits wl at time l specify a point a = M(yl , wl , zl ) in the constellation for a given subregion and a given subset. At the receiver, a Viterbi decoder with BIVA can be used to obtain the estimate of the transmitted j,i , one channel encoder Gc is information bits. To update the PCC condition W needed to re-encode the decoded bits xl at the receiver. It was shown in [7] that the free distance of block code is doubled by 1-D bootstrap structure. We note that the free distance is just determined by coded bits xl for such TCM schemes in which d2parallel is larger than d2f ree,c . Therefore, the shaping codes sl should not affect the distance property and the full shaping gains will be achieved when shaping techniques are combined with TCM systems using BIVA.

6

Numerical Results

In the previous sections, the BIVA for bootstrap TCM systems has been proposed and shaping technique integrated with such TCM was also discussed. In this section, we report some simulation results to compare the performance with the Viterbi algorithm. In our simulation, each superblock has 20×20 packets in 2-D bootstrap structure, i.e., mb = 20 and B = 20, and each packet includes 7ν symbols. The largest iteration number is set as 100. If the stop criterion is meted within 100 iterations, the decoding process will be stopped. The stop criterion in bootstrap decoding is straightforward. If the coded part of decoded symbols (part protection) in all mb packets (1-D bootstrap structure) or (mb × B) packets (2-D bootstrap structure)   =  satisfy the original PCC condition, i.e., W j i ⊕Vj,i = 0, then the iterative decoding process is stopped. Figure 5 shows the performance of 2-D BIVA

Near Optimal Decoding for TCM Using the BIVA and Trellis Shaping

199

−3

10

BER

VA 2−D BIVA without shaping 2−D BIVA with shaping

−4

10

−5

10

10.5

11

11.5

12.5

12

E /N b

13

13.5

14

0

Fig. 5. Performance of ν = 8 trellis codes at a spectral efficiency of 6 bits/T with part protection using 2-D BIVA.

with part protection for 2-D 256-state 6 bits/T TCM scheme, combined with trellis shaping technique. 16-state shaping code is applied in our simulation. The 16-state and 256-state TCM are Ungerboeck’s codes [2]. For comparison, the performance of such TCM scheme using VA is also reported. The results show that about 2.0 dB gross gain can be achieved beyond the VA without shaping and about 2.7 dB gross gain with shaping. However, noted that the shaping gain is smaller at low SNR’s and only about 0.7 dB shaping gain was obtained in such TCM scheme with 2-D bootstrap structure (similar case can be found in [11]). We also can see the error floor in Fig. 5. This error floor is mainly dominated by the parallel path error at relative low SNR’s. Such errors can be reduced through multilevel error protection techniques. It is noteworthy that each block includes some non-information bits which cause the real transmission rate to be lower than the nominal rate, namely, the parity bits introduced by the bootstrap structure offset a part of achieved gains and result in practical rate loss. In our case, the real rate now is 5.805 bits/T , Shannon limit for this rate is 9.76 dB. Therefore the performance of 1.25 dB away from Shannon limit at BER = 3 × 10−5 is achieved by such a TCM scheme. Finally, we have to point out that the scale factor α mentioned in section 3 is critical to the performance and convergent speed. The average iteration number varies according to the different SNR’s and different codes. Generally, less iterations are required at higher SNR’s or by larger memory length code. In our simulation, for ν = 8 code, the average numbers of iteration are 45 and 13 for SNR=11.01 dB and 11.1 dB, respectively. It is shown that the peak and average complexity can be significantly cut down if we increase SNR by 0.1 dB.

200

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Qi Wang, Lei Wei, and Rodney A. Kennedy

Conclusions

A bootstrap iterative Viterbi algorithm has been proposed and new metric functions which take into account the extrinsic information based on the bootstrap TCM systems have also been derived. Additional gains for trellis codes using the BIVA instead of VA can be obtained with low complexity and reasonable computation. For TCM schemes in which d2f ree,c ≤ d2parallel , protecting the coded bits instead of the whole symbol can achieve most gain with less redundancy. 1-D and 2-D bootstrap structures are also designed to be suitable for packet transmission. We can see that such bootstrap decoding can be applied with existing systems such as ADSL, etc. Furthermore, trellis shaping technique is employed on bootstrap TCM systems and full shaping gain can be achieved. In the simulation, the performance of 1.25 dB away from Shannon limit is achieved by 2-D BIVA for 2-D 256-state 6 bits/T TCM scheme combined with trellis shaping.

Acknowledgements The authors would like to thank Prof. D. J. Costello Jr. for introducing us to this area, Dr. G. D. Forney, Jr. for pointing out the importance of continuous transmission, which has been the key motivation of this research.

References 1. G.Ungerboeck: “Channel coding with multilevel/phase signals”, IEEE Trans. on Inform. Theory, Vol. IT-28, pp. 55-67, Jan. 1982. 2. Ezio Biglieri, D. Divsalar, P. J. McLane and M. K. Simon, Introduction to Trelliscoded Modulation with Applications, Macmillan Publishing Company, New Vork, 1991. 3. G.D. Forney, Jr.: “The Viterbi Algorithm” Proc. of the IEEE, Vol. 61, No. 3, March 1973. 4. F. Jelinek, J. Cocke: “Bootstrap hybrid decoding for symmetrical binary input channels”, Information and Control, Vol. 18, pp. 261-298, April 1971. 5. F. Jelinek: “Bootstrap trellis decoding”, IEEE Trans. on Inform. Theory, Vol. 21, pp. 318-325, May 1975. 6. H. A. Cabral, D. J. Costello, Jr., P.R. Chevillat: “Bootstrap hybrid decoding using the multiple stack algorithm”, Proceeding of IEEE ISIT’97, pp. 494, Ulm, Germany, June 29 - July 4. 7. L.Wei: “Near Optimal Limited-Search Detection, Part II: Convolutional Codes”, Submitted to IEEE Trans. on Inform. Theory, 8. L.Wei: “On bootstrap iterative Viterbi algorithm”, IEEE ICC’99, Vancouver, Canada. 9. G. Solomon, H. C. A. van Tilborg: “A connection between block and convolutional coded”, SIMA J. of Appl. Math, Vol. 37, No. 2, Oct. 1979. 10. G. D. Forney, Jr.: “Trellis shaping”, IEEE Trans. on Inform. Theory, Vol. 38, pp. 281-300, Mar. 1992. 11. S. Couturier, D. J. Costello, Jr., and Fu-Quan Wang, “Sequential decoding with trellis shaping”, IEEE Trans. on Inform. Theory, Vol. IT-41, No.6, pp. 2037-2040, Nov. 1995.

An Optimality Testing Algorithm for a Decoded Codeword of Binary Block Codes and Its Computational Complexity Yuansheng Tang1 , Tadao Kasami2, and Toru Fujiwara1 1

2

Graduate School of Information Science Nara Institute of Science and Technology, Ikoma, Nara, Japan Department of Computer Science, Faculty of Information Science Hiroshima City University, Hiroshima, Japan

Abstract. Based on h reference words, Kasami et. al. proposed an integer programming problem (IPP) whose optimal value of the objective function gives a sufficient condition on the optimality of a decoded codeword. The IPP has been solved for 1 ≤ h ≤ 3. In this paper, an algorithm for solving the IPP for h = 4 is presented. The computational complexity of this algorithm is investigated.

1

Introduction

Let IDA denote a soft-decision iterative decoding algorithm for a binary block code. For most IDA’s, in each successful decoding step a candidate codeword is generated by a simple decoder and an optimality testing condition is tested on the candidate codeword. When the testing condition is satisfied, the decoding iteration process is terminated and the optimal (or most likely) codeword is obtained. A number of testing conditions have been derived, such as those proposed in [1] and [2]. Recently, based on h reference words, Kasami et. al. in [3,4] proposed an integer programming problem (IPP), whose optimal value of the objective function gives an optimality testing condition and can be incorporated in any IDA which is based on the generation of a sequence of candidate codewords. This testing condition with h = 3 was used effectively in the iterative decoding algorithm presented in [5]. It was shown that this testing condition can provide fast termination of the decoding iteration without degrading the error performance. It is pointed out in [6] that the approach used in the derivation of this testing condition can also be used to derive other two important conditions for the IDA’s. One is the ruling-out condition, which can be used to skip useless decoding steps. If the ruling-out condition holds in some decoding step, then the output of the next decoding step can not be better (or more likely) than the best candidate codeword obtained so far. Another is a stronger version of the ruling-out condition, called early termination condition, if the early termination condition holds in some decoding step, then all the successive decoding steps can not generate any candidate codeword which is better than the best candidate codeword obtained so far, that is, there is no improvement on error performance Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 201–210, 1999. c Springer-Verlag Berlin Heidelberg 1999


202

Yuansheng Tang, Tadao Kasami, and Toru Fujiwara

by any further iteration. In [7], the approach is also used to find a good sequence of search centers around which algebraical decodings are iterated. In [3,4], the computational complexity for solving the IPP was investigated for 1 ≤ h ≤ 3, the number of additions and comparisons of real numbers, which is the majority of the computational complexity, for solving the IPP was shown to be of order N , where N is the code length. For larger h, this testing condition is stronger while the computational complexity for solving the IPP grows. It is desirable to give effective methods to solve the IPP for relatively large h. In this paper, we consider to solve the IPP for h = 4. In Section 2, we will review briefly the IPP proposed by Kasami et. al. in [3,4] and [6]. In Section 3, an algorithm for solving the IPP for h = 4 is presented. The IPP is split into at most 9 sub-IPP’s, the number of the variables of each sub-IPP is half of that of the original IPP. The number of additions and comparisons of real numbers for this algorithm is shown to be of order N 2 . In Section 4, each sub-IPP is split further into a few simpler subsub-IPP’s that are solved by simple iterations. The proofs of the theorems and lemmas appeared in Sections 3 and 4 can be found in [8].

2

The Testing Condition of Optimality

For a positive integer N , let V N denote the set of all binary N -tuples over V
GF(2). Suppose a binary block code C ⊆ V N is used for error control over the AWGN channel with BPSK signaling, and r = (r1 , r2 , . . . , rN ) is a received N -tuple at the output of a matched filter in the receiver. Let z = (z1 , z2 , . . . , zN ) be the binary hard-decision N -tuple obtained from r using the hard-decision function: zi = 1 for ri > 0 and zi = 0 for ri ≤ 0. |ri | indicates the reliability of zi for each i. For u = (u1 , u2 , . . . , uN ) ∈ V N , define
D1 (u)
{i : ui = zi , 1 ≤ i ≤ N } and D0 (u)
{1, 2, . . . , N}\D1 (u). L(u)
i∈D1 (u) |ri | is called the correlation discrepancy of u with respect to the hard-decision tuple z [3,4]. For any subset T of V N , let L[T ]
minu∈T L(u), and write L[∅]
+∞. The maximum likelihood decoding (MLD) can be stated in terms of the correlation discrepancy as follows: The decoder decodes the received tuple r into the optimal codeword copt ∈ C with L(copt ) = L[C]. For u ∈ V N and positive integer d, let Od (u)
{v ∈ V N : dH (u, v) < d}, where dH (u, v) is the Hamming distance between u and v. Let h be a positive integer and u1 , u2 , . . . , uh be h reference words in V N . Let R
∪hj=1 Odj (uj ), where dj is called the preassigned radius for the reference word uj . Write VdN1 ,d2 ,... ,dh (u1 , u2 , . . . , uh )
V N \R, that is VdN1 ,d2 ,... ,dh (u1 , u2 , . . . , uh ) = {v ∈ V N : dH (uj , v) ≥ dj for 1 ≤ j ≤ h}. (1) Lemma 1. If a codeword cbest in R satisfies L(cbest ) ≤ L[VdN1 ,d2 ,... ,dh (u1 , u2 , . . . , uh )],

(2)

Optimality Testing Algorithm

203

then the optimal codeword copt must belong to the set R. Furthermore, we have cbest = copt if L(cbest ) = L[C ∩ R]. Lemma 1 not only defines a region which contains the optimal codeword copt but also provides a sufficient testing condition for the optimality of candidates which have been generated. The selection of the reference words and the preassigned radiuses effect quite the efficiency of the results of Lemma 1. If we incorporate it in a soft-decision iterative decoding algorithm, we can select the reference words from the previously generated candidate codewords, which are trusted to possess low correlation discrepancies, and take “the covering distance around uj ” assured by the decoding algorithm as the preassigned radius for the reference word uj . Here we do not consider the problem of selection of the reference words and the preassigned radiuses further. The main problem we concern in this paper is to evaluate L[VdN1 ,d2 ,... ,dh (u1 , u2 , . . . , uh )]. Below we will introduce an IPP proposed by Kasami et. al. in! [3,4], the optimal value of this IPP was shown to be equal to L[VdN1 ,d2 ,... ,dh (u1 , u2 , . . . , uh )]. Let B h denote the set of all h-tuples over B
{0, 1}. For simple notation, the h-tuple α = (α1 , α2 , . . . , αh ) may be represented as α1 α2 · · · αh . Define Dα


h 

Dαi (ui ), nα
|Dα |.

(3)

i=1

Let Q∗h denote the set of all the 2h -tuples over nonnegative integers. For q ∈ Q∗h , each of the 2h components of q is referred as qα with α ∈ B h . Let Qh denote the set of all the 2h -tuples q ∈ Q∗h which satisfy 0 ≤ qα ≤ nα , for all α ∈ B h ,  qα (−1)αi ≥ δi
di − |D1 (ui )|, for i = 1, 2, . . . , h. ∆(q)i


(4) (5)

α∈B h

Without loss of generality, we assume further that the components of the received tuple r are ordered in the increasing order of their absolute values |r1 | ≤ |r2 | ≤ · · · ≤ |rN |.

(6)

For convenience, we define r∗
+∞. For any subset X ⊆ {1, 2, . . . , N, ∗} and integer j, let X (j) denote the set of j smallest integers in X\{∗} if 1 ≤ j ≤ |X\{∗}|, the set X ∪ {∗} if j > |X\{∗}|, the empty set ∅ otherwise. For q ∈ Q∗h ,
(q ) let D(q)
∪α∈B h Dα α , L (q)
i∈D(q) |ri |. For nonempty subset Q ⊆ Q∗h , let L [Q ] denote the optimal value of the following IPP: P(Q ): Minimize {L (q) | q ∈ Q }, i.e. L [Q ]
minq∈Q
L (q). For convenience, we write L [∅]
+∞. A 2h -tuple q ∈ Q is called a Q -optimum if L (q) = L [Q ]. It was shown in [3,4] that L[VdN1 ,d2 ,... ,dh (u1 , u2 , . . . , uh )] is equal to the optimal value L [Qh ] of the IPP P(Qh ) if the received tuple r satisfies (6), i.e.

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Yuansheng Tang, Tadao Kasami, and Toru Fujiwara

Theorem 1. If the received tuple r satisfies (6), we have the following formula: L [Qh ] = L[VdN1 ,d2 ,... ,dh (u1 , u2 , . . . , uh )].

(7)

If δi ≤ 0 for i = 1, 2, . . . , h, then the all zero 2h -tuple q = 0 belongs to Qh and L [Qh ] = 0, this implies that the hard-decision tuple z belongs to the set VdN1 ,d2 ,... ,dh (u1 , u2 , . . . , uh ). Without loss of generality, hereafter we assume that the components of δ
(δ1 , δ2 , . . . , δh ) satisfy δ1 ≥ δ2 ≥ · · · ≥ δh , δ1 > 0.

(8)

For each h with 1 ≤ h ≤ 3, the IPP P(Qh ) was solved in [3,4]. The main results on the IPP P(Qh ) obtained in [3,4] can be summarized as the following three results: Result 1. If h = 1 and (6) hold, then 

L [Q1 ] =

|ri |.

(9)

i∈D0 (u1 )(δ1 )

Result 2. If h = 2 and (6) and (8) hold, then  L [Q2 ] =

|ri |.

(10)

((δ −δ )/2) (δ ) i∈(D00 ∪D01 1 2 ) 1

Result 3. If h = 3 and (6) and (8) hold, then L [Q3 ] = min



min

1≤j≤2 0≤k≤kj

|ri |,

(11)

i∈Yj (k)

where k1
min{δ1 , (δ1 − δ2 )/2, (δ1 − δ3 )/2}, k2
min{(δ2 + δ3 )/2, (δ1 + δ3 )/2, (δ1 + δ2 )/2} and (k)

((δ −δ2 )/2 −k)

Y1 (k)
D011 ∪ (D000 ∪ D010 1 Y2 (k)


(k) D000

∪

((δ +δ )/2 −k) D100 2 3

∪

((δ −δ3 )/2 −k) (δ1 −k)

∪ D001 1

((δ +δ )/2 −k) D010 1 3

)

∪

,

((δ +δ )/2 −k) D001 1 2 .

(12) (13)




Furthermore, for each j, there is a kj with 0 ≤ kj ≤ kj such that i∈Yj (k) |ri | is non-increasing in [0, kj ], non-decreasing in [kj , kj ], respectively. Since with at most 4 operations of additions

and comparisons of real numbers we can determine whether i∈Yj (k+1) |ri |− i∈Yj (k) |ri | > 0 holds or not, the number of operations of additions and comparisons of real numbers for solving the IPP P(Q3 ) is of order N .   For greater h, the testing codition is stronger while the computational complexity for solving the IPP P(Qh ) grows. It is pointed out in [9] that, under the assumption of correlation uniqueness, i.e. L(u) = L(v) for any different tuples u and v in V N , the inequality L [Qh+1 ] > L [Qh ] holds if and only if the Qh -optimum is not in Qh+1 . We will solve the IPP for h = 4 in this paper.

Optimality Testing Algorithm

3

205

Algorithm for Solving the IPP P(Q4 )

In this section, we will present an algorithm to solve the IPP P(Q4 ) by splitting the IPP P(Q4 ) into at most 9 sub-IPP’s, the number of variables of each sub-IPP is only half of that of the IPP P(Q4 ). For any q ∈ Q∗4 , write S(q)
{α ∈ B 4 : qα ≥ 1}. For α ∈ B 4 , let wH (α) denote the Hamming weight of α. Let Y
{α ∈ B 4 : wH (α) ≤ 1}. For any with 1 ≤ ≤ 4, we call C
{α ∈ B 4 : α = 0} and D
Y ∪ {α ∈ 4 B : wH (α) = 2, α = 1} and E
Y ∪ {α ∈ B 4 : wH (α) = 2, α = 0} are M-sets. For each M-set Ξ, there is one and only one sequence in Ξ, denoted α(Ξ), such that ρ(Ξ) = (ρ(Ξ)1 , ρ(Ξ)2 , ρ(Ξ)3 , ρ(Ξ)4 )
δ + α(Ξ) is a 4-tuple over even integers or a 4-tuple over odd integers. Let Q(C ) denote the set of those 24 -tuples q ∈ Q4 which satisfy S(q) ⊆ C and ∆(q) = ρ(C ) = δ . Let Q(D ) denote the set of those 24 -tuples q ∈ Q4 which satisfy S(q) ⊆ D and ∆(q)j = ρ(D )j for all j with j = . Let Q(E ) denote the set of those 24 -tuples q ∈ Q4 which satisfy S(q) ⊆ E and ∆(q)j = ρ(E )j for j = 1, 2, 3, 4. Theorem 2. If the received N -tuple r satisfies (6), we have L [Q4 ] = L [Qmin ],

(14)

where Qmin
∪4=1 (Q(C ) ∪ Q(D ) ∪ Q(E )). In general, for some M-sets Ξ the sets Q(Ξ) are empty or covered by the others and thus should be excluded from further consideration. Theorem 3. If the 4-tuple δ satisfies (8), then we have  Qmin = Q(Ξ), Ξ∈{C1 }



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ℵ∗

where ℵ∗ is a set of M-sets defined as follows:
4 (i). ∅ if δ2 + δ3 + 1 < 0 and j=1 ρ(E1 )j < 0;
4 (ii). {E1 } if δ2 + δ3 + 1 < 0 and j=1 ρ(E1 )j ≥ 0; (iii). {D4 } if δ2 + δ3 + 1 ≥ 0 and δ1 + δ4 + 1 < 0; (iv). {E1 , D4 } if δ2 + δ3 + 1 ≥ 0 and δ1 + δ4 + 1 ≥ 0 and δ2 + δ4 + 1 < 0; (v). {E1 , E2 , D3 , D4 } if δ2 + δ4 + 1 ≥ 0 and δ3 + δ4 + 1 < 0;
4 (vi). {E1 , E2 , E3 , E4 , D2 , D3 , D4 } if δ3 + δ4 + 1 ≥ 0 and ρ(D1 )1 > j=2 ρ(D1 )j ;
4 (vii). {E1 , E2 , E3 , E4 , D1 , D2 , D3 , D4 } if ρ(D1 )1 ≤ j=2 ρ(D1 )j . If both of (6) and (8) are valid, from Theorems 2 and 3, we see easily that to solve the IPP P(Q4 ) it is needed only to solve the IPP P(Q(Ξ)) for the M-sets Ξ in {C1 } ∪ ℵ∗ . For each M-set Ξ ∈ {C1 , D1 , D2 , D3 , D4 , E1 , E2 , E3 , E4 }, we give a Sub-algorithm-Q(Ξ) to solve the IPP P(Q(Ξ)) in the next section. Using these sub-algorithms we can solve the IPP P(Q4 ) by the following algorithm. Algorithm for Solving the IPP P(Q4 ) Input. The N -tuple r with (6). The 4-tuple δ = (δ1 , δ2 , δ3 , δ4 ) defined by (5) and

206

Yuansheng Tang, Tadao Kasami, and Toru Fujiwara

satisfying (8). Dα and nα defined by (3) for all α ∈ B 4 . Output. L [Q4 ] = L[VdN1 ,d2 ,d3 ,d4 (u1 , u2 , u3 , u4 )]. Step 1. By the definitions, determine the 4-tuples ρ(D1 ) and ρ(E1 ). And then generate the set ℵ∗ by Theorem 3, and solve the IPP P(Q(Ξ)) for each Ξ ∈ {C1 } ∪ ℵ∗ by the Sub-algorithm-Q(Ξ). Step 2. Output minΞ∈{C1 }∪ℵ∗ L [Q(Ξ)] and END.   Since for all of the sub-algorithms the numbers of additions and comparisons of real numbers are of order δ12 , we see easily that the numbers of additions and comparisons of real numbers of above algorithm is of order δ12 , and thus is of order N 2 .

4 4.1

Sub-algorithms for Solving the IPP’s P(Q(Ξ)) Sub-algorithm for the Evaluation of L
[Q(C1 )]

Let δiC
max{0, (δ1 + δi + 1)/2}, i = 2, 3, 4. For integers d, k with 0 ≤ d ≤ δ1 , 0 ≤ k < δ4C , let Rd,k be the set of the 2h -tuples q in Q∗4 which satisfy (4) and    q0000 + q0001 = d, qα = δ1 − d,     α∈C1
(16) q0010 + q0011 ≥ δ2C − d, q0100 + q0101 ≥ δ3C − d,    q + q0010 + q0100 + q0110 ≥ k,   0000 qα = 0, for α ∈ C1 , 0001}. Clearly, Rd,k ⊆ Rd,k−1 ⊆ · · · ⊆ Rd,0 ⊆ Q∗4 . Let where C1
C1 \{0000,
l d
max{0, δ1 − α∈C
nα , δ2C − n0011 − n0010 , δ3C − n0101 − n0100 , δ2C + δ3C − δ1 } 1

and dr
min{δ1 , n0000 + n0001 }, then Rd,0 is not empty if and only if d satisfies dl ≤ d ≤ dr . We can show easily that  C Rd,δ4 . (17) Q(C1 ) = dl ≤d≤dr

For any d with dl ≤ d ≤ dr , let v(d)
δ1 − d − max{δ2C − d, 0} − max{δ3C − d, 0} ≥ 0, (δ2C −d)

V (d)
(D0011 ∪ D0010 )

(δ3C −d)

∪ (D0101 ∪ D0100 )

,

V ∗ (d)
V (d) ∪ (D0000 ∪ D0001 )(d) ∪ ((∪α∈C1
Dα )\V (d))(v(d)),

(18) (19) (20)

and let q d,0 denote the 24 -tuple which satisfies D(q d,0 ) = V ∗ (d). Then q d,0 is an C Rd,0 -optimum. We will show an iterative method to find an Rd,δ4 -optimum, or C determine Rd,δ4 = ∅, from q d,0 . For any 24 -tuple q ∈ Q∗4 and two sequences α, α ∈ B 4 , let ϕ(q, α, α ) denote the 24 -tuple which satisfies

ϕ(q, α, α )α = qα + 1, ϕ(q, α, α )α
= qα
− 1, (21) ϕ(q, α, α )α

= qα

, for all α ∈ B 4 \{α, α }.

Optimality Testing Algorithm

207

Lemma 2. Assume q d,k is an Rd,k -optimum which satisfies d,k d,k d,k d,k + q0010 + q0100 + q0110 = k. q0000 d,k

d,k





(22)

d,k+1

Let ξ(q )
{ϕ(q , α, α ) : (α, α ) ∈ A}∩R , where A
{(0000, 0001)}∪ ({0010, 0100, 0110} × {0011, 0101, 0111}). Then Rd,k+1 = ∅ if and only if ξ(q d,k ) = ∅, and the ξ(q d,k )-optimum must be an Rd,k+1 -optimum if ξ(q d,k ) = ∅. Assume q d,k is an Rd,k -optimum which satisfies (22) and ξ(q d,k ) = ∅. Since between L (ϕ(q d,k , α, α )) and L (ϕ(q d,k , β, β  )) with α = β or α = β we can find the smaller one with no operations of real numbers, we see easily that to determine the ξ(q d,k )-optimum it is enough to consider L (q) − L (q d,k ) for at most 4 tuples q of ξ(q d,k ) and thus with at most 7 operations of additions and comparisons of real numbers we can find an Rd,k+1 -optimum from q d,k . Hence, by iteration from q d,0 with at most 7δ4C operations of additions and comparisons C C of real numbers we can find an Rd,δ4 -optimum, or determine Rd,δ4 = ∅. With C respect to (17) and the definition of the Rd,δ4 -optimums, we can give the following Sub-algorithm-Q(C1) to compute L [Q(C1 )]. Sub-algorithm-Q(C1) Input. The N -tuple r with (6). The 4-tuple δ = (δ1 , δ2 , δ3 , δ4 ) defined by (5) and satisfying (8). Dα and nα defined by (3) for all α ∈ C1 . Output. L [Q(C1 )]. Step 1. Compute δjC , j = 2, 3, 4 and dl and dr . Step 2. If dl > dr , then output +∞ (i.e. Q(C1 ) = ∅) and END, otherwise, for each integer d with dl ≤ d ≤ dr , generate the set V ∗ (d) and determine the 24 -tuple qd,0 with D(q d,0 ) = V ∗ (d), and then, by using of Lemma 2, find an C C C Rd,δ4 -optimum q d,δ4 , or determine Rd,δ4 = ∅. C Step 3. Output mindl ≤d≤dr L (q d,δ4 ) and END.   The number of d with dl ≤ d ≤ dr is at most δ1 + 1. For each d with d ≤ d ≤ dr , it needs at most 7δ4C ≤ 7δ1 operations of additions and comparisons C C C of real numbers to find an Rd,δ4 -optimum q d,δ4 or to determine Rd,δ4 = ∅. To C compute L (q d,δ4 ) it needs δ1 − 1 additions of real numbers. Hence the total number of operations of additions and comparisons of real numbers for Subalgorithm-Q(C1) is not more than 8δ1 (δ1 + 1). l

4.2

Sub-algorithm for the Evaluation of L
[Q(D )]

For any integer k with 0 ≤ k ≤ 15, let b(k) = (b(k)1 , b(k)2 , b(k)3 , b(k)4 ) denote
4 the sequence in B 4 which satisfies j=1 b(k)j 2j−1 = k. Clearly, Q(D ) consists of the tuples q in Q∗4 which satisfy (4) and  3     (qsj + qs
j ) − 2(qsi + qs
i ) = ρ(D )ai , for i = 1, 2, 3,   j=0 (23) 3      (qsj − qs
j ) ≥ δ , and qα = 0 for α ∈ D ,  j=0

208

Yuansheng Tang, Tadao Kasami, and Toru Fujiwara

where s0 = 0000, s0 = b(2−1 ) and for i = 1, 2, 3

i+ , if i + ≤ 4, , si
b(2ai −1 ), si
b(2ai −1 + 2−1 ). (24) ai
i + − 4, otherwise,

For i = 1, 2, 3, let δiD
j∈Ii ρ(D )aj /2 =  j∈Ii δaj /2, where Ii
{1, 2, 3} \
3 {i} for i = 1, 2, 3. Let δ D
max{0, (δ + j=1 ρ(D )aj )/2} = max{0, (δ +
3 D ˙ d,k denote the set of the 2h j=1 δj )/2}. For nonnegative integers d, k, let R ∗ tuples q in Q4 which satisfy (4) and   qs0 + qs
0 = d, qs0 + qs1 + qs2 + qs3 ≥ k, qs + qs
i = δiD − d, for i = 1, 2, 3, (25)  i qα = 0, for α ∈ D . Clearly, R˙ d,k ⊆ R˙ d,k−1 ⊆ · · · ⊆ R˙ d,0 ⊆ Q∗4 . Let d˙r
min{ns0 + ns
0 , δ1D , δ2D , δ3D } and d˙l
max{0, max1≤i≤3 (δiD − nsi − ns
i )}. Then we can prove easily that  D Q(D ) = (26) R˙ d,δ −d . d˙l ≤d≤d˙r

For any d with d˙l ≤ d ≤ d˙r , let q˙ d,0 denote the 24 -tuple which satisfies  D D(q˙ d,0 ) = (Ds0 ∪ Ds
0 )(d) (Dsi ∪ Ds
i )(δi −d) .

(27)

1≤i≤3

Then q˙ d,0 must be an R˙ d,0 -optimum. The following lemma suggests an iterative D D method for finding an R˙ d,δ −d -optimum, or determining R˙ d,δ −d = ∅, from q˙ d,0 . Lemma 3. Assume q˙ d,k is an R˙ d,k -optimum which satisfies q˙sd,k + q˙sd,k + q˙sd,k + q˙sd,k = k. 0 1 2 3

(28)

˙ q˙ d,k )
{ϕ(q˙ d,k , sj , s ) : 0 ≤ j ≤ 3} ∩ R˙ d,k+1 . Then R˙ d,k+1 = ∅ if and only Let ξ( j ˙ q˙ d,k ) = ∅, and the ξ( ˙ q˙ d,k )-optimum is an R˙ d,k+1 -optimum if ξ( ˙ q˙ d,k ) = ∅. if ξ( Similar to the Sub-algorithm-Q(C1), we can devise a Sub-algorithm-Q(D) to compute L [Q(D )]. The detail of Sub-algorithm-Q(D) is given in [8]. The number of operations of additions and comparisons of real numbers for Subalgorithm-Q(D) is not more than 25δ1 (δ1 + 1)/2. 4.3

Sub-algorithm for the Evaluation of L
[Q(E )]

Clearly, Q(E ) consists of the 24 -tuples q in Q∗4 which satisfy (4) and   (qsj − qs∗j ) − qsi + qs∗i = ρ(E )ai , for i = 1, 2, 3,   qs0 + qs
0 +   j∈Ii 3   
+ − q (qsj + qs∗j ) = ρ(E ) , and qα = 0 for α ∈ E . q  s s  0 0 j=1

(29)

Optimality Testing Algorithm

209

where s∗i
b(15 − 2ai −1 − 2−1 ), i = 1, 2, 3. For i = 1, 2, 3, let δiE
(ρ(E ) + ρ(E )ai )/2 = (δ + δai )/2. Let  E 4 δ1 + δ2E + δ3E − δ , if δ Σ is even, 1 E ρ(E )j = 1 Σ δ
(30) (δ + 1), otherwise, 2 j=1 2
4 Σ 4 where δ
j=1 δj . Then the 2 -tuples q in Q(E ) can be given by    qs∗i = δiE − xi , qsi = xj − w, for i = 1, 2, 3, (31) j∈Ii  qs
0 = δ E − w, qs0 = 2w − x1 − x2 − x3 , and qα = 0, for α ∈ E . where w, x1 , x2 , x3 satisfy max{0, δ E − ns
0 } ≤ w ≤ δ E and 0 ≤ 2w − x1 − x2 − x3 ≤ ns0 ,  xj − w ≤ nsi , for i = 1, 2, 3. max{0, δi − ns∗i } ≤ xi ≤ δiE , 0 ≤ E

(32) (33)

j∈Ii

For max{0, δ E − ns
0 } ≤ w ≤ δ E , let Ω(w) denote the set of pairs π
(x1 , x2 , x3 ) which satisfy (32) and (33). For π ∈ Ω(w), let q w (π) be the 24 -tuple of Q(E ) defined by (31). We write Lw (π)
L (q w (π)). For any subset Ω  of Ω(w), let Lw [Ω  ]
minπ∈Ω
Lw (π), and write Lw [∅]
+∞. If a pair π ∈ Ω  satisfies Lw (π) = Lw [Ω  ], we call it Ω  -pair. We will show a method to evaluate L [Q(E )] by finding an Ω(w)-pair for each w with Ω(w) = ∅. For integers w, x, let Ω(w, x)
{π ∈ Ω(w) : π = (·, ·, x)}. At first, we consider to determine Ω(w, x)-pairs. Let Υ denote the set of pairs π  = (x1 , x2 , x3 ) with xi ∈ {1, 0, −1}. For π ∈ Ω(w) and π  ∈ Υ , we say that we can grow π in the π  -direction to π + π  if Lw (π) > Lw (π + π  ). For any nonempty set Ω(w, x), we can find an Ω(w, x)-pair by the following growth procedure. Growth Procedure of an Ω(w, x)-pair τ (w, x) Preparation. Select an arbitrary pair π of Ω(w, x) as the seed. Step 1. We consider to grow π in the (1, 0, 0)-direction and (−1, 0, 0)-direction till we can not do anymore and then goto Step 2. Step 2. If we can grow π in π  -direction for some π  of {(1, 0, 0), (−1, 0, 0), (0, 1, 0), (0, −1, 0)}, we grow it in π  -direction step by step till we can not do anymore and then goto Step 2, otherwise goto Step 3. Step 3. We consider to grow π in (1, −1, 0)-direction and (−1, 1, 0)-direction till we can not do anymore and then output τ (w, x) = π and END.   Lemma 4. The growth procedure outputs an Ω(w, x)-pair τ (w, x) with at most 16w operations of additions and comparisons of real numbers. Sometimes it is not easy to select a seed for this growth procedure. A concrete method for giving a seed for the growth procedure is shown in [8]. However, it is not needed to find all the Ω(w, x)-pairs by the growth procedure. Indeed, the following lemma suggests a simple method for finding a Ω(w)-pair from a known Ω(w, x)-pair.

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Lemma 5. Let τ (w, x) be an Ω(w, x)-pair and Υ ∗
{(0, 0, 0), (−1, 0, 0), (0, −1, 0), (1, −1, 0), (−1, 1, 0), (−1, −1, 0)}. Then 1. For ε ∈ {1, −1}, if Ω(w, x + ε) = ∅, then there exists a pair π  in Υ ∗ such that τ (w, x) + (0, 0, ε) + ε · π  is an Ω(w, x + ε)-pair. 2. τ (w, x) is an Ω(w)-pair if and only if Lw (τ (w, x)) ≤ min{Lw [Ω(w, x − 1)], Lw [Ω(w, x + 1)]}. For ε ∈ {1, −1}, according to Lemma 5, from an Ω(w, x)-pair τ (w, x) we can get an Ω(w, x + ε)-pair τ (w, x + ε) with at most 20 operations of additions and comparisons of real numbers, and determine whether Lw (τ (w, x)) ≤ Lw (τ (w, x+ ε)) holds or not with 5 more operations of additions and comparisons of real numbers. Then, with respect to Lemmas 4 and 5 and Ω(w, x) = ∅ for x > w, we can find an Ω(w)-pair τ (w) with at most 16w+25w/2 operations of additions and comparisons of real numbers. Furthermore, since the number of additions and comparisons of real numbers for computing Lw (τ (w)) is δ E +δ1E +δ2E +δ3E −2w −1 and Ω(w) = ∅ for w > δ E , we can devise a Sub-algorithm-Q(E) to evaluate
δ E L [Q(E )] with at most w=0 (53w/2 + δ E + δ1E + δ2E + δ3E ) ≤ 63δ1 (2δ1 + 1)/2 operations of additions and comparisons of real numbers. The detail of the Subalgorithm-Q(E) is given in [8].

References 1. D. J. Taipale and M. B. Pursley, “An Improvement to Generalized MinimumDistance Decoding,” IEEE Trans. Inform. Theory, vol. 37, pp. 167–172, Jan. 1991. 2. T. Kaneko, T. Nishijima, H. Inazumi and S. Hirasawa, “An Efficient Maximum Likelihood Decoding Algorithm for Linear Block Codes with Algebraic Decoder,” IEEE Trans. Inform. Theory, vol. 40, pp. 320–327, Mar. 1994. 3. T. Kasami, T. Takata, T. Koumoto, T. Fujiwara, H. Yamamoto and S. Lin, “The Least Stringent Sufficient Condition on Optimality of Suboptimal Decoded Codewords,” Technical Report of IEICE, Japan, IT94-82, Jan. 1995. 4. T. Kasami, T. Koumoto, T. Takata and S. Lin, “The Effectiveness of the Least Stringent Sufficient Condition on Optimality of Decoded Codewords,” Proc. 3rd Int. Symp. Commun. Theory & Appl., Ambleside, UK, pp. 324–333, July 1995. 5. T. Koumoto, T. Takata, T. Kasami and S. Lin, “A Low-Weight Trellis Based Iterative Soft-Decision Decoding Algorithm for Binary Linear Block Codes,” IEEE Trans. Inform. Theory, vol. 45, pp. 731–741, Mar. 1999. 6. T. Kasami, Y. Tang, T. Koumoto and T. Fujiwara, “Sufficient Conditions for Ruling-Out Useless Iterative Steps in a Class of Iterative Decoding Algorithms,” to appear in IEICE Trans. Fundamentals, vol. E82-A, Oct. 1999. 7. T. Koumoto and T. Kasami, “An Iterative Decoding Algorithm Based on Information of Decoding Failure,” Proc. of the 20th Symp. Inform. Theory & Appl., Matsuyama, Japan, pp. 325–328, Dec. 1997. 8. Y. Tang, T. Kasami and T. Fujiwara, “An Optimality Testing Algorithm for a Decoded Codeword of Binary Block Codes,” Technical Report of NAIST, Japan, NAIST-IS-TR98012, Oct. 1998. 9. T. Kasami, “On Integer Programming Problems Related to Soft-decision Iterative Decoding Algorithms,” to appear in Proc. of the 13th Int. Symp. AAECC, Honolulu, HI, USA, Nov. 1999.

Recursive MDS-Codes and Pseudogeometries Elena Couselo1, Santos Gonzalez1 , Victor Markov2, and Alexandr Nechaev2 1

2

University of Oviedo, Spain Center of New Informational Technologies of Moscow State University, Russia [email protected]

Abstract. In [2,4] the notion of a recursive code was introduced and some constructions of recursive MDS codes were proposed. The main result was that for any q
∈ {2, 6} (except possibly q ∈ {14, 18, 26, 42}) there exists a recursive MDS-code in an alphabet of q elements of length 4 and combinatorial dimension 2 (i.e. a recursive [4, 2, 3]q -code). One of the constructions we used there was that of pseudogeometries; it enabled us to show that for any q > 126 (except possibly q = 164) there exists a recursive [4, 2, 3]q -code that contains all the “constants”. One part of the present note is the further application of the pseudogeometry construction which shows that for any q > 164 (resp. q > 26644) there exists a recursive [7, 2, 6]q -code (resp. [13, 2, 12]q -code) containing ”constants”. Another result presented here is a negative one: we show that there is no nontrivial pseudogeometry consisting of 14, 18, 26 or 42 points with no lines of order 2, 3, 4 or 6, so the pseudogeometry construction cannot be applied for settling the question mentioned in the above. In both cases the usage of computer is essential.

Introduction A code K ⊂ Ω n in an alphabet Ω of q elements is called k-recursive, 1 ≤ k < n, if there exists a function f : Ω k → Ω such that K consists of all the rows u(0, n − 1) = (u(0), . . . , u(n − 1)) ∈ Ω n with the property u(i + k) = f(u(i, i + k − 1)),

i ∈ 0, n − k − 1.

In other words K is the set of all output n-sequences of a feedback shift register with a feedback function f. We denote K = K(n, f) and investigate the existence problem for MDS-codes of such type, i.e. recursive [n, k, n − k + 1]q -codes. In connection with this, we consider the following three parameters: n(k, q) – maximum of lengths of MDS codes K of (combinatorial) dimension k (|K| = q k ) in alphabet Ω of cardinality q. r n (k, q) – maximum of lengths of k-recursive MDS codes of the same type. nir (k, q) – maximum of lengths of k-recursive MDS codes of the same type which contains all the “constants”, i.e. all the words (a, . . . , a) : a ∈ Ω. It is clear that nir (k, q) ≤ nr (k, q) ≤ n(k, q). Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 211–220, 1999. c Springer-Verlag Berlin Heidelberg 1999


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Here we study only the case k = 2. In this case our problem has the following simplification. Let us define the i-th recursive derivation f (i) (x, y) of the recursive law f(x, y) recursively: f (0) (x, y) = f(x, y), f (1) (x, y) = f(y, f(x, y)), f (i) (x, y) = f(f (i−2) (x, y), f (i−1) (x, y)). The necessary condition for K(n, f) to be MDS is that (Ω, f(x, y)) is a quasigroup. We say that a quasigroup Ω is t-stable if (Ω, f (i) ) is also a quasigroup for i ∈ 1, t, It was proved in [2, Theorem 4] that nr (2, q) ≥ m (resp., nir (2, q) ≥ m) if and only if there exists an (m − 3)-stable (idempotent) quasigroup f(x, y) of order q. The aim of this note is to show how the notion of pseudogeometries can be used to provide some estimations of nir (2, q). We show also some limitations of this technique. The term “pseudogeometry” was taken from [1] where pseudogeometries were used to construct Latin squares which are orthogonal to their transposes. Since the construction of the quasigroups under consideration may be reduced to the construction of orthogonal Latin squares with special properties, it was natural to use pseudogeometries for this purpose also. Now we remind that a pseudogeometry is a pair P = (P, L), where L is a set of non-empty subsets of the non-empty set P , if for any different x, y ∈ P there is a unique subset L(x, y) ∈ L such that {x, y} ⊆ L(x, y). The elements of the sets P and L are called points and lines, respectively. We consider only the finite pseudogeometries. The cardinality of a line will be often called its length. Note that if one adds to or removes from L any set of one-point sets, the pair (P, L) remains a pseudogeometry. We will say that a pseudogeometry P = (P, L) is nontrivial if P ∈ L. The standard examples of pseudogeometries are affine and projective planes over finite fields.

1

Idempotent Quasigroups of Large Orders

For the convenience of the reader we recall here the definition of the accompanying pseudogeometry for a mutually orthogonal set of Latin squares [2]. Let q be some positive integer and suppose that there exists a mutually orthogonal set of s Latin squares f1 , . . . , fs of order q, fi : (0, q − 1)2 → 0, q − 1. Consider the set P of all pairs (x, y), x ∈ 0, q − 1, y ∈ −1, s. Define lines of two types: (1) horizontal lines: Hy = {(x, y) : x ∈ 0, q − 1}, for any fixed y ∈ −1, s; (2) skew lines: Sij = {(i, −1), (j, 0)} ∪ {(fy (i, j), y) : y ∈ 1, s}, for any fixed i, j ∈ 0, q − 1. Let H = {Hy : y ∈ −1, s}, S = {Sij : i, j ∈ 0, q − 1}, L = H ∪ S. It is easy to see that (P, L) is a pseudogeometry with lines of length q and s+2. Later we use this construction mostly in the case of primary q, when one can take s = q − 1. Note that in this case one can obtain the accompanying pseudogeometry by removing one point from the projective plane over a field of order q.

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Let (P, L) be an arbitrary pseudogeometry, and P  an arbitrary non-empty subset of the set P . Let L = {L ∩ P  : L ∈ L, |L ∩ P  | > 1}. It is easy to see that (P  , L ) is also a pseudogeometry. We shall call (P  , L ) the reduced pseudogeometry for the pseudogeometry (P, L). The following result is very close to one of the main constructions of [1]: Theorem 1 Let (P, L) be a pseudogeometry. Then nir (2, |P |) ≥ min {nir (2, |L|) : L ∈ L}. ✷ The proof follows that of [2, Theorem 10]. Let t = min {nir (2, |L|) : L ∈ L} − 3 and define an operation fL : L2 → L for every L ∈ L in such a way that L becomes an idempotent t-stable quasigroup. Then for any x, y ∈ P such that x = y let f(x, y) = fL (x, y) where L = L(x, y) and let f(x, x) = x, for any x ∈ P . It is obvious that the operation is well-defined and (P, f) is an idempotent quasigroup. So if y = f(x, y) then x = y, x, y ∈ P . An easy induction argument (i−1) (i) shows that for any i ∈ 1, t, any L ∈ L and any x, y ∈ L, fL (x, y) = fL (x, y) (i) implies x = y. So if x = y, then f (i) (x, y) = fL (x, y) where L = L(x, y), (i) (i) and f (x, x) = x, for any x ∈ P , hence (P, f ) is again a quasigroup. The application of [2, Theorem 4] finishes the proof. ✷ Now let I(n) be the set of integers q such that nir (2, q) ≥ n. It is known that 2, 3, 4, 6 ∈ I(4) and that r ∈ I(4) for any q ≥ 127 (except possibly q = 164) [2, Corollary 8]. The authors do not know if 10 ∈ I(4) but any prime number p ≥ n and any primary number q > n belong to I(n) as well as any product of such numbers (see [2, Propositions 10, 11]). The following theorem is a slight generalization of [2, Theorem 11], but it gives many extra numbers in the search described below. Theorem 2 Let n be an integer, n ≥ 4, and s, t, l, m, d1 , . . . , dl such nonnegative integers that (a) there exist t + l + m − 2 mutually orthogonal Latin squares of order s; (b) s, t, t + 1, . . . , t + l, d1 , . . . , dl ∈ I(n); (c) if m > 0 then also t + l + 1, t + l + m ∈ I(n); (d) 1 ≤ di < s for i = 1, . . . , l. Then q = st + d1 + · · · + dl + m ∈ I(n). ✷ Consider the accompanying pseudogeometry (P, L) for the given set of mutually orthogonal Latin squares. It contains t+l +m horizontal lines which we shall consider as numbered from the first one to the (t + l + m)-th. Take one skew line and call it the distinguished vertical line. Delete s − dl points from the (t + l)-th horizontal line, s − dl−1 points from the (t + l − 1)-th one and so on in such a way that no point of the distinguished vertical line is removed. Then delete all the points of the remaining m horizontal lines except that of the distinguished vertical line.

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Let P  be the set of the remaining points and (P  , L ) the reduced pseudogeometry. The situation is illustrated by Fig. 1 (here s = 11, t = 7, l = 1, d1 = 5, m=3, empty circles represent deleted points, full circles represent the points of P  , the left column of points represents the distinguished vertical line). • • • • • • • • • • •

• • • • • • • • ◦ ◦ ◦

• • • • • • • • ◦ ◦ ◦

• • • • • • • • ◦ ◦ ◦

• • • • • • • • ◦ ◦ ◦

• • • • • • • ◦ ◦ ◦ ◦

• • • • • • • ◦ ◦ ◦ ◦

• • • • • • • ◦ ◦ ◦ ◦

• • • • • • • ◦ ◦ ◦ ◦

• • • • • • • ◦ ◦ ◦ ◦

• • • • • • • ◦ ◦ ◦ ◦

Fig. 1. Reduced pseudogeometry. If L is a horizontal line in L such that |L ∩ P  | > 1 then |L ∩ P  | is either s or one of di , i ∈ 1, l. on L ∩ P  . If L is a skew line in L, and it is not the distinguished vertical line, then t ≤ |L ∩ P  | ≤ t + l + 1 since L contains not more than one remaining point of each of l “shortened” horizontal lines and not more then one point of the distinguished vertical line lying on the last m horizontal lines. Finally, if L is the distinguished vertical line then |L∩P  | = |L| = t+l +m. Now the result follows from Theorem 1. ✷ As in [2] we obtain the following Corollary 3 If there exists a number k such that 2k − 1, 22k+1 ⊂ I(n) then q ∈ I(n) for any q ≥ 2k − 1. ✷ The proof is exactly the same as that of [2, Corollary 8] and therefore is omitted. ✷ Now we shall outline the idea of the “experimental” part of our work. First we fixed some reasonable value qmax and filled the array of current estimations of nir (2, q) for q ≤ qmax with initial values based on the known estimations for prime and primary q and on the quasigroup product construction. Then the program repeatedly checked the numbers q ∈ 10, qmax for the existence of numbers satisfying the conditions of Theorem 2 (with prime and primary s and l ≤ 3), taking into account the estimations numbers found on the previous repetitions. The usage of this algorithm was limited because we had to keep all the found numbers in the computer’s memory. When we obtained rather long series of consecutive numbers in I(n) for some n > 4 we passed the result to the second step of the algorithm which used the Theorem 2 only with prime and primary s and with l = 0 (we did not need to keep all the estimations now). We summarize the computation results in the following

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Theorem 4 nir (2, q) ≥ 7 for all q > 164; nir (2, q) ≥ 13 for all q > 26644. One can conjecture that for any integer n there exists an integer q0 = q0 (n) such that nir (2, q) ≥ n for all q ≥ q0 . For n(2, q), the corresponding fact is well known (see e.g. [5]): n(2, q) ≥ q 10/143 . Note however that this gives n(2, q) ≥ 7 only for q > 714.3 ≈ 1.2 · 1012 and n(2, q) ≥ 13 only for q > 1314.3 ≈ 8.5 · 1015 .

2

Nonexistence Theorem

In this section we will consider the question of existence of a pseudogeometry with lines whose lengths belong to a given set of integers. Recall that it is still an open question if there exists a 1-stable quasigroup of order q if q ∈ {14, 18, 26, 42}. On the other hand, there is no 1-stable idempotent quasigroups of order q if q ∈ {2, 3, 4, 6}. So it would be desirable, as the first step to the positive solution of the question by the pseudogeometry technique, to construct the pseudogeometries with 14, 18, 26 and 42 points without lines of length 2,3,4,6. But instead of that we have proved a negative result. First we give a direct proof that there is no pseudogeometry with 14, 18 or 26 points without lines of length 2,3,4,6. For the pseudogeometries with 42 points and the same restrictions on lines we deduce some properties that allow us to carry out an exhaustive computer search which shows that they do not exist also. Let π = (P, L) be a pseudogeometry. For any point x ∈ P denote by Lx the set of lines containing x, and for any x ∈
P and L ∈ L such that x ∈ L let M(x, L) = {L(x, y) : y ∈ L}, M(x, L) = {L : L ∈ M(x, L)}. Let m(π) and M(π) be the minimal and the maximal length of line in π, respectively. Denote by Li the set of lines of length i and let ki = |Li |. We begin with some rather elementary observations. Lemma 5 If π = (P, L) is a pseudogeometry then |P |(|P | − 1) = Σi i(i − 1)ki

(2.1)

✷ It is sufficient to note that the both parts of (2.1) are equal to the number of ordered pairs (x, y) ∈ P × P such that x = y, taking into account that every such pair belongs to L × L for exactly one line L ∈ L. ✷ Lemma 6 If π = (P, L) is a pseudogeometry with even (odd) number of points then every point in P belongs to odd (even) number of lines of even length. ✷ Let x ∈ P . Then P \ x is a disjoint union of the sets L \ {x}, where L ∈ Lx , so |P | − 1 = ΣL∈Lx (|L| − 1) ≡ k

mod 2,

where k is the number of lines of even length in Lx . ✷ Lemma 7 For any nontrivial pseudogeometry π = (P, L), (m(π) − 1)M(π) + 1 ≤ |P |

(2.2)

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✷ Let L be a line of length M(π). Take a point x ∈ P \ L and note that |M(x, L)| = |{x} ∪ (



L(x, y) \ {x})| = 1 + Σy∈L |L(x, y) − 1|,

y ∈L

since the union in the previous line is a disjoint one. So |P | ≥ |M(x, L)| ≥ 1 + (m(π) − 1)M(π). ✷ Note that for a projective plane the equality holds in (2.2). We will say for brevity that a pseudogeometry is admissible if it is nontrivial and does not contain a line whose length is 2, 3, 4 or 6. Now we arrive at the following Prop osition 8 If π = (P, L) is an admissible pseudogeometry, then either |P | > 28 or |P | ∈ {21, 25}. ✷ First suppose that m(π) ≥ 7. Then Lemma 7 gives |P | ≥ 7 · 6 + 1 = 43. So we can suppose that m(π) = 5. Again if M(π) ≥ 7 then |P | ≥ 7 · 4 + 1 = 29, and only the case remains when all the lines have length 5. If any two lines in π have a common point then there is equality in (2.2) and so |P | = 21. Suppose there are two “parallel” lines, say L1 and L2 . Consider any line L that intersects L1 at some point x. Suppose L ∩ L2 = ∅. Then we have a disjoint union {x} ∪ (L1 \ {x}) ∪ (L \ {x}) ∪ (M(x, L2 ) \ {x}) ⊆ P, so |P | ≥ 1+4+4+4·5 = 29. So if |P | ≤ 28 then for any point x ∈ L1 and any point y ∈ L1 the line L(x, y) intersects L2 . This means that P = (L1 \ {x}) ∪ M(x, L2 ) and |P | = 4 + 21 = 25. ✷ Proposition 8 means in particular that there is no hope to construct an idempotent 1-stable quasigroup of order 14, 18 or 26 using pseudogeometries. From this point till the end of the section we suppose that π = (P, L) is an admissible pseudogeometry with 42 points. We already know from Lemma 7 that m(π) = 5. The following lemma gives a rough estimation of M(π). Lemma 9 M(π) < 10. ✷ By virtue of Lemma 7, M(π) < 11, since 4 · 11 + 1 > 42. Let L be a line in π of length 10 and x ∈ P \ L. If some line in M(x, L) contains more than 5 (and hence more than 6) points, then as in the proof of Lemma 7 one has |P | ≥ 1 + 6 + 9 · 4 = 43 > 42, a contradiction. So all the lines L0 , . . . , L9 have length 5, so |M(x, L)| = 1 + 10 · 4 = 41. This means that there exists a point y ∈ P \ M(x, L). So there is a disjoint union L(x, y) ∪ (M(x, L) \ {x}) ⊆ P , so |P | ≥ |L(x, y)| + 10 · 4 ≥ 5 + 40 > 42, a contradiction again. ✷ Now the only even length of a line in P is 8, so by Lemma 6 every point in P belongs to 1, 3 or 5 lines of length 8 (7 lines of length 8 with one common point contain 50 points). Lemma 10 Every two lines of length 8 in π intersect each other.

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✷ Suppose L1 , L2 ∈ L, |L1 | = |L2 | = 8 and L1 ∩ L2 = ∅. Consider a point x ∈ P \ (L1 ∪ L2 ). Take a line L ∈ Lx of length 8. Suppose first that L ∩ L1 = ∅ and L ∩ L2 = ∅. As in the proof of Lemma 7, there is a disjoint union {x} ∪ (L1 \ {x}) ∪ (L \ {x}) ∪ (M(x, L2 ) \ {x}) ⊆ P, so |P | ≥ 1 + 2 · 7 + 4 · 8 = 47 > 42, a contradiction. If L ∩ L1 = ∅ and L ∩ L2 = ∅ then L ∈ M(x, L2 ), so again we have a disjoint union {x} ∪ (L1 \ {x}) ∪ (M(x, L2 ) \ {x}) ⊆ P, but now one of the lines in M(x, L2 ) has length 8, so |M(x, L2 )\{x}| ≥ 7·4+7 = 35 and |P | ≥ 1 + 7 + 35 = 43 > 42. So the only remaining possibility is that line L does not intersect lines L1 and L2 . Let L3 = L and consider any point y in P \ (L1 ∪ L2 ∪ L3 ). Again there is a line of length 8 that contains y and by the preceding argument it does not intersect lines L1 , L2 and L3 . Repeating this argument, we have that |P | is a multiple of 8, which is wrong. ✷ Lemma 11 There is no point in π that belongs to 5 lines of length 8. ✷ Suppose on the contrary that x ∈ P is such a
point and L1 , . . . , L5 are 5 lines 5 of length 8 such that x ∈ Li , i ∈ 1, 5. Let M = i=1 Li . Since |M| = 1+5·7 = 36 and the set P \ M is covered by disjoint sets L(x, y) \ {x}, where y spans P \ M, of cardinality not less then 4 each, the only possibility is that P \ M = L \ {x} for some line L of length 7. Take any point y ∈ L \ {x} and consider a line K of length 8 in Ly . Then K ∈ {L, L0 , . . . , L5 }, |K ∩ M| = 5 and |K ∩ L| = 1, so K contains 2 “extra points”. A contradiction. ✷ Now let us introduce some more notation. Let Ni be the set of points in P that belong to exactly i lines of length 8. Lemma 12 |N3 | = 7, |N1 | = 35, k8 = 7, k9 = 0. Moreover, the reduced pseudogeometry π  = (N3 , L ) is the projective plane P2 (F2 ). ✷ Let ni = |Ni |, i = 1, 3. Then by Lemma 6 and Lemma 11 we have n1 + n3 = 42.

(2.3)

Counting all the points of the lines in L8 as different ones and taking into account that each point of N3 will be counted 3 times, we obtain n1 + 3n3 = 6k8 .

(2.4)

Now let ∆ ⊆ L8 × L8 be the diagonal: ∆ = {(L, L) : L ∈ L8 } and consider the map ϕ : L8 × L8 \ ∆ → N3 defined by the rule: ϕ(L1 , L2 ) is the intersection point of the lines L1 and L2 . This map is well-defined by Lemma 10. By definition of N3 , |ϕ−1 (x)| = 6 for any x ∈ N3 . So 6n3 = k8 (k8 − 1).

(2.5)

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Solving the equation system (2.3)—(2.5) and using the condition n1 ≥ 0 we obtain N3 = 7, N1 = 35, k8 = 7. Since each point in P belongs to some of 7 lines in L8 , there is no line of length 9 in L, so k9 = 0. Now consider the reduced pseudogeometry π  = (N3 , L ). First check that every two different points x, y in N3 belong to one line of length 8 in L. Really, if (L8 ∩ Lx ) ∩ (L8 ∩ Ly ) = ∅ then three lines of L8 ∩Lx have 9 intersection points with three lines of L8 ∩Lx , which is impossible. So |N3 ∩ L| ≤ 1 for any line L ∈ L \ L8 ), and so |L | = |L8 | = 7. Note also that four points of N3 cannot belong to one line: every one of these 4 points would belong to two another lines of length 8 and all these 9 lines would be different, which contradicts the equality k8 =7. Using Lemma 5 for π  we have k2 + k3 = 7, 2k2 + 6k3 = 42, so k2 = 0 and k3 = 7. So we have finally that every line in L contains 3 points and every point is contained in 3 lines, while the numbers of points and lines in π  are equal to 7. So π  is isomorphic to the projective plane P2 (F2 ). ✷ The configuration described by Lemma 12 is presented in Fig. 2, where the lines of length 8 are drawn and the points of N3 and N1 are marked.

Fig. 2. Lines of length 8 in the pseudogeometry π.

Lemma 13 Every point of N3 belongs to 5 lines of length 5 and to no line of length 7. ✷ Let x ∈ N3 and L ∈ Lx \ L8 . Then L \ {x} is contained in the union of four disjoint sets K \ N3 , K ∈ L8 \ Lx , and each of these sets has one common point with L. So |L| = 5. On the other hand, each of these sets contains 5 points, so |Lx ∩ L5 | = 5. ✷

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Lemma 14 Every point of N1 belongs to some line of length 7. ˜ ∈ L8 \ Lx . Then M(x, L) ˜ = 36 and ✷ Let x ∈ N1 and L ∈ Lx ∩ L8 and L remaining set of 6 points cannot be the disjoint union of sets of cardinality 4. ✷ Now we pass again to the “experimental” part of our arguments. Applying Lemma 5 and Lemma 12 to π we obtain the equation 42 · 41 = 20k5 + 42k7 + 7 · 56. It has three solutions in non-negative integers: (a) k5 = 14, k7 = 25; (b) k5 = 35, k7 = 15; (c) k5 = 56, k7 = 5. The case (a) is impossible since by Lemma 13 k5 ≥ 35. In the case (b) it is evident that every point x ∈ N1 belongs to 1 line of length 8 and to 4 lines of length 5, so it must belong to 3 lines of length 7, and that every line of length 7 must intersect every line of length 8 in some point of N1 . If the points of L \ N1 are marked by numbers 0, . . . , 4 for every line L of length 8 then every line of length 7 may be presented as vector (a0 , . . . , a6 ), were ai ∈ 0, 4. In other words the lines of length 7 must form a code of length n = 7 over an alphabet of q = 5 elements having cardinality c = 15 and distance d = 6. Such codes seem to be of independent interest because their parameters lay on Plotkin boundary: d≤

q−1 c · ·n q c−1

(see, e.g., [9, Theorem 1.1.39]). So we first asked if such codes exist. The answer was affirmative: Proposition 15 There exist codes of length n = 7 over an alphabet of q = 5 elements having cardinality c = 15 and distance d = 6. ✷ The codes in question were constructed by a computer program. ✷ But the attempt to add the lines of length 5 failed: exhaustive search showed that it is impossible. In the case (c) it is easy to deduce from Lemma 14 that the lines of length 7 do not intersect each other, and that every point in N1 belongs to 4 lines of length 5 that connect it with points of N3 and to one line of length 5 that is contained in N1 . Again these restrictions make the exhaustive search possible and again it showed that the desired configuration does not exist. We can summarize the results on the pseudogeometries with 42 points as follows. Theorem 16 Any nontrivial pseudogeometry with 42 points must contain a line of length 2,3,4 or 6.

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References 1. R.K.Brayton, D.Coppersmith & J.Hoffman, “Self orthogonal Latin squares of all orders n
= 2, 3, 6”, Bull, Amer. Math. Soc., 80 (1974), 116–119. 2. Couselo E., Gonzalez S., Markov V., Nechaev A. “Recursive MDS-codes and recursively differentiable quasigroups”, Diskr. Math. and Appl., V.8, No. 3, 217-247, VSP, 1998. 3. Couselo E., Gonzalez S., Markov V., Nechaev A. “Recursive MDS-codes and recursively differentiable quasigroups-II”, Diskr. Math. and Appl., to appear. 4. Couselo E., Gonzalez S., Markov V., Nechaev A. “Recursive MDS-codes”. In: Workshop on coding and cryptography (WCC’99). January 11-14, 1999, Paris. Proceedings. 271-277. 5. J.D´enes & A.D.Keedwell, “Latin Squares and their Applications”. Akad´emiai Kiad´ o, Budapest; Academic Press, New York; English Universities Press, London, 1974. 6. J.D´enes & A.D.Keedwell, “Latin squares. New developments in the theory and applications”. Annals of Discrete Mathematics, 46. North-Holland, Amsterdam, 1991. 7. Heise W. & Quattrocci P. “Informations- und codierungstheorie”, Springer, 1995, Berlin–Heidelberg. 8. MacWilliams F.J. & Sloane N.J.A. “The theory of Error-Correcting Codes”. Elsevier Science Publishers, B.V., 1988. North Holland Mathematical Library, Vol. 16. 9. Tsfasman M.A., Vlˇ adut¸ S.G., “Algebraic-geometric codes”, Kluwer Acad. Publ., Dordrecht–Boston–London, 1991.

Strength of MISTY1 without FL Function for Higher Order Differential Attack Hidema Tanaka, Kazuyuki Hisamatsu, and Toshinobu Kaneko Department of Electrical Engineering, Science University of Tokyo 2641 Yamazaki Noda city 278-8510 Japan

Abstract. The encryption algorithm MISTY is a ”provably secure” one against Linear and Differential cryptanalysis. Since the designer showed 3 round MISTY1 without FL function is provable secure, we omit FL to estimate the strength for Higher Order Differential Attack. This attack is a chosen plain text attack and uses the value of higher order differential of output to derive an attacking equation for sub-keys. The value depends on the degree of output and the degree depends on the choice of plain texts. We show that the effective chosen plain text and 5 round MISTY1 without FL is attackable using 11 different 7th order differentials. And we show the attacks to remaining sub-keys by the determined sub-keys and intermediate values. Intermediate value is a constant in the process of encryption. As the result, we can determine all sub-key in 5 round MISTY1 without FL.

1

Introduction

Linear and Differential cryptanalysis are the powerful attack to DES-like cryptosystems. As a counter measure, the concept of ”provably secure” against them is proposed [5][7]. The encryption algorithm MISTY, proposed by Matsui in 1996, is a block cipher designed under such concept [1]. There are two types of algorithm, MISTY1 and MISTY2. MISTY has F-function named FO. MISTY1 has DES-like structure with 8 rounds FO function. The designer claimed that 3 round MISTY1 without FL function is enough to have provable security against Linear and Differential cryptanalysis. Since ”provable security” is guaranteed only by FO, we analyzed modified MISTY1, which has no FL and reduced number of rounds, by Higher Order Differential Attack. This is a chosen plain text attack which uses the fact that the value of higher order differential of the output does not depend on sub-keys. The order for the attack depends on the chosen plain text and it affects the number of plain texts and the computational cost. We show the outline of the attack in Section 2 and 3. In Section 4, we show the effective chosen plain text which enables the attack to 5 round MISTY1 without FL. The attack is consisted of 3 main parts. Section 4 shows an attack using 7th order differentials to determine 4 sub-keys in 5th round. Section 5 shows the estimation of intermediate values. Intermediate value is a constant in the process of encryption, which is a function of fixed part of plain text and Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 221–230, 1999. c Springer-Verlag Berlin Heidelberg 1999


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P : Plain text 64[bit]

❄ ❄ FO ✲ ❤  ✘✘✘ ✘✘  ❄ FO ✲ ❤  ✘✘✘ ✘✘  ❄ FO ✲ ❤  ✘✘✘ ✘✘ 

❄ FI

❤✛  ✘✘✘ ✘✘ 

❤✛ ❄

FI

FI

❤✛truncate  ✘✘✘ ✘✘  Kijk✲ ❤

❤✛  ✘✘✘ ✘✘ 

 ✘✘✘ ✘✘ 

❄

K

C : Cipher text 64[bit] (i) MISTY without FL

❄

❤✛zero-extend  ✘✘✘ ✘✘  Kijk✲ ❤

❤✛  ✘✘✘ ✘✘ 

·· ·

·· ·

Kijk✲ ❤ S9

(ii) Equivalent FO

S7

S9

❤✛zero-extend  ✘✘✘ ✘✘  ❄

(iii) Equivalent FI

Fig. 1. The modified MISTY (where K and Kijk denotes the eqivalent sub-keys).

sub-keys. Section 6 shows an attack using differentials of intermediate values to determine remaining sub-keys.

2

Modified MISTY1

In the following, we discuss one type of the algorithm, MISTY1. The successful attack of Linear Attack or Differential attack depends on maximum linear or differential probability. Let p be the average probability of them for F function. From the theorem shown by Nyberg and Knudsen [5], the probability for 3 round F functions equals to p2 . If the probability p2 is low enough, they call such property as ”provable secure” against Linear and Differential Attack. The designer showed that F function named FO has the probability with p ≤ 2−56 and 3 round FO function is provable secure. Though the main part of security is guaranteed by FO function, the designer added the auxillary function FL, expecting the higher security. FO function is consisted of 3 rounds FI functions. FI function is consisted of two kinds of S-boxes called S7 and S9. The degree of S7 is 3 and the degree of S9 is 2. We denote FO function in i-th round as FOi, j-th FI function in FOi as FIij . We attack modified MISTY1 by Higher Order Differential Attack. The modification assumes omitting FL functions and reducing number of rounds motivated by the statement of the designer that MISTY1 without FL is enough to attain a provable security assuming more than or equals to 3 rounds. To simplify

Strength of MISTY1 without FL Function Xij

❄ FIij Yij

❄ ❤✛ ❄

Hij

(i) FIij

223

Zijk

❄ ❤✛ Xijk

Kijk

❄

K (i)

❄

Yijk

❄ ❤✛ ❄

❄

CL (X)

Hijk

H (i) (X)

❄ FO ✲ ❤

S

❄

CR (X)

(ii) k-th S-box in FIij (iii)The last round of i round MISTY

Fig. 2. (i)(ii):The input and output variables for FIij and k-th S-box, (iii):The last round of i round MISTY.

the attacking equation, we deduce the equivalent FO function and FI function (Figure 1). We denote equivalent sub-keys as K and Kijk . In the following, we use input and output variables for FIij and k-th S-box as shown in Figure 2(i) and (ii).

3 3.1

Higher Order Differential Attack Higher Order Differential

Let F (X; K) be a function : GF(2)n × GF(2)s → GF(2)m . Y = F (X; K),

(X ∈ GF(2)n , Y ∈ GF(2)m , K ∈ GF(2)s )

(1)

Let (A1 , A2 , . . . , Ai ) be a set of linear independent vectors in GF(2)n and V (i) be a subspace spanned by the set. We define ∆(i) F (X; K) as the i-th order differential of F (X; K) with respect to X as follows.
∆(i)F (X; K) = F (X + A; K) (2) A∈V (i)

If degX F (X; K) = N , we have the following properties. Let symbol ”]d ” be the operation which omits terms whose degree is smaller than d. Property 1.  degX {F (X; K)} = N ⇒

∆(N +1)F (X; K) = 0 ∆(N )F (X; K) = ∆(N )F (X; K)]N

(3)

Property 2. Let F (X) : GF(2)n → GF(2)m . If V (n)=GF(2)n , then for any fixed value f ∈ GF(2)n , ∆(n)F (X + f; K) = ∆(n)F (X; K).

224

3.2

Hidema Tanaka, Kazuyuki Hisamatsu, and Toshinobu Kaneko

Attacking Equation

Figure 2(iii) shows the last round of i round MISTY1. H (i) (X) can be calculated by cipher text CL (X), CR (X)∈ GF(2)32 and sub-key K (i), as follows. H (i) (X) = F O(CL (X); K (i) ) + CR (X)

(4)

If degX H (i) (X) = N , following equation holds. ∆(N )H (i) (X) = ∆(N )F (X; K (1···(i−2)))]N

(5)

where F (·) denotes the function GF(2)32 × GF(2)75×(i−2) → GF(2)32 . K (1···(i−2)) denotes the set of keys for previous (i − 2) rounds. From equations (4) and (5), we can derive the following equation.
{F O(CL (X + A); K (i) ) + CR (X + A)} = ∆(N )F (X; K (1···(i−2)) )]N (6) A∈V (N )

If the right hand of this equation can be determined for some analytical method, we can use this equation (6) as the attacking equation for K (i).

4 4.1

Attack of MISTY1 Effective Chosen Plain Text

The order for Higher Order Differential Attack depends on the chosen plain text. Since the order affects the number of chosen plain texts and the computational cost, it is important to search for the effective chosen plain text. The plain text can be divided into 8 sub-blocks according to the S-boxes to be inputed to.  GF(2)7 , i = even P = (X7 , X6 , . . . , X1 , X0 ), Xi ∈ (7) GF(2)9 , i = odd, (i = 0 ∼ 7) The degree of output depends on which sub-block we choose as a variable. We searched for the effective choice which makes the slowest increase of degree. As the result, the effective one is to keep all the sub-blocks fixed except right most 7[bit] sub-block X0 . For this chosen plain text, the increase of degree by the formal analysis is shown in Figure 3. The symbol < i|j > denotes that the degree of left block is i and the right block is j. 4.2

Attacking Equation Using 7th Order Differential

We have 7[bit] variable for the chosen plain text P . Let’s discuss the attack using 7th order differential. We use a subspace V (7) as follows. V (7) = (A1 , A2 , . . . , A7 ), Ai = (0, 0, . . . , 1, . . . , 0)) ∈ GF(2)64 ↑ (i − 1)th bit

(8)

Strength of MISTY1 without FL Function < 3|3 >

❄ X0 + H133 S9 Y221

< 2|2 >

❄ Z322

❄

Function F(·)

225

S9

S9

S7 < 3 >❤✛truncate

S7 < 9 >❤ ✛truncate

S7 < 6 >❤ ✛truncate

S9

S9 < 12 >

S9

❤✛zero-extend  ✘✘✘ ✘✘ 

❤✛zero-extend  ✘✘✘ ✘✘ 

 ✘✘✘ ✘✘ 

 ✘✘✘ ✘✘ 

❤✛zero-extend  ✘✘✘ ✘✘ 

❤✛ H21

< 3 >❄< 3 > < 3|3 > (i) FI22

❤✛zero-extend  ✘✘✘ ✘✘ 

H312 < 9 >❄< 12 > < 9|12 > (ii) FI31

❤✛zero-extend  ✘✘✘ ✘✘   ✘✘✘ ✘✘  ❤✛zero-extend  ✘✘✘ ✘✘ 

H322 < 6 >❄< 8 > < 6|8 > (ii) FI32

Fig. 3. The increase of degree by the formal analysis (To simplify the expression, we omit sub-keys in these figures.)

L7 Let H32 be the left 7[bit] of the output from FO3. L7 = H312 + H322 + Z322 H32

(9)

From Property 1, the following holds. L7 ∆(7)H32 = ∆(7)(H312 + H322 + Z322 )]7 = ∆(7)H312 ]7

(10)

Let F (·) be the function GF(2)7 × GF(2)9 → GF(2)7 shown in Figure 3. H312 = F (X0 + H133 + K222 , Y221 )

(11)

Note that Y221 is a constant for the chosen plain text P . As X0 spans GF(2)7 , from Property 2, the following holds. ∆(7)H312 = ∆(7)F (X0 + H133 + K222 , Y221 ) = ∆(7)F (X0 , Y221 )

(12)

L7 From equation (10) and (12), we have 7th order differential of H32 as follows. L7 ∆(7)H32 = ∆(7)F (X0 , Y221 )]7

(13)

We calculated the Boolean expressions of H312 by using the computer algebra software REDUCE. As the result, we found the followings.

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Hidema Tanaka, Kazuyuki Hisamatsu, and Toshinobu Kaneko

Table 1. The Boolean expression of H312 . ˆ h0 ˆ h1 ˆ h2 ˆ h3 ˆ h4 ˆ h5 ˆ h6

x0 x1 x2 x3 x4 x5 x6 + (y0 + y3 + y5 + y6 + y8 )x0 x1 x2 x3 x4 x5 + · · · + 1 (y0 + y2 + y4 + y7 )x0 x1 x2 x3 x4 x5 + · · · + y5 y7 + y5 y8 + y6 y8 + y6 x0 x1 x2 x3 x4 x5 x6 + (y0 + y2 + y4 + y5 + y7 + y8 + 1)x0 x1 x2 x3 x4 x5 + · · · + 1 x0 x1 x2 x3 x4 x5 x6 + (y0 + y3 + y4 + y6 + y8 )x0 x1 x2 x3 x4 x5 + · · · + 1 (y0 + y2 + y3 + y6 + y7 )x0 x1 x2 x3 x4 x5 + · · · + y6 y7 y8 + y7 + y8 + 1 x0 x1 x2 x3 x4 x5 x6 + (y1 + y6 + y8 + 1)x0 x1 x2 x3 x4 x5 + · · · + y8 x0 x1 x2 x3 x4 x5 x6 + (y0 + y2 + y5 + y7 + 1)x0 x1 x2 x3 x4 x5 + · · · + y6 + y7

1. The degree of H312 equals to 7. L7 2. The value of 7th order differential of H32 equals to 0x6D. 3. The coefficients of terms whose degree is 6, are functions of elements in Y221 . We show a part of them in Table 1. X222 = (x6 , . . . , x0 ), (X222 = X0 + H133 + K222 ) ˆ 6 , . . . , ˆh0 ) Y221 = (y8 , . . . , y0 ), H312 = (h L7 By using ∆(7)H32 = 0x6D, the following attacking equation, with respect to K522 , K521 , K512 and K511 (32[bit] out of 75[bit]) can be derived.
F O(CL (P + A); K522 , K521 , K512 , K511 ) + CR (P + A) = 0x6D (14) A∈V (7) L7 , the resultant equation is As we construct the attacking equation for 7[bit] H32 7 the vector equation on GF(2) . Note that the appropriate bit of CL (P ), CR (P ) and FO(·) is selected for the attacking equation (15).

4.3

Number of Chosen Plain Texts and Computational Cost

We adapted the algebraic method [8] for solving the attacking equation (15). We regard all the variable terms with respect to K522 , K521 , K512 and K511 as the independent variables. The attacking equation has two 9[bit] unknowns (K521 and K511 ) whose degree is 1, and two 7[bit] unknowns (K522 and K512 ) whose degree is 2. The equation can be regarded as the linear equation which has 2 × (9 + 7 + 7 C2 ) = 74 unknowns. We can deduce 7 linear equations from one 7th order differential. To solve the equation, we need 74/7  11 different 7th order differentials. Thus we need 27 × 11 = 1, 408 chosen plain texts. To solve the attacking equation, we calculate the coefficient matrix. The size 7 17 of matrix is 74 × 74, so it needs 74 × 74 times FO function operations. 7 ×2  2 7 If we make a brute force search by using 2 chosen plain texts, the computational cost is 27 × 232 = 239 . The algebraic method is 222 times faster than the brute force search (Table 2). The computer simulation took about 0.5[s] for this attack (SONY NEWS5000: CPU R4400 150[Mhz], Memory 32[M]).

Strength of MISTY1 without FL Function

227

Table 2. The compalison of the algebraic method and the blute force sarch. Number of texts Computatinal cost CPU time Algebraic method 1,408 217 about 0.5[s] Brute force sarch 256 239 -

Table 3. The Boolean expression of H322 . h0 h1 h2 h3 h4 h5 h6

5 5.1

(h0 h1 + h0 h4 + h1 h3 + h1 h4 + · · · + h4 h5 + h4 h6 + h5 h6 + 1)x0 x1 x2 x3 x4 x5 · · · (h0 h3 + h0 h5 + h0 h6 + h1 h2 + h2 h3 + h2 h4 + h2 h5 + h3 h4 )x0 x1 x2 x3 x4 x5 · · · (h0 h1 + h0 + h1 h2 + h1 h3 + h1 h4 + h1 h5 + h1 h6 + · · · + h6 )x0 x1 x2 x3 x4 x5 · · · (h0 h2 + h0 h3 + h0 h5 + h1 h4 + · · · + h3 h6 + h4 h5 + h5 + h6 )x0 x1 x2 x3 x4 x5 · · · (h0 h1 + h0 h3 + h0 h4 + h0 h5 + h0 h6 + h0 + h1 h5 + · · · + h6 )x0 x1 x2 x3 x4 x5 · · · (h0 h1 + h0 h2 + h0 h3 + h0 h4 + · · · + h3 h6 + h4 h5 + h4 h6 + 1)x0 x1 x2 x3 x4 x5 · · · (h0 h2 + h0 h3 + h0 h5 + h0 h6 + h0 + h1 h2 + h1 h5 + · · · + 1)x0 x1 x2 x3 x4 x5 · · ·

Estimation of Intermediate Value Intermediate Value

There are constants which are functions of the fixed part of plain text and subkeys. We call such constants as intermediate values. Since we have 11 different 7th order differentials mentioned before, we have 11 different constants in fixed part in chosen plain texts. Thus each intermediate value has 11 different values. In the following, we use the differentials of these to derive attacking equations for remaining sub-keys. Next, we estimate intermediate values using 6th order differentials. 5.2

Attacking Equation Using 6th Order Differential

L7 By decoding the cipher texts using the derived sub-keys, we can calculate H32 . From Figure 3, the degree of H322 will be 6. Let’s consider 6th order differential of equation (9). The terms whose degree is greater than 5 should be counted. L7 H32 ]6 = (H312 + H322 + Z322 )]6

(15)

We calculate Boolean expressions of H322 by REDUCE. As the result, we found the followings. 1. The degree of H322 equals to 6. 2. The coefficients of the terms whose degree is 6, are functions of H213 . We show a part of them in Table 3. X222 = (x6 , . . . , x0 ), H213 = (h6 , . . . , h0 ), H322 = (h6 , . . . , h0 )

(16)

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Hidema Tanaka, Kazuyuki Hisamatsu, and Toshinobu Kaneko

L7 L7 L7 From Table 1, 3 and equation (15), hL7 i (i = 0 ∼ 6) in H32 = (h6 , . . . , h0 ) is as follows.

hL7 i ]6 = cj · x0 · · · x6 +

6


([Y221 ]j + [H213 ]j )x\{j}

(17)

j=0

where cj ∈ GF(2) is a coefficient. And where x\{j} denotes the product of x0 , · · · , x6 except xj , [Y221 ]j denotes the coefficient of x\{j} in Table 1, and [H213 ]j in Table 4. We can calculate 6th order differential of hL7 i as follows. ∆

L7 (6) h V\{j} i (6)

= cj ·xj +[Y221 ]j +[H213 ]j ,

(6)

V\{j} = (A1 , A2 , . . . , Aj−1 , Aj+1 , . . . , A7 )

(18) By solving this, we can determine the intermediate values of Y221 , H213 and one bit xj of X222 . Since X222 is 7 [bit], to determine all bit, we need 7 different 6th order differentials. One 7th order differential can derive 7 different 6th order differentials. The equation (18) has 7 [bit] and 9[bit] unknowns whose degree is 1 and one 7 [bit] unknown whose degree is 2. So the equation has 7+9+7+7C2 = 44 unknowns. Thus we can solve equation (18) by using one 7th order differential

6

Attack to the Remaining Sub-keys

We can determine intermediate values X222 , Y221 and H213 . The attack to the remaining sub-keys is ; (1)Estimation of the intermediate values X212 and X213 , (2)Attack of FO1, (3)Attack of FI21 , (4)Attack of FI22 . Since we have 11 different 7th order differentials, fixed sub-block Xi (i = 0), has some different values. 6.1

Estimation of the Intermediate Values X212 and X213

In FI21 , following holds. H213 = Y213 + Y212 + Y211 + Z212

(19)

The value of Z212 is a constant determined by the plain text sub-block X2 . Z212 = X2 + H122

(20)

Since H122 is a constant, the value of 1st order differential of Z212 equals to the value of 1st order differential of X2 . The value of H213 is known, we can calculate the value of 1st order differential of (19) as follows. ∆H213 = ∆Y213 + ∆Y212 + ∆Z212

(21)

Since Y213 = S9(X213 ), we can regard ∆Y213 as linear equation. In the same way, ∆Y212 as 2nd order equation. The equation (21) has 7 [bit] unknown whose degree is 2 and 9 [bit] unknown whose degree is 1. So the equation has 7 + 9 + 7 C2 = 37 unknowns. To solve this, we need 37/7  6 different 1st order differentials. As the result, we can determine X212 and X213 .

Strength of MISTY1 without FL Function

6.2

229

Attack of FO1

The intermediate value X213 is calculated as follows. L7 L7 X213 = Z212 + K213 + S9(H11 + H12 + K211 )

(22)

Since Z212 and K213 are constants and X213 is known, we can calculate 1st order differential of this as follows. L7 L7 ∆X213 = ∆S9(H11 + H12 + K211 )

(23)

We can determine sub-keys K111 , K112 , K121 , K122 and K211 by solving this equation. The intermediate value X222 is calculated as follows. L7 L7 L7 X222 = Y11 + Y12 + Y13 + K222 + X0 + X4

(24)

Since X222 , X0 and X4 are knowns, we have following. L7 L7 L7 ∆X222 = ∆Y11 + ∆Y12 + ∆Y13 + ∆X0 + ∆X4

(25)

By solving equation (26), we can determine sub-keys K113 , K123 , K131 , K132 and K133 . As the result, we can determine all sub-key in FO1 . So we can calculate the values of inputs to FO2. 6.3

Attack of FI21

In FI21 , X213 can be calculated as follows. X213 = Y211 + Z212 + K213

(26)

Since all sub-key in FO1 and K211 are determined, the values of Z212 and Y211 can be calculated. Thus we can determine sub-key K213 from equation (26). The following equation holds. X212 = K212 + Z212

(27)

The value of Z212 is known, we can determine sub-key K212 . Since K211 is determined in the attack of FO1 , we can determine all sub-key in FI21 . In the same way, we can attack to all sub-key in FI22 . Due to the limitation of space, we omit the details.

7

Conclusion

We showed that 5 round MISTY1 without FL function, which is secure against Linear and Differential cryptanalysis, is attackable by Higher Order Differential Attack using the effective chosen plain texts. Our attack is consisted of 2 attacking phases. The 1st phase is an attack using 7th order differentials to determine

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Hidema Tanaka, Kazuyuki Hisamatsu, and Toshinobu Kaneko

4 sub-keys in 5th round. This attack needs 1,408 chosen plain texts and 217 of computational cost. Our computer simulation for this phase took about 0.5[s]. The 2nd phase is an attack using intermediate values, to determine another 15 sub-keys. The chosen plain texts for the 1st phase are sufficient for this phase. After sub-keys mentioned above are determined, it is far easier to estimate the rest sub-keys. We conclude that at least 6 rounds is necessary for resistance against Higher Order Differential Attack.

References 1. Matui, ”Block Encryption Algorithm MISTY”, ISEC96-11(1996-07) 2. Shimoyama, Moriai, Kaneko, ”Improving the Higher Order Differential Attack and Cryptanalysis of the KN Cipher”, ISEC97-29(1997-09) 3. Moriai, Shimoyama, Kaneko, ”Higher Order Attack of a CAST Cipher (II)”, SCIS98-12E 4. Jakobsen, Knudsen, ”The Interpolation Attack on Block Cipher”, FSE-4th International Workshop, LNCS.1008 5. Nyberg, Knudsen, ”Provable Security against Differential Cryptanalysis”, Journal of Cryptology, Vol.8-no.1 (1995) 6. Shimoyama, Moriai, Kaneko, ”Improving the Higher Order Differential Attack and Cryptanalysis of the KN Cipher”, 1997 Information Security Workshop 7. Matsui, ”New Structure of Block Ciphers with Provable Security against Differential and Linear cryptanalysis”, FSE-3rd International Workshop, LNCS.1039 8. Moriai, Shimoyama, Kaneko, ”Higher Order Attack of a CAST Cipher”, FSE-4th International Workshop, LNCS.1372

Quantum Reed–Solomon Codes Markus Grassl, Willi Geiselmann, and Thomas Beth Institut f¨ ur Algorithmen und Kognitive Systeme Arbeitsgruppe Quantum Computing Universit¨ at Karlsruhe, Am Fasanengarten 5, 76 128 Karlsruhe, Germany

Abstract. We introduce a new class of quantum error–correcting codes derived from (classical) Reed–Solomon codes over finite fields of characteristic two. Quantum circuits for encoding and decoding based on the discrete cyclic Fourier transform over finite fields are presented.

1

Introduction

During the last years it has been shown that computers taking advantage of quantum mechanical phenomena outperform currently used computers. The striking examples are integer factoring in polynomial time (see [18]) √ and finding pre–images of an n–ary Boolean function (“searching”) in time O( 2n ) (see [12]). Quantum computers are not only of theoretical nature—there are several suggestions how to physically realize them (see, e. g., [6,7]). On the way towards building a quantum computer, one very important problem is to stabilize quantum mechanical systems since they are very vulnerable. A theory of quantum error–correcting codes has already been established (see [15]). Nevertheless, the problem of how to encode and decode quantum error–correcting codes has hardly been addressed, yet. In this paper, we present the construction of quantum error–correcting codes based on classical Reed–Solomon (RS) codes. For RS codes, many classical decoding techniques exist. RS codes can also be used in the context of erasures and for concatenated codes. Encoding and decoding of quantum RS codes is based on quantum circuits for the cyclic discrete Fourier transform over finite fields which are presented in Section 4, together with the quantum implementation of any linear transformation over finite fields. We start with some results about binary codes obtained from codes over extension fields, followed by a brief introduction to quantum computation and quantum error–correcting codes.

2 2.1

Binary Codes from Codes over F2k Bases of Finite Fields

First, we recall some facts about finite fields (see, e. g., [13]). Any finite field of characteristic p, i. e., a finite field Fq where q = pk , is a vector space of dimension k over Fp . For a fixed basis B of Fq over Fp , any element of Fq can thus be represented by a row vector of length k over Fp . To Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 231–244, 1999. c Springer-Verlag Berlin Heidelberg 1999


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stress the dependence on the choice of the basis B, we will denote this Fp vector space homomorphism by B: Fq = Fpk → Fpk ,

a → B(a).

(1)

The multiplication with a fixed element a ∈ Fq defines an Fp –linear mapping. Thus it can be written as a k × k matrix MB (a) over Fp where B(a · a) = B(a ) · MB (a). The trace of MB (a) is independent of the choice of the basis and defines an Fp –linear mapping tr: Fq → Fp ,

a → tr(a) :=

k−1


i

ap = tr(MB (a))

i=0

(for the last equality see, e. g., [9, Satz 1.24]). To be able to proceed further, we recall the definition of the dual basis. Given a basis B = (b1 , . . . , bk ) of a finite field Fq over Fp , the dual basis of B is a basis B ⊥ = (b1 , . . . , bk ) with (2) ∀i, j: tr(bi bj ) = δij . For any basis there exists a unique dual basis (see [13, Theorem 4.1.1]). Furthermore, for any finite field of characteristic two there exists a self–dual basis, i. e., a basis B with B ⊥ = B (see [13, Theorem 4.3.5]). For a self–dual basis B, the matrix MB (a) is symmetric. This follows from [9, Satz 1.22], where MB⊥ (a) = MB (a)t

(3)

is shown. Finally, any linear transformation A ∈ GL(n, Fpk ) can be written as a linear transformation B(A) ∈ GL(nk, Fp ) by replacing each entry aij of A by MB (aij ). Moreover, the diagram (4) is commutative, i. e., the change to the ground field can be done after or before the linear transformation—a fact that will be essential later. A Fpnk −−−→ Fpnk     B  (4)  B B(A)

Fpkn −−−→ Fpkn 2.2

Subfield Expansion of Linear Codes

In the following, we restrict ourselves to the case p = 2, but the results are valid for any characteristic p > 0. Definition 1. Let C = [N, K, D] denote a linear code of length N , dimension K, and minimum distance D over the field F2k , and let B = (b1 , . . . , bk ) be a basis of F2k over F2 . Then the binary expansion of C with respect to the basis B, denoted by B(C), is the linear binary code C2 = [kN, kK, d ≥ D] given by      ∈C . C2 = B(C) := (cij )i,j ∈ F2kN  c = j cij bj i

Quantum Reed–Solomon Codes

233

The following theorem relates the dual codes of a code and its binary expansion. Theorem 1. Let C = [N, K] be a linear code over the field F2k and let C ⊥ be its dual. Then the dual code of the binary expansion B(C) of C with respect to the basis B is the binary expansion B ⊥ (C ⊥ ) of the dual code C ⊥ with respect to the dual basis B ⊥ , i. e., the following diagram is commutative: C −→ C ⊥     basis B   dual basis B⊥ B(C) −→ B ⊥ (C ⊥ ) = B(C)⊥ Proof. Let c ∈ C and d ∈ C ⊥ be arbitrary elements of the code and its dual, resp. Then

k  k  N N



c i di = cij bj dil bl , 0= (5) i=1

i=1

j=1

l=1

where B = (b1 , . . . , bk ) is a basis of F2k over F2 and B ⊥ = (b1 , . . . , bk ) is the corresponding dual basis. Taking the trace in (5) and rewriting the summation yields N
N
k
k k

cij dil tr(bj bl ) = cij dij 0= i=1 j=1 l=1

i=1 j=1

(the last equality follows from Eq. (2)). Hence the binary expansions of the codewords c and d are orthogonal which proves that the binary expansion of C ⊥ is contained in B(C)⊥. The theorem follows from the observation that both  sets have 2k(N −K) elements. Corollary 1. Let C = [N, K] be a weakly self–dual linear code over the field F2k . Then the binary expansion B(C) of C with respect to a self–dual basis B is weakly self–dual, too.

3 3.1

Quantum Error–Correcting Codes Qubits and Quantum Circuits

In this section, we give a brief introduction to quantum computation (for a more comprehensive introduction see, e. g., [2,21]). The basic unit of quantum information, a quantum bit (or short qubit), is represented by the normalized linear combination |q = α|0 + β|1,

where α, β ∈ C, |α|2 + |β|2 = 1.

(6)

Here |0 and |1 are orthonormal basis states written in Dirac notation (see [8]). The normalization condition in Eq. (6) stems from the fact that when extracting

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classical information from the quantum system by a measurement, the results “0” and “1” occur with probability |α|2 and |β|2 , resp. A quantum register of length n is obtained by combining n qubits modeled by the n–fold tensor product (C 2 )⊗n . The canonical orthonormal basis of (C 2 )⊗n is 

B := |b1  ⊗ . . . ⊗ |bn  =: |b1 . . . bn  = |b  bi ∈ {0, 1} . Hence the state of an n qubit register is given by
|ψ = cb |b, where cb ∈ C and b∈{0,1}n

 b∈{0,1}n

|cb |2 = 1.

(7)

All operations of a quantum computer are linear. Furthermore, in order to preserve the normalization condition in Eq. (7), the operations have to be unitary. Basic operations are single qubit operations and two qubit operations. A single qubit operation on the jth qubit is given by U = I2j−1 ⊗ U2 ⊗ I2n−j , where U2 ∈ U(2) is a 2 × 2 unitary matrix. Important examples for single qubit operations are the Hadamard transform H and the Pauli matrices σx , σy , σz         1 0 1 0 −i 1 0 1 1 H := √ , σy := , σz := , (8) , σx := 1 0 i 0 0 −1 2 1 −1 where i2 = −1. The most important example for a two qubit gate is the so–called controlled NOT gate (CNOT ) since any unitary operation on a 2n –dimensional space can be implemented using only single qubit operations and CNOT gates (see [1]). The transformation matrix of the CNOT gate is given by: 

1  0 CNOT :=  0 0

0 1 0 0

0 0 0 1

 0 0  1  0

|00 → |00 |01 → |01 |10 → |11 |11 → |10

|x1 

•

|x1 

|x2 

✐

|x1 ⊕ x2 

(9)

The CNOT gate corresponds to the classical XOR gate since CNOT |x1 |x2  = |x1 |x1 ⊕ x2 . (For the graphical notation on the right hand side see, e. g., [1].) 3.2

Error Model

In the following, we briefly summarize some results about quantum error–correcting codes. For a more comprehensive treatment, we refer to, e. g., [3,15]. One common assumption in the theory of quantum error–correcting codes is that errors are local, i. e., only a small number of qubits are disturbed when transmitting or storing the state of an n qubit register. The basic types of errors are bit–flip errors exchanging the states |0 and |1, phase–flip errors changing the relative phase of |0 and |1 by π, and their combination. The bit–flip error corresponds to the Pauli matrix σx , the phase–flip error to σz , and their combination to σy . It is sufficient to consider only this discrete set of errors in order to cope with any possible local error (see [15]). The important duality of bit–flip errors and phase–flip errors is shown by the following lemma.

Quantum Reed–Solomon Codes

235

Lemma 1. Bit–flip and phase–flip errors are conjugated to each other by the Hadamard transform, i. e., Hσx H −1 = σz and Hσz H −1 = σx . Errors operating on an n qubit system are represented by tensor products of Pauli matrices and identity. The weight of an error e = e1 ⊗ . . . ⊗ en , where ei ∈ {id, σx , σy , σz } is the number of local errors ei that differ from identity. 3.3

Quantum Codes

Analogously to the notation C = [N, K, d] for a classical error–correcting code encoding K information symbols using N code symbols and being capable of detecting up to d − 1 errors, a quantum error–correcting code encoding K qubits using N qubits is denoted by C = [[N, K, d]]. The code C is a 2K –dimensional subspace of the 2N –dimensional complex vector space (C 2 )⊗N such that any error of weight less than d can be detected or, equivalently, any error of weight less than d/2 can be corrected. The construction of quantum Reed–Solomon codes is based on the construction of quantum error–correcting codes from weakly self–dual binary codes presented in [5] and [19,20] as summarized by the following definition and theorem. Definition 2. Let C = [N, K] be a weakly self–dual linear binary code, i. e., C ≤ C ⊥ . Furthermore, let {w j | j = 1, . . . , 2N −2K } be a system of representatives of the cosets C ⊥ /C. Then the basis states of a quantum code C = [[N, N − 2K]] are given by 1
|c + w j . (10) |ψj  =  |C| c∈C Theorem 2. Let d be the minimum distance of the dual code C ⊥ in Definition 2. Then the corresponding quantum code is capable of detecting up to d − 1 errors or, equivalently, is capable of correcting up to (d − 1)/2 errors. Proof. (Sketch) A general state of the quantum code is a linear combination of the states in Eq. (10), i. e.,


1
αj |ψj  = αj  |c + w j  = βc |c. (11) |ψ = |C| c∈C j j c∈C ⊥ A combination of bit–flip and phase–flip errors can be written as e

e

e = (σxb,1 σzep,1 ) ⊗ . . . ⊗ (σxb,n σzep,n ),

(12)

where eb and ep are binary vectors. The effect of this error on the state (11) is
βc (−1)c·ep |c + eb . (13) e|ψ = c∈C ⊥

Computing the syndrome with respect to the binary code C ⊥ using auxiliary qubits, we obtain the state
βc (−1)c·ep |c + eb |s(c + eb ). (14) c∈C ⊥

236

Markus Grassl, Willi Geiselmann, and Thomas Beth

As the syndrome s(c + eb ) depends only on the error eb , the state (14) is a tensor product and we can measure the syndrome without disturbing the first part of the quantum register. Using a classical decoding algorithm for the code C ⊥ , the error vector eb is computed from the measured syndrome s(eb ). For each non–zero position of eb , a σx gate is applied to correct the error. From Lemma 1 follows that the Hadamard transform exchanges the rˆole of eb and ep in Eq. (12). Furthermore, computing the Hadamard transform of the states in (10) yields
1 H ⊗N |ψj  =  (−1)c·wj |c. (15) |C ⊥| c∈C ⊥ Hence, the Hadamard transform changes the state (13) into
γc (−1)c·eb |c + ep . H ⊗N e|ψ = c∈C ⊥



The error vector ep can be determined as before. The general outline of decoding is shown in Fig. 1.

H . . .

H correction of phase–flip errors

. . .

syndrome computation

correction of bit–flip errors

H syndrome computation

     . (erroneous) . encoded state   .      . . auxiliary qubits .  

. . . H . . .

Fig. 1. General decoding scheme for a quantum error–correcting code constructed from a weakly self–dual binary code. Before we will be ready to present quantum Reed–Solomon codes, we need to show how to implement a discrete Fourier transform over a finite field on a quantum computer.

4

Quantum Implementation of the Cyclic DFT over F2k

Recall from Section 3.1 that the state of an n qubit system can be written as
 |ψ = cx |x, where cx ∈ C and |cx |2 = 1. (16) x∈F2n

x∈F2n

Hence any invertible linear transformation A ∈ GL(n, F2 ) on the binary vector a linear transformation Q(A) ∈ GL(2n , C) on the complex space F2n induces 2n vector space C = (C 2 )⊗n . The transformation Q(A) permutes the basis states |x according to Q(A) : |x → |xA. In the following we will show how this transformation can be implemented efficiently using only CN OT gates.

Quantum Reed–Solomon Codes

237

Theorem 3. Let π ∈ Sn be a permutation and let P ∈ GL(n, F2 ) be the corresponding permutation matrix acting on the binary vector space F2n . Then the quantum transformation Q(P ) ∈ GL(2n , C) defined by Q(P ) : |x → |xP  is a permutation matrix permuting the n tensor factors C 2 of the complex vector n space C 2 = (C 2 )⊗n . It can be implemented using at most 3(n − 1) CNOT gates. Proof. Any permutation π ∈ Sn on n letters can be written as product of at most n − 1 transpositions. Each transposition (i, j) can be implemented by a quantum circuit with three CNOT gates, see Fig. 2. 

(i, j) = 

... .. . ...

•

❣

❣

•

... .. ❣ . ...

•

i j

Fig. 2. Implementing a transposition of two qubits using three CNOT gates. Theorem 4. Let A ∈ GL(n, F2 ) be an invertible linear mapping on the binary vector space F2n . Then the quantum transformation Q(A) ∈ GL(2n , C) defined by Q(A) n: |x → |xA is a permutation matrix acting on the complex vector space C 2 . It can be implemented using at most n(n − 1) + 3(n − 1) CNOT gates. Proof. Any matrix A ∈ GL(n, F2 ) can be decomposed as A = P · L · U, where P is a permutation matrix and L (resp. U) is a lower (upper) triangular matrix. By Theorem 3 we need at most 3(n − 1) CNOT gates for the implementation of Q(P ). For the implementation of the lower diagonal matrix L, we use the factorization L = L1 · . . . · Ln , where Li is almost an identity matrix, but the ith row equals the ith row of L. Hence multiplication of a binary vector x with Li is given by xLi = (x1 + xi Li1 , . . . , xi−1 + xi Li,i−1 , xi , xi+1 , . . . , xn ), i. e., the jth position of x is inverted iff both xi and Lij are equal to one. This translates into a sequence of at most i − 1 CNOT gates with control qubit i and target qubit j whenever Lij equals one. In total, the implementation of Q(L) needs at most n(n − 1)/2 CNOT gates. The quantum transformation Q(U) can be implemented similarly.  The quantum implementation of linear mappings over an extension field F2k can be reduced to implementing linear mappings over F2 . First, we fix a basis B of F2k . By extending the homomorphism B given in Eq. (1) we obtain a homomorphism F2nk → F2kn . Vectors v ∈ F2nk are mapped to binary vectors of length kn represented by kn qubits. Similarly to Eq. (16), we get

cv |v = cv |B(v1 ) ⊗ . . . ⊗ |B(vn). (17) |ψ = v∈F nk 2

v∈F nk 2

238

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In this representation, a linear mapping A ∈ GL(n, F2k ) corresponds to a linear mapping B(A) ∈ GL(nk, F2 ) (see Eq. (4)). In the context of quantum Reed–Solomon codes, we will use the cyclic discrete Fourier transform over F2k which can be implemented efficiently as a quantum circuit. Theorem 5. Let n be a divisor of 2k − 1 and let α ∈ F2k be an element of order n. Then the cyclic DFT of length n over the field F2k , given by the matrix   (18) DFT = αij i,j=0,...,n−1 can be implemented on a quantum computer using O(k 2 n2 ) CNOT gates. Proof. The condition n|(2k − 1) ensures that the field F2k contains a primitive nth root of unity α. Thus, we have DFT ∈ GL(n, F2k ). Fixing a basis B of F2k , we obtain a linear transformation B(DFT) ∈ GL(nk, F2 ) which can be implemented  using O(k 2 n2 ) CNOT gates using Theorem 4.

5 5.1

Quantum Reed–Solomon Codes Definition of Quantum Reed–Solomon Codes

First, we recall the definition of Reed–Solomon codes (see [16, Fig. 10.6]). Definition 3. A (classical) Reed–Solomon (RS) code of length N = 2k − 1 over the field F2k is a cyclic code with generator polynomial g(X) = (X − αb )(X − αb+1 ) . . . (X − αb+δ−2 ), where α is a primitive element of F2k , i. e., an element of order 2k − 1 = N . The dimension of the code is K = N − δ + 1 and the minimum distance is δ. Alternatively, an RS code can be described by the spectrum with respect to the cyclic discrete Fourier transform of length N over F2k , see Eq. (18). For any vector v ∈ F2Nk , the spectrum is defined by    := v · DFT = v(αi ) i=0,...,N −1 , v N −1 c has δ − 1 where v(X) = j=0 vj X j . Then for any codeword c of an RS code,  consecutive (possibly cyclically wrapped around) zeros starting at position b. Fixing the zeros in the spectrum, all codewords can be obtained by the inverse Fourier transform, i. e., the set of codewords is given by

 ∈ F2Nk ,   · DFT−1 | v vb = vb+1 = . . . =  vb+δ−1 = 0 , (19) C= v where the indices are computed modulo N . Lemma 2. For b = 0 and δ > N/2 + 1, RS codes are weakly self–dual.

Quantum Reed–Solomon Codes

239

Proof. The generator polynomial of C is g(X) = (X − 1)(X − α) . . . (X − αδ−2 ). The generator polynomial of the dual code C ⊥ is the reciprocal polynomial of (X N − 1)/g(X), i. e., g ⊥ (X) = (X − α−(δ−1) )(X − α−δ ) . . . (X − α−(N −1)) = (X − α1 )(X − α2 ) . . . (X − αN −δ+1) ). For δ > N/2 + 1, N − δ + 1 ≤ δ − 2. Thus g(X)⊥ is a divisor of g(X) which  proves C ≤ C ⊥ . The relation between the spectra of an RS code C and its dual is illustrated in Fig. 3. The spectrum of any codeword c ∈ C is zero at the first δ − 1 positions, 0

1 ...

δ−2

0 ∗

0 0 0

∗

spectrum  c of c ∈ C

❍ ✟ ❍ ✡✟ ❍✟  ✡ ✟ ❍ ✟ ❍ ❄✟ ✢ ❄ ✙ ✡ ❥ ~❍

∗ 0 

0 ∗

 ! N −δ+1

∗ ∗

 δ−2

!

spectrum  c of c ∈ C ⊥

Fig. 3. Relation between the spectra of a Reed–Solomon code C and its dual. Positions taking arbitrary values (marked with ∗) and positions being zero are interchanged. whereas the spectrum of any codeword c ∈ C ⊥ may take any value at the corresponding positions, the first one and the last δ − 2 positions. In contrast, the last N − δ + 1 positions of the spectrum of c ∈ C are arbitrary, and positions 1 to N − δ + 1 in the spectrum of c ∈ C ⊥ are zero. Combining Lemma 2 and Corollary 1, we are ready to define quantum Reed– Solomon codes. Definition 4. Let C = [N, K, δ] where N = 2k − 1, K = N − δ + 1, and δ > N/2 + 1 be a Reed–Solomon code over F2k (with b = 0). Furthermore, let B be a self–dual basis of F2k over F2 . Then the quantum Reed–Solomon (QRS) code is the quantum error–correcting code C of length kN derived from the weakly self–dual binary code B(C) according to Definition 2. The parameters of the QRS code are given by the following theorem. Theorem 6. The QRS code C of Definition 4 encodes k(N − 2K) qubits using kN qubits. It is able to detect at least up to K errors, i. e., the parameters are C = [[kN, k(N − 2K), d ≥ K + 1]]. Proof. The weakly self–dual binary code B(C) has length kN and dimension kK. Hence, by Definition 2 the corresponding quantum code encodes kN − 2kK = k(N − 2K) qubits. The dual code B(C ⊥) has dimension k(δ − 1) and minimum distance d ≥ K + 1. From Theorem 2 follows that the QRS code can detect up to d − 1 ≥ K errors. 

240

5.2

Markus Grassl, Willi Geiselmann, and Thomas Beth

Encoding Quantum Reed–Solomon Codes

Encoding of QRS codes is based on the quantum version of the cyclic discrete Fourier transform over F2k presented in Section 4. In the sequel, let C be an RS code over F2k and let B be a self–dual basis of F2k over F2 . Furthermore, we fix a primitive element α ∈ F2k . Theorem 7. Let C = [[kN, k(N −2K), d > K]] where N = 2k −1, K = N −δ+1, and δ > N/2 + 1 be a quantum Reed–Solomon code constructed from the Reed– Solomon code C = [N, K, δ] over F2k . The transformation   ⊗k(δ−1) ⊗ H ⊗kK E = Q(B(DFT−1 )) · I2 operating on states of the form |φ1  ⊗ . . . ⊗ |φk  ⊗ |0 ⊗ . . . ⊗ |0 ⊗ |φk+1  ⊗ . . . ⊗ |φk(N −2K) ⊗ |0 ⊗ . . . ⊗ |0  !   !   !   !  k

kK

k(N −2K−1)

kK

is an encoder for the QRS code. The corresponding quantum circuit is shown in Fig. 4. |φ1 |φk |0 |0 |φk+1 |φk(N −2K) |0 |0

.. .

.. .

. . .

. . . DFT−1

. . . . . .

H

. . . . . .

...

& k qubits " kK qubits     

k(N − 2K − 1) qubits

" kK qubits

H

Fig. 4. Encoder for a quantum Reed–Solomon code. Proof. Similarly to Eq. (10), any basis state of the QRS code can be written as 1
|ψj  =  |B(c + w j ), |C| c∈C

(20)

where the coset representatives wj ∈ C ⊥ will be specified later. The first δ − 1 positions of the spectrum of c are zero. From Eq. (19), the other positions may take any value. Thus computing the Fourier transform of the state (20) yields
1 'j ). |B(0, . . . , 0, i1 , . . . , iK ) + B(w Q(B(DFT))|ψj  =    ! |C| i∈FK 2k

δ−1

'j can be chosen to be zero. Without loss of generality, the last K positions of w Hence applying the Hadamard transform to the last kK qubits yields   ⊗k(δ−1) 'j ). I2 ⊗ H ⊗kK · Q(B(DFT))|ψj  = E −1 |ψj  = |B(w

Quantum Reed–Solomon Codes

241

'j are zero, too, since w j ∈ C ⊥ (see Furthermore, positions i = 1, . . . , K of w Fig. 3). For any set of values for the remaining positions, we get a different coset  of C in C ⊥ . 5.3

Decoding Quantum Reed–Solomon Codes

Decoding procedures for quantum Reed–Solomon codes follow the scheme of Fig. 1. The syndrome of a vector v ∈ F2Nk are positions i = 1, . . . , K of the  of v which is obtained by computing the DFT of v. This syndrome, spectrum v indicating bit–flip–errors, is “copied” to kK auxiliary qubits using CNOT gates. Computing the inverse Fourier transform DFT−1 returns to the original basis. After a Hadamard transform, the same circuit is used to compute the syndrome of the phase–flip errors. The whole quantum circuit is shown in Fig. 5. Both the .. .

. . .

H

H

.

.

•

❝

.

.

.

❝

.. .

•

H

.. .

H

.

.

.

•

H

DFT−1

.

. . .

. . .

H

DFT

•

DFT−1

. . .

H

.. .

DFT

                (erroneous) encoded   state                 |0 kK qubits  |0   |0 kK qubits  |0

H

.. .

H H

. . .

. . .

H

H

.. . . . .

.

.

.

❝

k qubits " kK qubits

  .  k(N − K − 1) . .  qubits  . . .

❝

"

. . .

"

"

syndrome of bit–flip errors syndrome of phase–flip errors

Fig. 5. Computation of the syndrome for a quantum Reed–Solomon code. syndrome of bit–flip errors and the syndrome of phase–flip errors are measured yielding classical syndrome vectors. Then the most likely positions of errors are computed using a classical algorithm, e. g., the Berlekamp–Massey algorithm or the Euclidean algorithm (see [16]). The quantum circuit in Fig. 5 can be simplified using the following theorem. Theorem 8. Let DFT denote the cyclic discrete Fourier transform of length n over the field F2k and be B a self–dual basis of F2k over F2 . Then the following identities hold: Q(B(DFT−1 )) · H ⊗kn · Q(B(DFT)) = H ⊗kn · Q(B(π)) = Q(B(π)) · H

⊗kn

(21)

,

where π is the permutation x → −x mod n. Using the factorization on the right hand side of Eq. (21) instead of the factorization on the left hand side for the implementation as a quantum circuit reduces the complexity from O(k 2 n2 ) to O(kn).

242

Markus Grassl, Willi Geiselmann, and Thomas Beth

Proof. Let D := B(DFT) denote the binary matrix obtained by replacing each entry αij of DFT by MB (αij ). For a self–dual basis, MB (αij ) is symmetric (see Eq. (3)), and the Fourier matrix is symmetric, too. Hence the matrix D is also symmetric. Using Dirac notation, the matrices read


|xDx| = |xxD−1 | and H ⊗kn = (−1)x·y |xy|. Q(D) = x∈F2kn

x∈F2kn

x,y∈F2kn

Multiplying the matrices results in Q(D−1 )H ⊗kn Q(D)

=




(−1)x·y |uD−1  u|x y|vvD−1 |   !  ! kn

u∈F2kn x,y∈F2kn v∈F2




=

=δux =δyv

(−1)x·y |xD−1 yD−1 | =

x,y∈F2kn




(−1)xD·yD |xy|.

x,y∈F2kn

The inner product of xD and yD is the same as the inner product of xDDt and y. Since D is symmetric, DDt = D2 = B(DFT2 ) = B(π). Finally, we obtain Q(D−1 )H ⊗kn Q(D)
(−1)x·y |xyB(π)| = = x,y∈F2kn

=










(−1)x·y |xy|

x,y∈F2kn

(−1)x·y |xB(π)y| =

x,y∈F2kn




|vB(π)v|

v∈F2kn

|vvB(π)|

v∈F2kn




(−1)x·y |xy|.

x,y∈F2kn

 From Eq. (21) in the preceding theorem it follows that Q(B(DFT−1 )) · H ⊗kn = Q(B(π)) · H ⊗kn Q(B(DFT−1 )).

(22)

Using the identities (21) and (22), and conjugating the CNOT gates by the permutation of qubits Q(B(π)), we obtain the simplified quantum circuit shown in Fig. 6.

6

Example

We construct a quantum Reed–Solomon code from an RS code over the field F8 . We choose δ = 5 and obtain an RS code C = [7, 3, 5] with generator polynomial g(X) = (X − α0 )(X − α1 )(X − α2 )(X − α3 ), where α is a primitive element of F8 fulfilling α3 + α + 1 = 0. The dual code C ⊥ = [7, 4, 4] is generated by g ⊥ (X) = (X − α−4 )(X − α−5 )(X − α−6 ) = (X − α3 )(X − α2 )(X − α1 ). ¿From Theorem 6, the resulting QRS code has parameters C = [[21, 3, d ≥ 4]]. As self–dual basis of F8 we choose B = (α3 , α6 , α5 ). The binary expansions of C and C ⊥ yield binary codes C2 = B(C) = [21, 9, 8] and C2⊥ = B(C ⊥) = [21, 12, 5]. Thus the QRS code has parameters C = [[21, 3, 5]].

Quantum Reed–Solomon Codes

•

. . . .. .

.

.

.

•

H

. . . . . . . . .

DFT−1

.. .

DFT

                 (erroneous) encoded   state                  |0 kK qubits  |0   |0 kK qubits  |0

.. .

❝

•

.

H

.

.

H

.

.

.

❝

•

.. .

.. k qubits . " . . kK qubits . .. . . . .

H

. . .

❝

.

.

.

❝

243

. . .

&

k(N − 2K − 1) qubits

" kK qubits "

"

syndrome of bit–flip errors syndrome of phase–flip errors

Fig. 6. Simplified quantum circuit for computing the syndrome for a QRS code.

7

Conclusion

Most quantum error–correcting codes known so far are based on classical binary codes or codes over GF (4) = F22 (see [4]). In this paper, we have demonstrated how codes over extension fields of higher degree can be used. They might prove useful, e. g., for concatenated coding. The spectral techniques for encoding and decoding presented in this paper do not only apply to Reed–Solomon codes, but in general to all cyclic codes. The main advantage of Reed–Solomon codes is that no field extension is necessary. The same is true for all BCH codes of length n over the field F2k where n|2k − 1. In addition to the spectral techniques, cyclic codes provide a great variety of encoding/decoding principles, e. g., based on linear shift registers that can be translated into quantum algorithms (see [10]). The quantum implementation of linear mappings over finite fields presented in Section 4 enlarges the set of efficient quantum subroutines. In contrast, the transforms used in most quantum algorithms—such as cyclic and generalized Fourier transforms—are defined over the complex field (see, e. g., [17]). It has to be investigated how efficient fully quantum algorithms for error– correction can be obtained, e. g., using quantum versions of the Berlekamp– Massey algorithm or of the Euclidean algorithm.

Acknowledgments The authors would like to thank Martin R¨ otteler and Rainer Steinwandt for numerous stimulating discussions during the process of writing this paper. Part of this work was supported by Deutsche Forschungsgemeinschaft, Schwerpunktprogramm QIV (SPP 1078), Projekt AQUA (Be 887/13-1).

244

Markus Grassl, Willi Geiselmann, and Thomas Beth

References 1. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Physical Review A, 52 (1995), pp. 3457–3467. 2. A. Berthiaume, Quantum Computation, in Complexity Theory Retrospective II, L. A. Hemaspaandra and A. L. Selman, eds., Springer, New York, 1997, pp. 23–51. 3. T. Beth and M. Grassl, The Quantum Hamming and Hexacodes, Fortschritte der Physik, 46 (1998), pp. 459–491. 4. A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum Error Correction Via Codes over GF(4), IEEE Transactions on Information Theory, IT–44 (1998), pp. 1369–1387. 5. A. R. Calderbank and P. W. Shor, Good quantum error–correcting codes exist., Physical Review A, 54 (1996), pp. 1098–1105. 6. J. I. Cirac and P. Zoller, Quantum Computation with Cold Trapped Ions, Physical Review Letters, 74 (1995), pp. 4091–4094. 7. D. G. Cory, A. F. Fahmy, and T. F. Havel, Ensemble Quantum Computing by Nuclear Resonance Spectroscopy, Technical Report TR–10–96, B. C. M. P., Harvard Medical Medical School, Boston, Dec. 1996. 8. P. M. A. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 4th ed., 1958. 9. W. Geiselmann, Algebraische Algorithmenentwicklung am Beispiel der Arithmetik in endlichen K¨ orpern, Shaker, Aachen, 1994. Zugleich Dissertation, Universit¨ at Karlsruhe, 1993. 10. M. Grassl and T. Beth, Codierung und Decodierung zyklischer Quantencodes, in Fachtagung Informations– und Mikrosystemtechnik, B. Michaelis and H. Holub, eds., Magdeburg, 25–27 Mar. 1998. 11. M. Grassl, T. Beth, and T. Pellizzari, Codes for the Quantum Erasure Channel, Physical Review A, 56 (1997), pp. 33–38. 12. L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings 28th Annual ACM Symposium on Theory of Computing (STOC), New York, 1996, ACM, pp. 212–219. 13. D. Jungnickel, Finite Fields, BI–Wissenschaftsverlag, Mannheim, 1993. 14. E. Knill and R. Laflamme, Concatenated Quantum Codes. LANL preprint quant–ph/9608012, 1996. , Theory of quantum error–correcting codes, Physical Review A, 55 (1997), 15. pp. 900–911. 16. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error–Correcting Codes, North–Holland, Amsterdam, 1977. ¨schel, M. Ro ¨tteler, and T. Beth, Fast Quantum Fourier Transforms 17. M. Pu for a Class of Non–abelian Groups, in Proceedings AAECC–13, 1999. 18. P. W. Shor, Polynomial–Time Algorithms for Prime Factorization and Discrete Logarithms, in Proceedings 35th Annual Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society Press, Nov. 1994, pp. 124–134. 19. A. Steane, Error Correcting Codes in Quantum Theory, Physical Review Letters, 77 (1996), pp. 793–797. , Multiple Particle Interference and Quantum Error Correction, Proceedings 20. of the Royal Society London Series A, 452 (1996), pp. 2551–2577. , Quantum computing, Reports on Progress in Physics, 61 (1998), pp. 117– 21. 173.

Capacity Bounds for the 3-Dimensional (0, 1) Runlength Limited Channel Zsigmond Nagy and Kenneth Zeger Department of Electrical and Computer Engineering University of California, San Diego, La Jolla CA 92093-0407 {nagy,zeger}@code.ucsd.edu

(3)

Abstract. The capacity C0,1 of a 3-dimensional (0, 1) runlength constrained (3)

channel is shown to satisfy 0.522501741838 ≤ C0,1 ≤ 0.526880847825.

1 Introduction A binary sequence satisfies a 1-dimensional (d, k) runlength constraint if there are at most k zeros in a row, and between every two consecutive ones there are at least d zeros. An n-dimensional binary array is said to satisfy a (d, k) runlength constraint, if it satisfies the 1-dimensional (d, k) runlength constraint along every direction parallel to a coordinate axis. Such an array is called valid. The number of valid n-dimensional (d,k) arrays of size m1 × m2 × . . . × mn is denoted by Nm1 ,m2 ,...,mn and the corresponding capacity is defined as (d,k)

(n)

Cd,k =

log2 Nm1 ,m2 ,...mn . m1 ,m2 ,...mn →∞ m1 m2 · · · mn lim

(n)

(n)

By exchanging the roles of 0 and 1 it can be seen that C0,1 = C1,∞ for all n ≥ 1. A simple proof of the existence of the 2-dimensional (d, k) capacities can be found in [1], and the proof can be generalized to n-dimensions. It is known (e.g. see [2]) that the 1-dimensional (0, 1)-constrained capacity is the logarithm of the golden ratio, i.e. √ 1+ 5 (1) = 0.694242 . . . C0,1 = log2 2 and in [3] very close upper and lower bounds were given for the 2-dimensional (0, 1)constrained capacity. The bounds in [3] were calculated with greater precision in [4] and are further slightly improved here by us (see Remark section at end for more details), now agreeing in 9 decimal positions: (2)

0.587891161775 ≤ C0,1 ≤ 0.587891161868 .

(1)

(2)

A lower bound of C 0,1 ≥ 0.5831 was obtained in [5] by using an implementable encod(2)

ing procedure known as “bit-stuffing”. The known bounds on C 0,1 have played a useful Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 245–251, 1999. c Springer-Verlag Berlin Heidelberg 1999 

246

Zsigmond Nagy and Kenneth Zeger

role in [1] for obtaining bounds on other (d, k)-constraints in two dimensions. The 3-dimensional (0, 1)-constrained bounds given in the present paper can play a similar role for obtaining different 3-dimensional bounds, and are also of theoretical interest. In fact, a recent tutorial paper [6] discusses an interesting connection between run length constrained capacities in more than one dimension and crossword puzzles (based on work of Shannon from 1948). In the present paper we consider the 3-dimensional (0, 1) constraint, and by extending ideas from [3] our main result is to derive (in Sections 2 and 3) the following bounds on the 3-dimensional (0, 1) capacity. Theorem 1 (3)

0.522501741838 ≤ C0,1 ≤ 0.526880847825 It is assumed henceforth in this paper that d = 0 and k = 1. Two valid m 1 × m2 rectangles can be put next to each other in 3 dimensions without violating the 3dimensional (0, 1) constraint if they have no two zeros in the same positions. Define a (0,1) (0,1) transfer matrix Tm1 ,m2 to be an Nm1 ,m2 × Nm1 ,m2 binary matrix, such that the rows and columns are indexed by the valid 2-dimensional m 1 × m2 patterns, and an entry of Tm1 ,m2 is 1 if and only if the corresponding two rectangles can be placed next to each other in 3 dimensions without violating the (0, 1) constraint. Then, (0,1) m3 −1 m2 −1 m1 −1 Nm = 1 · Tm 1 = 1 · Tm 1 = 1 · Tm 1 1 ,m2 ,m3 1 ,m2 1 ,m3 2 ,m3

where 1 is the all ones column vector and prime denotes transpose. The matrix T m1 ,m2 meets the conditions of the Perron-Frobenius theorem [7], since it has nonnegative real elements and is irreducible (since the all one’s rectangle can be placed next to any valid rectangle without violating the (0, 1) constraint). Therefore the largest magnitude eigenvalue Λm1 ,m2 , of Tm1 ,m2 , is positive, real, and has multiplicity one. This implies that (0,1) lim (Nm )1/m3 = Λm1 ,m2 , 1 ,m2 ,m3 m3 →∞

and (0,1)

(3)

C0,1 =

log2 Nm1 ,m2 ,m3 m1 ,m2 ,m3 →∞ m1 m2 m3 lim

(0,1)

log2 limm3 →∞ (Nm1 ,m2 ,m3 )1/m3 m1 ,m2 →∞ m1 m2 log2 Λm1 ,m2 = lim m1 ,m2 →∞ m1 m2 =

lim

1/m

2 log2 limm2 →∞ Λm1 ,m 2 m1 →∞ m1 log2 Λm1 = lim , m1 →∞ m1

= lim

1/m

log Λ

(2) log Λ

2 2 m1 ,m2 2 m1 and can be where Λm1 = limm2 →∞ Λm1 ,m 2 . The quantities m1 m2 m1 viewed as capacities corresponding to 3-dimensional arrays with two fixed sides (lengths m1 and m2 ), and one fixed side (length m1 ), respectively.

Capacity Bounds for the 3-Dimensional (0, 1) Runlength Limited Channel

247

Upper and lower bounds on the 3-dimensional capacity can be computed directly from the inequalities (similar to the 2-dimensional case, as noted in [4]) log2 Λm1 ,m2 log2 Λm1 ,m2 (3) ≤ C 0,1 ≤ (m1 + 1)(m2 + 1) m1 m2 but these do not yield particularly tight bounds for values of m 1 and m2 that result in reasonable space and time complexities (e.g. Table 1 shows that the eigenvalues Λm1 ,m2 correspond to matrices with more than 40 million elements when roughly m 1 m2 ≥ 20). The upper and lower capacity bounds derived in this paper agree to within ±0.002 and were computed using less than 100 Mbytes of computer memory. (3)

2 Lower Bound on C0,1

(3)

To derive a lower bound on C 0,1 we generalize a method of Calkin and Wilf [3]. Since Tm1 ,m2 is a symmetric matrix, the Courant-Fischer Minimax Theorem [8, pg. 394] implies that p x · Tm x 1 ,m2 Λpm1 ,m2 ≥ (3) x · x q 1 for any integer for any nonzero vector x and any integer p ≥ 0. Choosing x = T m 1 ,m2 q ≥ 0 gives m2 −1 p+2q 1 1 · Tm 1 1 · Tm 1 ,m2 1 ,p+2q+1 = . (4) Λpm1 ,m2 ≥ 2q m2 −1   1 · Tm1 ,2q+1 1 1 · Tm1 ,m2 1 Thus, 2

(3)

pC0,1


=

lim Λ1/(m1 m2 ) m1 ,m2 →∞ m1 ,m2


≥ lim

m1 →∞

Λm1 ,p+2q+1 Λm1 ,2q+1

p


= lim

1/m1

m1 →∞

lim Λp/m2 m →∞ m1 ,m2

1/m1

2

1/m

=

limm1 →∞ Λm1 ,p1+2q+1 1/m limm1 →∞ Λm1 ,21q+1

=

Λp+2q+1 (5) Λ2q+1

and therefore for any odd integer r ≥ 1 and any integer z > r,
 1 Λz (3) C 0,1 ≥ . log2 z−r Λr

(6)

(3)

This lower bound on C0,1 is analogous to a 2-dimensional bound in [3], but Λz and Λr are not eigenvalues associated with transfer matrices of 2-dimensional arrays here, and cannot easily be computed as in the 2-dimensional case. Instead, we obtain a lower bound on Λz and an upper bound on Λ r . From (4) and (5) a lower bound on Λ z is  Λz =

lim Λ1/m2 m →∞ z,m2 2

≥ lim

m2 →∞

m2 −1 1 · Tz,v 1

m2 −1 1 · Tz,u 1

1/((v−u)m2 )


=

Λz,v Λz,u

1/(v−u) ,

248

Zsigmond Nagy and Kenneth Zeger

where u is an arbitrary positive odd integer, v > u, and Λ z,v and Λz,u are the largest eigenvalues of the transfer matrices Tz,v and Tz,u , respectively. To find an upper bound on the quantity Λ r for a given r, we apply a modified version of a method in [3]. We say that a binary matrix satisfies the (0, 1) cylindrical constraint if it satisfies the usual 2-dimensional (0, 1) constraint after joining its leftmost column to its rightmost column (i.e. the left and right columns can be put next to each other without violating the (0, 1) constraint). A binary matrix satisfies the (0, 1) toroidal constraint if it satisfies the usual 2-dimensional (0, 1) constraint after both joining its leftmost column to its rightmost column, and its top row to its bottom row. Proposition 1 Let s be a positive even integer and let Tm1 ,m2 be the transfer matrix whose rows and columns are indexed by all (0, 1)-constrained m1 × m2 rectangles. Let Bm1 ,s denote the transfer matrix whose rows and columns are indexed by all cylindrically (0, 1)-constrained m1 × s rectangles. Then, s m2 −1 Trace[Tm ] = 1  · Bm 1. 1 ,m2 1 ,s

m1

111 000 000 111 111 000 111 000 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 000 111 00 11 00 11 000 111 00 11 00 11 000 111 00 11 11 00 000 111 00 11 000 111 00 11 000 111 00 11 00 11 000 111 00 0011 11 111 000 00 111 000 11 00 11 11 00 0 1 0 1 000 111 0 1 000 111 0 1 111 000 111 000

11 00 00 11 00 11 00 11

m2

s Fig. 1. Cylindrically (0, 1)-constrained m 1 × s rectangles used to build cylindric m 1 × m2 × s arrays.

For every positive integer m 1 and m2 , and every even positive integer s, the mas trix Tm has nonnegative eigenvalues and thus any one of its eigenvalues is upper 1 ,m2 bounded by its trace. Hence,  s 1/s 1/s   m2 −1 Λm1 ,m2 ≤ Trace Tm = 1 · Bm 1 (7) 1 ,m2 1 ,s which gives the following upper bound on Λ r :  1   m2 −1 1/s 2 Λr = lim Λ1/m 1 sm2 = ξr,s , 1 · Br,s r,m2 ≤ lim m2 →∞

m2 →∞

(8)

Capacity Bounds for the 3-Dimensional (0, 1) Runlength Limited Channel

249

where ξr,s is the largest eigenvalue of Br,s (note that Br,s satisfies the Perron-Frobenius theorem for the same reasons as for Tm1 ,m2 in Section 1). (3) The lower bound on C0,1 in (6) can now be written as  (3)

C 0,1 ≥

1  log2  z−r

Λz,v Λz,u

1/(v−u)  1/s

ξr,s

 

r and u odd, s even z>r≥1 v>u≥1 s≥2

(9)

To obtain the best possible lower bound, the right hand side of (9) should be maximized over all acceptable choices of r, z, u, v, and s, subject to the numerical computability of the quantities Λz,v , Λz,u , and ξr,s . Table 1 shows the largest eigenvalues of various transfer matrices which were numerically computable. From this table, the best parameters we could find for the lower bound in (9) on the capacity were r = 3, z = 4, u = 5, v = 6, and s = 10, yielding (3)

C0,1 ≥

9346.35893701 1 2102.73425568 ≥ 0.522501741838. log2 4−3 (80481.0598379)1/10

(3)

3 Upper Bound on C0,1

Proposition 2 Let s1 and s2 be positive even integers and let Bs∗1 ,s2 denote the transfer matrix whose rows and columns are indexed by all toroidally (0, 1)-constrained s1 × s2 (3) rectangles. If ξs∗1 ,s2 is the largest eigenvalue of Bs∗1 ,s2 , then C0,1 ≤ s11s2 log2 ξs∗1 ,s2 . ∗ ∗ Note that B2,s2 = B2,s and thus ξ2,s2 = ξ2,s . The best parameters we were able 2 2 to find (from Table 1) were s1 = 4 and s2 = 6, and the resulting eigenvalue gave the following upper bound: (3)

C0,1 ≤

1 log2 6405.69924332 ≤ 0.526880847825. 24

4 Remark Direct computation of eigenvalues using standard linear algebra algorithms generally requires the storage of an entire matrix. This severely restricts the matrix sizes allowable, due to memory constraints on computers. By exploiting the fact that our matrices are all binary, symmetric, and easily computable, we were able to obtain the largest eigenvalues of much larger matrices. Specifically, the eigenvalues used to obtain the capacity bounds in Theorem 1 were computed using the “power method” [8, pg. 406]. Similarly, we obtained the upper bound in (1) with the power method (computing Λ 1,21 , Λ1,23, and ξ1,24 ). Originally these bounds were computed in [3] as 0.587891161 ≤ (2) C0,1 ≤ 0.588339078 (computing Λ 1,13, Λ1,15 , and ξ1,6 ) and were later improved in [4] (2)

(computing Λ 1,13 , Λ1,14 , and ξ1,14 ) to 0.587891161775 ≤ C0,1 ≤ 0.587891494943. The lower bound in (1) is from [4].

250

Zsigmond Nagy and Kenneth Zeger ∗ ∗ Table 1. Largest eigenvalues of Ta,b , Ba,b, and Ba,b are Λa,b, ξa,b , and ξa,b .

∗ ∗ a b Λa,b rows of Ta,b ξa,b rows of Ba,b ξa,b rows of Ba,b 1 1 1.61803398875 2 2 2.41421356237 3 2.41421356237 3 3 3.63138126040 5 4 5.45770539597 8 5.15632517466 7 5 8.20325919376 13 6 12.3298822153 21 11.5517095660 18 7 18.5324073775 34 8 27.8550990963 55 26.0579860919 47 9 41.8675533183 89 10 62.9289457252 144 58.8519350815 123 11 94.5852312050 233 12 142.166150393 377 132.947794048 322 13 213.682559741 610 14 321.175161677 987 300.345852027 843 15 482.741710897 1597 16 725.584002895 2584 678.525669346 2207 17 1090.58764423 4181 18 1639.20566742 6765 1532.89283597 5778 19 2463.80493521 10946 20 3703.21728345 17711 3463.03987027 15127 21 5566.11363689 28657 22 8366.13642876 46368 7823.53857819 39603 23 12574.7053170 75025 24 18900.3867144 121393 17674.5747630 103682 2 2 5.15632517466 7 5.15632517466 7 5.15632517466 7 3 11.1103016575 17 4 23.9250625386 41 21.9287654025 35 21.9287654025 35 5 51.5229210280 99 6 110.954925971 239 100.236549238 199 100.236549239 199 7 238.942175857 577 8 514.563569622 1393 463.203410887 1155 463.203410887 1155 9 1108.11608218 3363 10 2386.33538059 8119 2146.04060032 6727 2146.04060032 6727 11 5138.98917320 19601 12 11066.8474924 47312 9949.63685703 39203 9949.63685703 39203 3 3 34.4037405361 63 4 106.439377528 227 94.2548937790 181 5 329.331697608 827 6 1018.97101980 2999 884.498791440 2309 7 3152.75734322 10897 8 9754.81971205 39561 8421.60680806 30277 9 30181.9963196 143677 10 93384.9044989 521721 80481.0598378 398857 4 4 473.069084944 1234 404.943621498 933 355.525781764 743 5 2102.73425567 6743 6 9346.35893702 36787 7799.87080772 26660 6405.69924332 18995

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References 1. A. Kato and K. Zeger, “On the Capacity of Two-Dimensional Run Length Constrained Channels,” IEEE Trans. Info. Theory, 1999 (to appear). 2. D. Lind and B. H. Marcus, An Introduction to Symbolic Dynamics and Coding. New York: Cambridge University Press, 1995. 3. N. J. Calkin and H. S. Wilf, “The Number of Independent Sets in a Grid Graph,” SIAM Journal on Discrete Mathematics, vol. 11, pp. 54–60, February 1998. 4. W. Weeks and R. E. Blahut, “The Capacity and Coding Gain of Certain Checkerboard Codes,” IEEE Trans. Info. Theory, vol. 44, pp. 1193–1203, May 1998. 5. P. H. Siegel and J. K. Wolf, “Bit Stuffing Bounds on the Capacity of 2-Dimensional Constrained Arrays,” in Proceedings of ISIT98, (MIT, Cambridge, MA), p. 323, August 1998. 6. K. A. Immink, O. H. Siegel, and J. K. Wolf, “Codes for Digital Recorders,” IEEE Trans. Info. Theory, vol. 44, pp. 2260–2299, October 1998. 7. R. B. Bapat and T. E. S. Raghavan, Nonnegative Matrices and Applications. Cambridge, United Kingdom: Cambridge University Press, 1997. 8. G. H. Golub and C. F. van Loan, Matrix Computations (3rd edition). Baltimore and London: Johns Hopkins University Press, 1996.

Rectangular Codes and Rectangular Algebra
V. Sidorenko1, J. Maucher2, and M. Bossert2 1

Institute for Information Transmission Problems, Russian Academy of Science B. Karetnyi per. 19, 101447, Moscow GSP-4, Russia [email protected] 2 University of Ulm, Dept. of Information Technology Albert-Einstein-Allee 43, 89081 Ulm, Germany {joma,boss}@it.e-technik.uni-ulm.de

Abstract. We investigate general properties of rectangular codes. The class of rectangular codes includes all linear, group, and many nongroup codes. We define a basis of a rectangular code. This basis gives a universal description of a rectangular code. In this paper the rectangular algebra is defined. We show that all bases of a t-rectangular code have the same cardinality. Bounds on the cardinality of a basis of a rectangular code are given.

1

Introduction

A block code C is a set of words c = (c1 , . . . cn) of length n over an alphabet Q = {0, 1, . . . , q − 1}. Given t ∈ 1, n − 1, split every codeword c into the head (past) p = (c1 , . . . ct ) and the tail (future) f = (ct+1 , . . . , cn ), i.e., c is the concatenation of the head p and the tail f: c = pf. A set C ⊂ Qn is called t-rectangular if the following implication is true [1] (in [2] such a set was called t-separable): p1 f1 , p1 f2 , p2 f1 ∈ C

→ p2 f2 ∈ C.

(1)

A set C ⊂ Qn is called rectangular if it is t-rectangular for each t. All group codes (and hence all linear codes) are rectangular. Many famous nonlinear codes are also rectangular. This includes Hadamard, Levenshtein, Delsarte-Goethals codes (and hence Kerdock and Nordstrom-Robinson codes) [3], Goethals codes (and hence Preparata codes) [4]. All codewords of a linear block code having fixed Hamming weight form a rectangular set.
= {(00), (01), (10)} is the simplest example of a nonThe binary code C rectangular code. As an example of a rectangular code consider the binary code  = {(0000), (0011), (0101), (1000), (1011), (1101)}. The minimal trellis of the C  is shown in Fig. 1. There is a one to one correspondence between codecode C  and paths in the trellis. A trellis is called minimal if it has words of the code C in each depth the minimum number of vertices among all code trellises of the given code.


The work was supported by Russian Fundamental Research Foundation (project No 99-01-00840) and by Deutsche Forschungs Gemeinschaft (Germany).

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 252–258, 1999. c Springer-Verlag Berlin Heidelberg 1999


Rectangular Codes and Rectangular Algebra vr t

0 1

✲t ✼

0

✲t

1

0

✲t

1

❘t

0

0

253

✲ tvg ✒

1

❘ ✲t

 Fig. 1. A trellis of a rectangular code C. Rectangular codes have the following nice property. The minimal trellis of a rectangular code is unique, biproper [1], and minimizes the number of vertices |V | (by definition) the number |E| of edges, and the cycle rank |E| − |V | + 1. As a result the Viterbi decoding complexity of a rectangular code is minimum when using the minimal trellis of the code. In addition, the minimal code trellis gives a universal compact representation of a rectangular code. If a rectangular code has no additional structure then perhaps the minimal code trellis is the only known compact description of the code. We present another universal compact description of a rectangular set using a suggested idea of rectangular basis. Given an arbitrary block code C, a rectangular set that includes C and has the minimum cardinality is called a rectangular closure of C and is denoted by [C]. A rectangular closure [C] is unique. We say that a set G generates a rectangular set S (G is a generating set for S) if [G] = S. A set G is called independent if for any g ∈ G g ∈ / [G \ g]. An independent set B generating a rectangular set S is called a basis of the rectangular set S. In [8] we obtained the following results. Given a rectangular set S, the Coloring algorithm was proposed to design a basis B having cardinality |B| = |E| − |V | + 2,

(2)

where |E| and |V | is the number of edges and vertices in the minimal trellis of S respectively. Thus, a basis gives approximately the same compact description of a rectangular set as the minimal code trellis. Similar to trellis complexity, the cardinality of a code basis depends on the order of codewords positions. It was also shown in [8] that the Merging algorithm [5], [2] applied to a trellis of a set A generates the rectangular closure [A]. The complexity of the minimal trellis of a closure [A] is less than the one of any trellis of the nonrectangular set A. This fact can be used to simplify iterative decoding algorithms [6],[7]. A similar problem of constructing a set with smallest trellis complexity was considered in [7]. The Wolf bound is not valid for nonlinear codes (even for rectangular codes). In this paper we continue investigation of rectangular codes. In Section 2 we define the rectangular complement operation and introduce the rectangular algebra. We believe that the rectangular algebra is an interesting mathematical object. We show that the algebra has not Exchange of independent sets property. Despite of this fact we conjecture (Conjecture 12) that all bases of a rectangular code have the same cardinality (given by (2)). We propose an upper bound on the cardinality of rectangular closure of a given set. This bound was obtained independently also by Yu. Sidorenko [10]. In Section 3 we consider t−rectangular codes. This is equivalent to consider codes of length 2. We show that Conjecture 12 is true for t−rectangular codes.

254

V. Sidorenko, J. Maucher, and M. Bossert

In Section 4 using results from rectangular algebra we propose lower and upper bounds on the cardinality of a rectangular code. The codes attaining the upper bound are called prime. The sufficient condition for a code to be prime is presented.

2

Rectangular Algebra

A universal algebra or, briefly, algebra A is a pair < A; F >, where A is a nonvoid set and F is a family of finitary operations on A. A is called the base set. Define the base set A to be A = Qn = Q × Q . . . × Q, where Q is a finite alphabet . So, an element a ∈ A is a vector of length n over Q. For every t ∈ [1, n] we define a ternary partial operation of t-rectangular complement rt : A × A × A → A as follows. If a, b, c ∈ A can be represented as the following concatenations a = p2 f1 , b = p1 f1 , c = p1 f2 ,

(3)

where pi ∈ Qt , fi ∈ Qn−t , then rt (a, b, c) = p2 f2 , else rt (a, b, c) is undefined. Definition 1 The partial algebra Ret =< A; rt > is called a t-rectangular algebra. The rectangular algebra can be defined as < A; r1 , . . . , rn−1 > having n − 1 operations. However, we can simplify the definition of the algebra as follows. Extend the alphabet Q by joining a special zero element θ, Qθ = Q ∪ {θ} and define the partial operation sum as follows: ∀α, β, γ ∈ Qθ 1. α + α = θ; 2. α + β = β + α; 3. α + θ = α; 4. (α + β) + γ = α + (β + γ). The sum of words over Qθ is defined as componentwise sum. Lemma 2 If rt (a, b, c) is defined then rt (a, b, c) = a + b + c. Proof. If rt (a, b, c) is defined then there exist t-words p1 , p2 and (n − t)words f1 , f2 such that (3) is satisfied and rt (a, b, c) = p2 f2 . On the other hand a + b + c = (p2 + (p1 + p1 ))((f1 + f1 ) + f2 ) = (p2 + θ)(θ + f2 ) = p2 f2 . Corollary 3 If both ri (a, b, c) and rj (a, b, c) are defined then ri (a, b, c) = rj (a, b, c). This Corollary allows to define rectangular algebra using only one operation instead of n − 1 operations. Definition 4 The rectangular complement operation r(a, b, c) is defined as follows. If there exists t such that rt (a, b, c) is defined, then r(a, b, c) = rt (a, b, c), else r(a, b, c) is undefined.


All results are valid for the base set A = Q1 × Q2 . . . × Qn , where Qi is a finite alphabet.

Rectangular Codes and Rectangular Algebra

255

Definition 5 The partial algebra Re =< A; r > is called the rectangular algebra. Now we can give equivalent to (1) definitions of rectangular codes. Definition 6 A code C ⊆ A is called t-rectangular if < C; rt > is a subalgebra of Ret , i.e., C is closed under operation rt . Definition 7 A code C ⊆ A is called rectangular if < C; r > is a subalgebra of Re, i.e., C is closed under operation r. Definition 8 A rectangular code of minimum cardinality that includes a set G is called the rectangular closure of G and is denoted by [G]. Some properties of rectangular codes can be immediately obtained from the theory of universal algebra [9]. For example, the intersection of rectangular codes is rectangular [1], since the intersection of subalgebras is a subalgebra [9]. The Rectangular closure is unique [9]. We say that a set G generates a rectangular set C if [G] = C. How many words can be generated by a set G? The following theorem gives an upper bound for the rectangular closure of the set G. Theorem 9 |[G]| ≤ 2|G|−1 . Proof. Let G = {g1 , . . . , gm }. Every word c ∈ C = [G] can be generated by words from the set G using a chain of rectangular complement operations. So, c = r(y1 , y2 , y3 ), where

(i)

(i)

(i)

yi = r(z1 , z2 , z3 ), i = 1, 2, 3, and so on until we get the rectangular complement of words from the set G. Using Lemma 2 for this chain of rectangular complements we get c = y1 + y2 + y3 , c=

(1) z1

(1)

(1)

+ z2 + z3 + y2 + y3 , ...

c = gi1 + gi2 + . . . + gil ,

(4)

where l is odd since one item is always replaced by 3 items. Since the sum is commutative and distributive we can rewrite 4 as c = k1 g 1 + k2 g 2 + . . . + km g m , where ki is integer and kg = g + . . . + g (k times) by definition. Σki = l is odd. Using properties 1 and 3 of the sum operation we get c = j 1 g1 + j 2 g2 + . . . + j m gm , where ji ∈ {0, 1}, Σji is odd. So, to each word c ∈ [G] corresponds a binary sequence j1 , . . . , jm of odd Hamming weight. Hence, the number 2m−1 = 2|G|−1 of such sequences gives an upper bound for |[G]|. Q.E.D.

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Definition 10 A set G ⊆ A is called independent if for every g ∈ G : g ∈ / [G\g]. Definition 11 A set B ⊆ A is called a rectangular basis of a rectangular code C if B is independent and [B] = C. An important question for any universal algebra is: “Do all bases of a closed set have the same cardinality?”. Conjecture 12 All bases of a rectangular code have the same number of words. The invariance of the number of elements in a basis follows from the following exchange property: let y, z ∈ / [X] and z ∈ [X ∪y], then y ∈ [X ∪z]. Unfortunately, the rectangular algebra does not have this property. To prove, consider A = {0, 1}3 , X = {(100), (010), (001)}, y = (000), z = (111). However we still think that Conjecture 12 is true. In the next section we show that the conjecture is true for length 2 codes.

3

Rectangular Codes of Length 2

Assume that we are interested in t-rectangularity of a code C of length n for a particular t only. Denote by Qp the set of all t-heads of the code and by Qf the set of all t-tails. Each codeword of the code can be represented as a word pf, p ∈ Qp , f ∈ Qf . In this section we consider only length 2 codes over alphabets Qp , Qf . Every rectangular code can be represented as a union of disjoint subcodes [1,2] m  C= Pi × Fi , (5) 

i=1

 where Pi ⊆ Qp , Fi ⊆ Qf , Pi Pj = ∅, Fi Fj = ∅, i = j, and every subcode Ci = Pi × Fi is a direct product: Ci = {pf : p ∈ Pi , f ∈ Fi }. The following theorem gives the cardinality of a basis of a direct product. Theorem 13 Let B be a basis of length 2 code C = P × F , then |B| = |P | + |F | − 1.

(6)

Proof. Consider a |P |×|F | rectangular array. Rows of the array are labeled by words of the set P, columns are labeled by words from F. A word c = pf from the code C will be represented by a cross in the array in row p and column f. The code C is represented by the array A(C) completely filled with crosses. Let us fill the empty array with words of a basis B of the code C, denote the array by A(B). We say that three words a, b, c that satisfy (3) form a triangle because they occupy 2 columns and 2 rows in the array. These words generate the fourth word d = r(a, b, c) situated in the same lines. Using this procedure of cross generating the complete array must be filled starting from A(B).

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257

1. Every line (row or column) of the array A(B) must contain a cross. Otherwise this line will not be filled. 2. Let the first column of the array A(B) have n1 ≥ 1 crosses. These crosses occupy n1 + 1 lines (n1 rows and 1 column). Denote by Lr the set of these rows and by Lc the set consisting of the first column. The cells Lr × Lc of the array A(B) are filled with crosses. 3. Among the rest columns F \Lc there exists a column f having a cross in rows Lr . Otherwise the crosses in cells Lr × Lc are isolated and the array will not be filled; this contradicts the fact that B is a basis. The column f has exactly 1 cross in rows Lr , say in row p. If it has more than 1 cross then B is dependent because the set of crosses Lr × Lc and one cross pf will generate all crosses Lr × f. Join the column f to the set Lc and join rows occupied by crosses in column f to the set Lr . Totally we join nf new occupied lines, where nf is number of crosses in the column f. Using the column f the whole array Lr × Lc can be filled. 4. We will repeat step 3 until all columns of the array will be exhausted. After this we have Lc = F and Lr = P because the whole array must be filled. Hence the number of occupied lines will be |P | + |F |. On the other hand by construction the number of occupied lines is  nf = 1 + |B| = |P | + |F |, 1+ f ∈F

from where the statement of the theorem follows. Q.E.D. Since tailsets Fi and headsets Pi in (5) are disjoint from Theorem 13 we get Theorem 14 Let B be a basis of length 2 code C that satisfies (5), then |B| =



(|Pi | + |Fi | − 1).

(7)

i

The minimal trellis of a length 2 code has m + 2 vertices and edges. So, from Theorem 14 we have



i (|Pi | + |Fi |)

Theorem 15 Let B be a basis of length 2 rectangular code having the minimal trellis T = (V, E) then |B| = |E| − |V | + 2. Since the minimal trellis of a rectangular code is unique it follows from Theorem 15 that all bases of a rectangular length 2 code have the same cardinality.

4

Bounds on the Cardinality of a Basis

Now we return to the general case of rectangular codes of length n. What can we say about cardinality of a basis B(C) of a code C if only |C| and |Q| are known. From Theorem 9 we get

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Theorem 16 The cardinality of a basis B(C) of a binary rectangular code C is bounded by log2 |C| + 1 ≤ |B(C)| ≤ |C|. The lower bound can be attained for a binary alphabet. So in binary case the bounds can not be improved if only |C| is available. There exist a wide class of rectangular codes that attain the upper bound. We call these codes prime because they can be generated only by the complete code. Definition 17 A rectangular code C that satisfies B(C) = C is called prime code. The following theorem gives a sufficient condition for a code to be prime. Theorem 18 Let C be a rectangular code having minimum distance d(C) and diameter D(C) in the Hamming metric and 2d(C) > D(C),

(8)

then |B(C)| = |C|. The condition (8) in Theorem 18 can be replaced by d(C) > n/2. It follows from Theorem 17 and from the Coloring algorithm that |E| − |V | + 2 = |C| if (8) is satisfied. So |E| − |V | does not depend on reordering of codewords positions and one can find a permutation-minimal trellis that minimizes simultaneously |V | and |E|, when |E| − |V | is constant (this means minimization of the Viterbi decoding complexity).

References 1. F. R. Kschischang, “The trellis structure of maximal fixed-cost codes,” IEEE Trans. Inform. Theory, vol. 42, Part I, no. 6, pp. 1828 - 1838, Nov. 1996. 2. V. Sidorenko, “The Euler characteristic of the minimal code trellis is maximum,” Problems of Inform. Transm. vol. 33, no. 1, pp. 87-93, January–March. 1997. 3. V. Sidorenko, I. Martin, and B. Honary “On the rectangularity of nonlinear block codes,” IEEE Trans. Inform. Theory, vol. 45, no. 2, pp. 720 - 725, March 1999. 4. Y. Shany and Y. Be’ery, “On the trellis complexity of the Preparata and Goethals codes,” to appear in IEEE Trans. Inform. Theory. 5. A. Vardy and F. R. Kschischang, “Proof of a conjecture of McEliece regarding the optimality of the minimal trellis,” IEEE Trans. Inform. Theory, vol. 42, Part I, no. 6, pp. 2027 - 1834, Nov. 1996. 6. R. Lucas, M. Bossert, M. Breitbach, “Iterative soft-decision decoding of linear binary block codes,” in proceedings of IEEE International Symposium on Information Theory and its Applications, pp.811-814, Victoria, Canada, 1996. 7. S. Lin, T. Kasami, T. Fujiwara, and M. Fossorier, “Trellises and trellis-based decoding algorithms for linear block codes,” Boston: Kluwer Academic, 1998. 8. V. Sidorenko, J. Maucher, and M. Bossert, “On the Theory of Rectangular Codes,” in Proc. of 6th Intern. Workshop on Algebraic and Combinatorial Coding theory, Pskov, Russia, pp.207-210, Sept. 1998. 9. P.M. Cohn, Universal algebra, Harper and Row, New York, N.Y., 1965. 10. Yu. Sidorenko, “How many words can be generated by a rectangular basis”, preprint (in Russian).

Decoding Hermitian Codes with Sudan’s Algorithm T. Høholdt and R. Refslund Nielsen Technical University of Denmark Department of Mathematics, Bldg 303, DK-2800 Lyngby, Denmark {tom,refslund}@mat.dtu.dk http://www.student.dtu.dk/~p938546/public.html

Abstract. We present an efficient implementation of Sudan’s algorithm for list decoding Hermitian codes beyond half the minimum distance. The main ingredients are an explicit method to calculate so-called increasing zero bases, an efficient interpolation algorithm for finding the Qpolynomial, and a reduction of the problem of factoring the Q-polynomial to the problem of factoring a univariate polynomial over a large finite field.

1

Introduction

In 1997 M. Sudan [1] presented a conceptually easy algorithm for decoding ReedSolomon codes of rates less than 13 beyond half the minimum distance. The method was extended to algebraic geometry codes by M.A. Shokrollahi and H. Wassermann in [2], and M. Sudan and V. Guruswami in [3] further improved the algorithm to cover all rates both for Reed-Solomon codes and for general algebraic geometry codes. It is clear from [3] that the resulting algorithm has polynomial complexity, but also that some more work was needed to make the computations really efficient. In this paper we address that problem. The paper is organized as follows. Section 2 gives the necessary background on algebraic geometry codes, in particular the codes that come from the Hermitian curve. Section 3 gives the prerequisites on multiplicities and the concept of increasing zero bases, and Section 4 presents and proves Sudan’s algorithm. Section 5 gives an efficient method for calculating increasing zero bases and Section 6 gives a fast interpolation algorithm for finding the Q-polynomial. Section 7 treats the factorization problem and reduces it to factoring a univariate polynomial over a large finite field for which Berlekamp’s algorithm [4] can be used. Section 8 is the conclusion. A version including more detailed proofs and an example can be obtained at

2

Hermitian Codes

Let χ be a nonsingular absolutely irreducible curve over Fq and let P1 , . . . , Pn , P∞ be Fq -rational points on χ. The curve defines an algebraic function field, Fq (χ), with a discrete valuation, vPi , corresponding to each point (i = 1, . . . , n, ∞). More details can be found in [5]. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 260–269, 1999. c Springer-Verlag Berlin Heidelberg 1999 

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261

A class of algebraic geometry codes is given by CL (P1 + . . . + Pn , mP∞ ) = {(f(P1 ), . . . , f(Pn )) | f ∈ L(mP∞ )}

, m g − 1, a received word, w = (w1 , . . . , wn ) ∈ Fnq , and a parameter, s ≥ 1. Output: decτs (w).

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−1 – Set  ρ(z) := vP∞ (φk ) and calculate rs and t as in Lemma 1 with j := s+1 n . Now 2   (rs − 1)ρ(z) + t −1 τs = n − s

– Calculate Q(z) ∈ R[z]\{0} so that 1. For all i, Q has (Pi , wi ) as a zero of multiplicity at least s (see Definition 2). 2. ρ(Q) ≤ (rs − 1)ρ(z) + t – Factorize Q into irreducible factors. – If z−f divides Q and f ∈ L((k+g−1)P∞ ) and d((f(P1 ), . . . , f(Pn )), w) ≤ τs then add (f(P1 ), . . . , f(Pn )) to the set of candidates. That is decτs (w) := {(f(P1 ), . . . , f(Pn )) | f ∈ L((k + g − 1)P∞ ) ∧ (z − f)|Q ∧ d((f(P1 ), . . . , f(Pn )), w) ≤ τs } Any polynomial, Q, satisfying the two conditions in the algorithm will be called a Q-polynomial (with zero multiplicity s) in the following. The correctness of this algorithm must be shown by proving the existence of a Q-polynomial and proving that such a polynomial has the right factors. The existence is given by the following theorem: Theorem 2. A Q-polynomial does exist. Proof. By Lemma 3 it is clear that the first condition on a Q-polynomial can be written as a system of homogeneous linear equations. By Lemma 1 there is a non-zero solution satisfying the second condition.

The fact that a Q-polynomial has all the factors corresponding to codewords in decτs (w) is proved by the following lemma and theorem: Lemma 4. Let f ∈ L(mP∞ ) and suppose that f(Pi ) = wi . Then vPi (Q(f)) ≥ s. Proof. Follows from the definition of Q and Lemma 2 since vPi (f − wi ) ≥ 1.



Theorem 3. Let Q be a Q-polynomial and suppose that f ∈ L(k + g − 1) with d((f(P1 ), . . . , f(Pn )), w) ≤ τs . Then (z − f)|Q. Proof. Let h = Q(f). Then ρ(h) ≤ (rs − 1)ρ(z) + t. By Lemma 4 vPi (h) ≥ s for each value of i where f(Pi ) = wi . This happens at least n − τs times. So the total number of zero multiplicities of h is at least 

 (rs − 1)ρ(z) + t vPi (h) ≥ s(n − τs ) = s( + 1) > (rs − 1)ρ(z) + t ≥ vP∞ (h−1 ) s i=1

n 

So h = 0 and therefore z − f must divide Q.



Remark 1. Notice that by Lemma 1 the degree of Q must be less than rs . This means that Sudan’s algorithm gives at most rs − 1 codewords as output. So |decτs (w)| < rs

Decoding Hermitian Codes with Sudan’s Algorithm

5

265

Calculating Increasing Zero Bases

In principle, the method for calculating increasing zero bases, which is described in the proof of Theorem 1 is perfectly fine. However, in practice there are some unsolved problems since it is not trivial to calculate the standard representation of a function and evaluate the unit. In this section both problems are solved in the case of the Hermitian function field. Suppose that we want to calculate the standard representation of some polynomial, f ∈ OPi , with respect to the point Pi = (xi , yi ) (i ∈ {1, . . . , n}). In this case x − xi is a valid local parameter in Pi . Since f is regular in Pi the valuation of f in Pi can in principle be calculated by repeatedly dividing f by x − xi until a unit (a function which evaluates to a non-zero value in Pi ) is obtained. f can now be written as f=

q1  

fα,β (x − xi )α (y − yi )β

(4)

α=0 β∈N

 j where fα,β = j∈Z fα,β,j e with fα,β,j ∈ Fq12 , fα,β,j = 0 for all but a finite number of values of j, and e := (y − yi )q1 −1 + 1. It should be mentioned that this representation of f is not unique, however, that will not be a problem in this context. The idea of using this representation is that f can now be divided by x − xi in such a way that the result is a function which can be written on the same form. This is seen by noticing that in the Hermitian function field we have (x − xi )q1 + xi (x − xi )q1 −1 + xqi 1 y − yi = (5) x − xi e  e is a unit since e(Pi ) = 1 and furthermore f(Pi ) = j∈Z f0,0,j . In the case where f is not a unit (f(Pi ) = 0) let s := min{j | f0,0,j = 0}. Then  f0,0 = es f0,0,j ej−s j∈Z

where



j−s j∈Z f0,0,j e

is a polynomial in y, which is divisible by y − yi , since f

is not a unit. So with h :=



f0,0,j ej−s y−yi

j∈Z

we have

f0,0 y − yi = es h = es−1 ((x − xi )q1 + xi (x − xi )q1 −1 + xqi 1 )h x − xi x − xi

(6)

This leads to the following algorithm for calculating the standard representation of a polynomial in the Hermitian function field: Algorithm 2. Input: Polynomial, f, and point Pi = (xi , yi ). Output: u and m so that m = vPi (f) and f = u(x − xi )m .  1. Initialization: f (0) := f = α,β fα,β (x − xi )α (y − yi )β , m := 0.  2. If j∈Z f0,0,j = 0 then let u := f (m) and stop.

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3. Use equations (6) and (5) to calculate f (m+1) :=

f (m) = x − xi

  f0,0 (m) y − yi + fα,β (m) (x − xi )α−1 (y − yi )β + f0,β (m) (y − yi )β−1 x − xi x − xi α≥1,β

β≥1

4. Let m := m + 1 and go to step 2. The above algorithm for calculating standard representations provides all what is needed to implement the method for determining increasing zero bases in the proof of Theorem 1 since the unit of the standard representation is given on a form which can be evaluated directly.

6

Interpolation

The goal of the interpolation step is to determine a valid Q-polynomial. As mentioned in Theorem 2, such a polynomial must exist in the vector space   s+1 n span{Q0 , . . . , Q } ,  := 2 with Q0 , . . . , Q being as in Lemma 1, and a Q-polynomial can be found by solving a system of linear equations using Gaussian elimination (In the following each of these equations will be referred to as a zero-condition, see Lemma 3). However, the system has a special structure, and that can be used to make the calculations more efficient. One method for doing this is described in the following. The method is an application of the Fundamental Iterative Algorithm, along the same lines as the application in [7], Chapter 4. The Fundamental Iterative Algorithm was first presented in [8]. Let ord : R[z] → N∞ be given by ord(0) = −∞, ord(span{Q0 }\{0}) = {0} and ord(span{Q0 , . . . , Qi }\span{Q0 , . . . , Qi−1 }) = {i} for i > 0. If f ∈ R[z] then ord(f) will be called the order of the function f. The task of finding a Q-polynomial can now be rephrased as the task of finding a polynomial which satisfies the zero-conditions and which is minimal with respect to ord. It will be safe only to look for such a polynomial in the vector space V := {fz j | f ∈ R ∧ 0 ≤ j ≤ max{deg(Qi ) | 0 ≤ i ≤ }} The idea of the algorithm which is presented in the following is to make a partition of V and then consider the zero-conditions one by one while maintaining a list of polynomials – one from each partition class – which all satisfies the

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267

zero-conditions considered so far, and which all are minimal within the given partition class. To do this we need for each point a (small) element of R[z] which has a given pair, (Pi , wi ), as a zero of multiplicity at least s and which whenever it is multiplied by a polynomial gives a result within the same partition class as that polynomial. To make this work, the order of these elements must be the same for all i ∈ {1, . . . , n}. Therefore, choose t as the smallest integer so that ord(φj(i),i ) = t ∧ vPi (φj(i),i ) ≥ s , i = 1, . . . , n Now construct a partition of V doing the following: Let A := ord(V )\{i ∈ ord(V ) | ∃f ∈ ord−1 ({i})∃h ∈ ord−1 ({t}) : h|f} Furthermore, let h ∈ ord−1 ({t}) and define T :=

∞


ord−1 (ord(hi ))

i=0

Notice that this definition of T does not depend on the choice of h. Let A= {a1 , . . . , a|A| } and let Gj := ord−1 (ord({fj g | fj ∈ ord−1 ({aj }) ∧ g ∈ T })) Now G1 , . . . , G|A| is a partition of V , and furthermore, if a polynomial in some Gj is multiplied by a polynomial in ord−1 ({t}) then the result will remain in Gj . following notation is needed: Let f ∈ R[z] be written as f =  Finally, the j f φ z then j,k j,k k,i coef(f, φk,i z j ) := fj,k Now the algorithm can be stated: Algorithm 3. Input: Pairs, (P1 , w1 ), . . . , (Pn , wn ), and required zero multiplicity, s. Initialize by setting G := {g1 , . . . , g|A| } so that ord(G) = A (which means ord(gj ) = min ord(Gj ) for j = 1, . . . , |A|). For i = 1, . . . , n do the following: For each pair (j, k) ∈ N2 with j + k ≤ s and k ≥ 1 do the following: Let G := {g ∈ G | coef(g(z + wi ), φk,i z j ) = 0}. If G = ∅ then choose f ∈ G so that ord(f) = min ord(G ) and let G := {φj(i),if} ∪ {coef(f(z + wi ), φk,i z j )g − coef(g(z + wi ), φk,i z j )f} After this, the result is given by the polynomial in G which is smallest with respect to ord. The correctness of this algorithm is proved by induction over the iteration steps since G at any time holds one polynomial from each partition class satisfying the zero-conditions considered so far and minimal with respect to ord.

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Factorization

The Q-polynomial is a polynomial in R[z] and therefore, factorization is not so easy. However, fairly simple and efficient methods exist for factoring univariate polynomials over a finite field (see for example [4], Chapter 4). In this section the problem of factoring the Q-polynomial is transformed into a problem of factoring a univariate polynomial over a (large) finite field. In the case of Hermitian codes, R = Fq12 [X, Y ]/X q1 +1 − Y q1 − Y  and R = span{xα y β | 0 ≤ α ≤ q1 ∧ 0 ≤ β}. So seen this way, polynomials in span{Q0 , . . . , Q } have degree in x smaller than q1 + 1 and degree in y smaller than some integer, c. Now let f(Y ) ∈ Fq12 [Y ] with deg(f) ≥ c and D1 = Fq12 [Y ]/f(Y ) Furthermore, let g(X) := X q1 +1 − Y q1 − Y mod f and D2 = D1 [X]/g(X) Consider the mappings, φ : Fq12 [X, Y ][z] → D1 [X][z] and θ : D1 [X][z] → D2 [z], given by φ(h1 ) = h1 mod f θ(h2 ) = h2 mod g It is a well-known fact that these mappings are homomorphisms and that the composition of these mappings, θφ, is again a homomorphism. So suppose that h ∈ L(mP∞ ) and that z − h | Q then θφ(z − h) | θφ(Q) and furthermore, reducing z − h modulo f and g will not change z − h since the degree in Y of f and the degree in X of g is higher than the degrees of h in Y and X respectively. Therefore, in the factorization step, it will be sufficient to factorize θφ(Q), since this will still reveal the factors corresponding to codewords within distance τs from the received word. This will be very useful if f(Y ) is chosen so that D2 is a finite field. That will be the case if and only if D1 is a finite field (f is irreducible over Fq12 ) and g(X) is irreducible over D1 . Suppose that f is irreducible so that D1  Fq2c1 , where c1 = deg(f). Notice 1 that g(X) = X q1 +1 − (y q1 + y) then is a binomial in Fq2c1 [X]. The question is 1 now if f can be chosen with degree at least c so that g is irreducible. That this is in fact the case is shown in the following. The following theorem, which is a special case of theorem 3.75 of [4] is needed: Theorem 4. Let ω ∈ Fq12c \{0} and let e denote the order of ω. Then X q1 +1 − ω is irreducible over Fq12c if and only if each prime factor of q1 + 1 divides e but not

q12c −1 e .

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Notice that since q1 + 1 divides q12c − 1 the theorem states that X q1 +1 − ω is q2c −1 irreducible if and only if gcd(q1 + 1, 1 e ) = 1 with e being as in the theorem. The existence of a polynomial as the one called f above is given by the following theorem (the proof is omitted here): Theorem 5. Let c ≥ 1 be an integer. Then there exists a polynomial, f(Y ), so that deg(f) ≥ c and the order, e, of y q1 + y in Fq12 [Y ]/f(Y ) satisfies gcd(q1 + 1,

q12c − 1 )=1 e

It should be mentioned that experiments indicate that irreducible polynomials with the property described in Theorem 5 are rather common, so in practice it seems to be sufficient to generate random irreducible polynomials and check if they have the right property. However, we have no proof that this will always work.

8

Conclusion

We have demonstrated how to decode Hermitian codes efficiently beyond half the minimum distance using Sudan’s algorithm. The main steps are calculation of increasing zero bases, fast interpolation in order to determine the Q-polynomial, and a fast method of factorization. The actual complexity of the overall algorithm remains to be calculated and the extension to more general algebraic geometry codes is a subject for future work.

References 1. Madhu Sudan: “Decoding of Reed Solomon Codes beyond the Error-Correction Bound” Journal of Complexity 13, 180-193, 1997. 2. M.A. Shokrollahi and H. Wassermann: “List Decoding of Algebraic-Geometric Codes” IEEE Trans. Inform. Theory, vol 45, pp. 432-437, March 1999. 3. Venkatesan Guruswami and Madhu Sudan: “Improved Decoding of Reed-Solomon and Algebraic-Geometric Codes” Proc. 39th IEEE Symp. Foundations Comput. Sci., 1998. 4. Rudolf Lidl and Harald Niederreiter: “Finite fields” Addison-Wesley, 1983. 5. Henning Stichtenoth: “Algebraic function fields and codes” Springer-Verlag, 1993. 6. Weishi Feng and R. E. Blahut: “Some Results on the Sudan Algorithm”, Proceedings 1998 IEEE International Symposium on Information Theory, p.57, august 1998. 7. Ralf K¨ otter: “On Algebraic Decoding of Algebraic-Geometric and Cyclic Codes” Department of Electrical Engineering, Link¨ oping University, 1996. 8. G.-L. Feng and K. K. Tzeng: “A Generalization of the Berlekamp-Massey Algorithm for Multisequence Shift-Register Synthesis with Applications to Decoding Cyclic Codes”, IEEE Trans. Inform. Theory, vol. 37, pp. 1274-1287, Sept. 1991. 9. H. Elbrønd Jensen, R. Refslund Nielsen, and T. Høholdt: “Performance analysis of a decoding algorithm for algebraic geometry codes” IEEE Trans. on Inform. Theory, vol. 45, pp. 1712-1717, July 1999.

Computing a Basis of L(D) on an Affine Algebraic Curve with One Rational Place at Infinity Ryutaroh Matsumoto1 and Shinji Miura2 1

Sakaniwa Lab., Dept. of Electrical and Electronic Engineering Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8552 Japan [email protected] http://tsk-www.ss.titech.ac.jp/~ryutaroh/ 2 Sony Corporation Information & Network Technologies Laboratories 6-7-35, Kitashinagawa, Shinagawa-ku, Tokyo, Japan

Abstract. Under the assumption that we have defining equations of an affine algebraic curve in special position with respect to a rational place Q, we propose an algorithm computing a basis of L(D) of a divisor D from an ideal basis of the ideal L(D + ∞Q) of the affine coordinate ring
L(∞Q) of the given algebraic curve, where L(D + ∞Q) := ∞ i=1 L(D + iQ). Elements in the basis produced by our algorithm have pairwise distinct discrete valuations at Q, which is crucial in the construction of algebraic geometry codes. Our method is applicable to a curve embedded in an affine space of arbitrary dimension, and involves only the Gaussian elimination and the division of polynomials by the Gr¨ obner basis of the ideal defining the curve.

1

Introduction

For a divisor D on an algebraic curve, there exists the associated linear space L(D). Recently we showed how to apply the Feng-Rao bound and decoding algorithm [8] for the Ω-construction of algebraic geometry codes to the Lconstruction, and showed examples in which the L-construction gives better linear codes than the Ω-construction in certain range of parameters on the same curve [15]. In order to apply the Feng-Rao algorithm to an algebraic geometry code from the L-construction, we have to find a basis of the differential space Ω(−D + mQ) whose elements have pairwise distinct discrete valuations at the place Q. Finding such a basis of Ω(−D + mQ) reduces to the problem finding a basis of L(D
) whose elements have pairwise distinct discrete valuations at Q as described in [16]. Finding such a basis of Ω(−D + mQ) is also necessary in the precomputation of the efficient encoding method proposed in [16]. But there seems to be no algorithm capable to find such a basis of L(D
). In this paper we present an algorithm computing such a basis. 

2000 Mathematical Subject Classification. Primary 14Q05, 13P10; Secondary 94B27, 11T71, 14H05, 14C20.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 271–281, 1999. c Springer-Verlag Berlin Heidelberg 1999 

272

Ryutaroh Matsumoto and Shinji Miura

An affine algebraic curve with one rational place Q at infinity is easy to handle and used in many publications [9,17,18,19,20,21,22,23]. For a divisor D
∞ we define L(D+∞Q) := i=1 L(D+iQ). An affine algebraic curve is said to be in special position with respect to a place Q of degree one if its affine coordinate ring is L(∞Q) and the pole orders of coordinate variables generate the Weierstrass semigroup at Q (Definition 2). Under the assumption that we have defining equations of an affine algebraic curve in special position with respect to Q, we point out that a divisor can be represented as an ideal of L(∞Q), and we propose an efficient algorithm computing a basis of L(D). For effective divisors A and B with supp A ∩ supp B = ∅ and Q ∈ / supp A ∪ supp B, there is a close relation between the linear space L(A − B + nQ) and the fractional ideal L(A − B + ∞Q) of L(∞Q), namely L(A − B + nQ) = {f ∈ L(A − B + ∞Q) | vQ (f) ≥ −n}, where vQ denotes the discrete valuation at Q. When A = 0, by this relation we can compute a basis of L(−B + nQ) from a generating set of L(−B + ∞Q) as an ideal of L(∞Q) under a mild assumption. When A > 0, we find an effective divisor E such that −E + n
Q is linearly equivalent to A − B + nQ, then find a basis of L(−E + n
Q) from a generating set of the ideal L(−E + ∞Q), then find a basis of L(A − B + nQ) from that of L(−E + n
Q) using the linear equivalence. Computing an ideal basis of L(−E + ∞Q) from A − B + nQ involves computation of ideal quotients in the Dedekind domain L(∞Q), but by clever use of the properties of an affine algebraic curve in special position, our method involves only the Gaussian elimination and a small number of division of polynomials by the Gr¨ obner basis of the ideal defining the curve, thus it is efficient. Moreover while the other algorithms [2,3,4,6,11,12,13,25,26] except [10] are applicable only to a plane algebraic curve, our method is applicable to a curve embedded in an affine space of arbitrary dimension. The algorithm [10] is designed for an arbitrary projective nonsingular variety whose homogeneous coordinate ring satisfies Serre’s normality criterion S2 (a definition of S2 can be found in [7, Theorem 11.5]), and due to the wide applicability their method involves Buchberger’s algorithm that sometimes takes very long computation time. Due to the page limitation we had to omit all examples and most of proofs. For the complete version, please wait for the journal paper version or send email to the first author.

2

Theoretical Basis for Computation

First we fix notations used in this paper. K denotes an arbitrary perfect field. We consider an algebraic function field F/K of one variable. IPF denotes the set of places in F/K. For a place P , OP (resp. vP ) denotes the discrete valuation ring (resp. discrete valuation) corresponding to P . Other notations follow those in Stichtenoth’s textbook [24] unless otherwise specified.

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273

In this section, we introduce theorems which play important roles in ideal computation in the affine coordinate ring of an affine algebraic curve and computation of a basis of L(D). Hereafter we fix a place Q of degree one in F/K. 2.1

Relation between Fractional Ideals of Nonsingular Affine Coordinate Ring and Divisors in Function Field
∞ Definition 1. For a divisor D in F/K, we define L(D + ∞Q) := i=0 L(D + iQ).  Then we have L(∞Q) = Q=P ∈IPF OP , and we can see that L(∞Q) is a Dedekind domain by [24, p.71]. By [24, Proposition III.2.9], the set of maximal ideals in L(∞Q) is {L(−P + ∞Q) | P ∈ IPF \ {Q}}. Proposition 1. For a divisor D in F/K with Q ∈ / supp(D), L(−D + ∞Q) is a fractional ideal in L(∞Q). We have  L(−D + ∞Q) = L(−P + ∞Q)vP (D).  P ∈IP F

Corollary 1. For two divisors D, E with support disjoint from Q, L(−D + ∞Q)L(−E + ∞Q) = L(−(D + E) + ∞Q),  L(−D + ∞Q) + L(−E + ∞Q) = L(− min{vP (D), vP (E)}P + ∞Q), P =Q

L(−D + ∞Q) ∩ L(−E + ∞Q) = L(−



max{vP (D), vP (E)}P + ∞Q),

P =Q

L(−D + ∞Q) : L(−E + ∞Q) = L(−



max{0, vP (D) − vP (E)}P + ∞Q).

P =Q

Proof. The assertion follows from Proposition 1 and the unique factorization of an ideal in a Dedekind domain [27, Theorem 11, Section 5.6].

 By the facts described so far, we can show a preliminary version of our method for a basis of L(D). Let D := A − B + nQ, where A and B are effective, supp A ∩ supp B = 0 and Q ∈ / supp A ∪ supp B. Suppose that generating sets of the ideal L(−A + ∞Q) and L(−B + ∞Q) are given. If A = 0, then L(D) = {x ∈ L(−B + ∞Q) | vQ (x) ≥ −n}. From the equation above if we have a basis of L(−B + ∞Q) as a K-linear space with pairwise distinct pole orders at Q, then finding a basis of L(D) from that

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of L(−B + ∞Q) is just picking up elements in the basis of L(−B + ∞Q) with pole orders ≤ n. We shall show how to compute such a basis of L(−B + ∞Q) from a generating set of the ideal L(−B + ∞Q) in Theorem 1. If A = 0, then choose a nonzero element f ∈ L(−A + ∞Q). Let f be the ideal generated by f in L(∞Q), and I = (fL(−B + ∞Q)) : L(−A + ∞Q). Then I = L(−(f) + ∞Q)L(−B + ∞Q) : L(−A + ∞Q) = L(A − B − (f) + ∞Q). Since L(A − B − (f) + ∞Q) is an ordinary ideal of L(∞Q), we can compute a basis of L(A − B + nQ − (f)), say b1 , . . . , bl . Then b1 /f, . . . , bl /f is a basis of L(D) = L(A − B + nQ). We have to compute an ideal quotient in the method above. We shall show how to compute an ideal quotient only with linear algebra techniques in Section 4. 2.2

Modules over L(∞Q)

In this subsection we shall study how we can represent an L(∞Q)-submodule of F. Proposition 2. For a K-subspace W of L(∞Q), suppose that there is a subset {αj }j∈vQ (W \{0}) ⊂ W such that vQ (αj ) = j. Then {αj }j∈vQ (W \{0}) is a K-basis of W . 

Let a ∈ −vQ (L(∞Q) \ K). Then a > 0. Fix an element x ∈ F such that (x)∞ = aQ. Proposition 3. For an L(∞Q)-submodule M of L(∞Q), we set bi := min{j ∈ −vQ (M \ {0}) | j mod a = i} for i = 0, . . . , a − 1. Choose elements yi ∈ M such that vQ (yi ) = −bi . Then {y0 , y1 , . . . , ya−1 } generates M as a K[x]-module.  Proposition 4. We retain notations from the previous proposition. If a Ksubspace W generates M as a K[x]-module, that is, M = K[x]W , then we can find the elements yi in W for i = 0, . . . , a − 1. 

3

Gr¨ obner Bases of an Affine Algebraic Curve with a Unique Rational Place at Infinity

An affine algebraic curve with a unique rational place at infinity is convenient and has been treated by several authors independently [9,17,18,19,20,21,22,23]. we review and extend results in [19,20,23].

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275

Definition 2. [23, Definition 11] Let an ideal I ⊂ K[X1 , . . . , Xt ] define an affine algebraic curve, R := K[X1 , . . . , Xt ]/I, F the quotient field of R, and Q a place of degree one in F/K. Then the affine algebraic curve defined by I is said to be in special position with respect to Q if the following conditions are met: 1. The pole divisor of Xi mod I is a multiple of Q for each i. 2. The pole orders of X1 mod I, X2 mod I, . . . , Xt mod I at Q generate the Weierstrass semigroup {i | L(iQ) = L((i − 1)Q)} at Q. In other words, for any j ∈ {i | L(iQ) = L((i − 1)Q)} there exists nonnegative integers l1 , . . . , lt such that j=

t 

−li vQ (Xi mod I).

i=1

The Weierstrass form of elliptic curves can be considered as a special case of curves in special position. Proposition 5. We retain notations from Definition 2. R = L(∞Q) and the affine algebraic curve defined by I is nonsingular. 

If an algebraic curve is not in special position, then the proposed method cannot be applied to it. We can put an arbitrary algebraic curve into special position using Gr¨ obner bases if we know elements in the function field which have their unique pole at some place Q of degree one and their pole orders generate the Weierstrass semigroup −vQ (L(∞Q) \ {0}) [23, p.1739]. In another direction, it is convenient to have a class of algebraic curves known to be in special position. Miura found the necessary and sufficient condition for a nonsingular nonrational affine plane curve to be in special position [19,20]. An affine algebraic set defined by F (X, Y ) = 0 is a nonsingular nonrational affine algebraic curve in special position with respect to Q if and only if it is nonsingular and  F (X, Y ) = αb,0 X b + α0,a Y a + αi,j X i Y j ai+bj 1. This crossover probability is defined as the critical probability for iterative APP decoding, and represented by pcrit,AP P . It is important to notice that this definition simply implies that after iteration-1, an average probability PB,AP P (J, w) smaller than p is achieved, but without guarantee that the iterative algorithm will converge to a zero error probability with a sufficient number of iterations. As a result, this definition differs from that of [17] where convergence to an error free channel is considered. In other words, in this paper, the critical probability is defined as the largest p for which a better channel than the original one can be obtained, while in [17], the critical probability is defined as the largest p for which an error free channel can be obtained. Our definition can be justified by a crypto oriented motivation as even with residual errors, an information set decoding approach can complete the iterative decoding, resulting in an unsecured crypto system.

4

Convergence Analysis for Iterative BP Decoding

The iterative decoding based on BP updates J + 1 values for each bit-i, i = 1, 2, . . . , n [9,16]. As for the iterative APP decoding, for each bit-i, the a-posteriori values qi evaluated from the J check sums intersecting on bit-i are updated iteratively. However, for each check sum-l, l = 1, 2, . . . , J intersecting on bit-i, J values ql,i are also updated iteratively. The value ql,i is evaluated via the J − 1 check sums other than check sum-l intersecting on bit-i. Therefore, the value ql,i corresponds to the probability that bit-i is in error, given the information obtained via checks other than check-l. At iteration-j, the decision about bit-i is made based on qi , which has been computed from the values ql,i ’s evaluated at iteration-(j − 1). As a result, the BSC obtained by self-composition of the

Critical Noise for Convergence of Iterative Probabilistic Decoding with BP

287

original BSC and corresponding to the values ql,i ’s has to be considered after iteration-1. To evaluate the value ql,i , the result of check sum-l is discarded. For s < J , define c(s) as the event that an unsatisfied check sum-l is discarded and for s > 0, define c(s − 1) as the event that a satisfied check sum-l is discarded. Let C represent the corresponding random variable. It follows that for each bit-i and each value of s, 0 ≤ s ≤ J , the ratio of posterior error probabilities qi =

Pr(Ei = 1 | Si , C) 1 − Pr(Ei = 1 | Si , C)

(7)

can take one of the two possible following values qi,1 qi,2 =

Pr(Ei = 1 | Si = s, C = c(s)) p = = 1 − Pr(Ei = 1 | Si = s, C = c(s)) 1−p

Pr(Ei = 1 | Si = s, C = c(s − 1)) p = 1 − Pr(Ei = 1 | Si = s, C = c(s − 1)) 1−p

 

1 + (1 − 2p)w 1 − (1 − 2p)w 1 + (1 − 2p)w 1 − (1 − 2p)w

J−1−2s (8) , J+1−2s (9) .

Note that (8) is defined for s < J , while (9) is defined for s > 0. For a given value Si = s and a given value C = c, define the Bayes probability of error for each bit-i after the first iteration step of the decoding algorithm as PB (s, c) = min{Pr(Ei = 1 | Si = s, C = c), 1 − Pr(Ei = 1 | Si = s, C = c)},  qi if qi ≤ 1, i = 1+q (10) 1 1+qi if qi > 1, where qi is defined in (7). The average Bayes probability of error for each bit-i after the first iteration step of the decoding algorithm is given by  PB,BP (J, w) = PB (s, c) P (Si = s, C = c), s

=

J  s=0

c



P (Si = s) 

 J −s s PB (s, c(s − 1)) + PB (s, c(s)) , J J



qi,1 − 1 J −s P (Si = s) J qi,1 + 1 s: qi,1 >1   s qi,2 − 1  + P (Si = s) . J qi,2 + 1 s: q >1

= p−

(11)

i,2

As for (5), a necessary condition for convergence is that for j = 1, 2, there exists at least one qi,j > 1, so that PB,BP (J, w) < p in (11). In [15], it is shown that the necessary condition derived in Section 3 is also sufficient for the self composition model (j)

Pi

(j−1)

= Pi

− f(P(j−1) ).

(12)

288

Marc P.C. Fossorier, Miodrag J. Mihaljevi´c, and Hideki Imai

Unfortunately, this result no longer applies for the self composition model (j)

Pi

(0)

= Pi

− f˜(P(j−1) ),

(13)

which has to be considered for iterative BP decoding. This fact also explains the difference between our definition of the critical noise value and that of [17]. Note finally that (12) and (13) are equivalent for j = 1. As in Section 3, (11) is associated with a critical noise value and a corresponding critical probability pcrit,BP for iterative decoding based on BP. In fact, the following theorem shows that (11) can be simply obtained from (5) by considering J − 1 check sums of weight w + 1 each, as expected. Theorem 1. Let PB,AP P (J, w) and PB,BP (J, w) represent the average Bayes probabilities of error for each bit-i after the first iteration step of iterative APP and BP decodings, respectively. Then PB,BP (J, w) = PB,AP P (J − 1, w).

(14)

Proof. Let s1 ≤ J − 1 represent the largest value of s such that qi,1 > 1 in (11) and define for 0 ≤ s ≤ s1 , f(J, s) = (qi,1 − 1)/(qi,1 + 1). It follows that for 1 ≤ s ≤ s1 + 1, (qi,2 − 1)/(qi,2 + 1) = f(J, s − 1) and (11) can be rewritten as s s1 +1 1   s J −s PB,BP (J, w) = p −

s=0

=p−

P (Si = s) f (J, s) +

J

s1  J −s s=0

J

s=1

P (Si = s) +

J

P (Si = s) f (J, s − 1)



s+1 P (Si = s + 1) f (J, s). J

After some algebra, we obtain     s+1 J −s P (Si = s) + P (Si = s + 1) J J     J −1 s J −1 J−1−s =p pw (1 − pw ) + (1 − p) (1 − pw )s pJ−1−s w s s

(15)

(16)

By comparing (15) and (16) with (5) and (6), we conclude that PB,BP (J, w) is equal to the value obtained in (5) by considering J − 1 check sums of weight w + 1 for each bit, which completes the proof. Theorem 1 suggests that iterative APP decoding converges faster than iterative decoding based on BP. However, based on the model considered, we have no guarantee that convergence to the correct solution is achieved. Note finally that both iterative APP and iterative BP decodings yield the same decision at iteration-1, but different a-priori probabilities are considered to evaluate the a-posteriori probabilities which determine the decisions at subsequent iterations. From (14), we readily conclude that pcrit,BP < pcrit,AP P .

(17)

Critical Noise for Convergence of Iterative Probabilistic Decoding with BP

289

Since iteration-1 is common to any class of iterative decoding algorithm for the BSC model, pcrit,AP P can be viewed as an upper bound on the critical probability of such decoding methods. Note that this upper bound remains valid for the stronger convergence considered in [17]. Also, since the model considered for self concatenation in Sections 3 and 4 implicitly assumes a bit-flipping strategy, pcrit,BP can be interpreted as a more realistic estimate for practical BP based iterative decoding schemes. Note that pcrit,BP corresponds to an instance of the model considered in Example 5 of [17]. The values pcrit,AP P and pcrit,BP for different values J and w are given in Table 1. As expected, we observe that the Table 1. Values pcrit,AP P and pcrit,BP for different values J and w. J w + 1 pcrit,AP P pcrit,BP 3 6 0.091 0.039 4 8 0.080 0.055 5 10 0.071 0.057 3 5 0.127 0.061 4 6 0.124 0.091 3 4 0.191 0.107

values pcrit,AP P slightly overestimate the corresponding values derived in Table 4 of [17], while the values pcrit,BP are close to the values derived in Table 2 of [17].

5

Cryptanalysis of a Keystream Generator with LFSR’s

In this section, we apply the previously obtained results for improving the reported results regarding the analysis of a class of cryptographic pseudorandom bit generators for stream ciphering systems (see [13], pp. 191-197 and 205-207, for example). A number of the published keystream generators are based on binary LFSR’s assuming that parts of the secret key are used to load the LFSR’s initial states. The unpredictability request, which is one of the main cryptographic requests, implies that the linearity inherent in LFSR’s should not be “visible” in the generator output. One general technique for destroying the linearity is to use several LFSR’s which run in parallel, and to generate the keystream as a nonlinear function of the outputs of the component LFSR’s. Such keystream generators are called nonlinear combination generators. A central weakness of a nonlinear combination keystream generator is demonstrated in [18]. Assuming certain nonlinear functions it is shown in [18] that there is possible to reconstruct independently initial states of the LFSR’s, i.e. parts of the secret key (and accordingly the whole secret as well) based on the correlation between the keystream generator output and the output of each of the LFSR’s. The reported approach is based on exhaustive search through all possible nonzero initial states of each LFSR. Due to the exponential complexity of this approach it is not feasible when the lengths of employed LFSR’s are sufficiently long. Substantial improvements of the previous approach which yield

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complexity linear with the LFSR length are proposed in [12] and [21]. Further extensions and refinements of this approach called fast correlation attack, as well as its analysis are presented in a number of papers including [2,4,10,14]. 5.1

Fast Correlation Attack and Decoding

The problem of the LFSR initial state reconstruction based on the keystream generator output sequence can be considered as decoding a punctured simplex code after a BSC with crossover probability uniquely determined by the correlation between the generator output and the component LFSR output. This correlation means that the modulo-2 sum of the corresponding output of the LFSR and the generator output can be considered as a realization of a binary random variable which takes value 0 and 1 with the probabilities 1 − p and p, respectively, p = 0.5. Accordingly, the problem of the LFSR initial state reconstruction given the segment of the generator output can be considered as follows: (1) The n-bit segment of output sequence from the k-length LSFR is a codeword of an (n, k) punctured simplex code; and (2) The corresponding nbit segment of the nonlinear combination generator output is the corresponding noisy codeword obtained through BSC with crossover probability p. Also, following the approaches from [18] and [12], note that, if the ciphertext only attack is considered, then the influence of the plaintext in the previous model yields only some increase of the parameter p. The main underlying ideas for the fast correlation attacks are based on the iterative decoding principle introduced in [6]. Accordingly, all the fast correlation attacks mentioned in the previous section could be considered as variants of iterative decoding based on either simple flipping or extensions of the well known APP decoding [11]. Due to the established advantages of BP based iterative decoding over iterative APP, the main objective of this section is to report results of applications of BP based iterative decoding for realizations of the fast correlation attack. 5.2

Belief Propagation Based Fast Correlation Attack

In this section, the advantages of BP based correlation attacks with respect to APP based ones are verified by simulation. We consider the decoding of the punctured simplex code of length n = 980 and dimension k = 49 defined from its cyclic dual code generated by the primitive polynomial 1 + X 9 + X 49 . It follows that w + 1 = 3, while J varies between 5 and 12 depending on the position of the bit considered. We consider the BSC associated with BPSK transmission over an additive white Gaussian noise (AWGN) channel and hard decision decoding of  the received values, so that p = Q( 2Eb /N0 ), where Eb and N0 /2 represent the average energy per transmitted BPSK signal and the variance of the AWGN, respectively. Note that since the keystream sequence is “superposed” to the plaintext sequence during encryption, no redundancy is added to the message text and therefore, no normalization of the transmitted average energy with respect to the channel code rate is required, as opposed to conventional channel

Critical Noise for Convergence of Iterative Probabilistic Decoding with BP

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coding schemes. Figures 1 and 2 depict the simulation results for iterative APP decoding 0

−0.5

−1 log10(Pe)

* : Iteration 1 + : Iteration 2 . : Iteration 3

−1.5

x : Iteration 5 o : Iteration 20 −2

−2.5 −13

−12

−11

−10

−9 −8 Eb/N0 (in dB)

−7

−6

−5

−4

Fig. 1. Iterative APP decoding of the (980,49) truncated simplex code.

0

−0.5

−1

log10(Pe)

From top to bottom at SNR = −6 dB: −1.5 * : Iteration 1 + : Iteration 2 −2 . : Iteration 3 x : Iteration 5 o : Iteration 10

−2.5

* : Iteration 15 + : Iteration 20 −3

−3.5 −16

−14

−12

−10 Eb/N0 (in dB)

−8

−6

−4

Fig. 2. Iterative BP decoding of the (980,49) truncated simplex code. and BP decoding, respectively. In both cases, a maximum of 20 iterations was performed. In these figures, the word (or key) error probability is represented as a function of the SNR Eb /N0 in dB. Each key of 49 bits is encoded in systematic form into a codeword of length 980, and at the receiver, the key is retrieved from the decoded codeword based on the systematic encoding. Indeed, the error performance could be further improved by information set decoding methods, but such improvements are beyond the scope of this paper. Based on Figures 1 and 2, we observe that the iterative BP algorithm converges slower than the iterative APP algorithm, but achieves a much better error performance. Figure 3 compares APP and BP iterative decodings after 20 iterations at SNR values for which the key starts having a non-zero probability of being recovered. We observe that this event occurs for SNR ≈ -15 dB for BP decoding and SNR ≈ -12 dB for APP decoding. The corresponding crossover probabilities are p ≈ 0.40

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Marc P.C. Fossorier, Miodrag J. Mihaljevi´c, and Hideki Imai

and p ≈ 0.36, respectively. For w + 1 = 3, we compute pcrit,BP = 0.454 from (11) and pcrit,AP P = 0.458 from (5) for J = 12, and pcrit,BP = 0.372 and pcrit,AP P = 0.399 for J = 5. Similar results were observed for n = 490, in which case J varies between 4 and 9. For w + 1 = 3, we compute pcrit,BP = 0.437 and pcrit,AP P = 0.444 for J = 9, and pcrit,BP = 0.327 and pcrit,AP P = 0.372 for J = 4, while from simulations, we record p ≈ 0.36 for iterative BP decoding and p ≈ 0.33 for iterative APP decoding. In both cases, the theoretical values provide a relatively good estimate of the critical probabilities for the iterative BP decoding algorithm. 0

−0.005

log10(Pe)

−0.01 o : APP decoding + : BP decoding −0.015

−0.02

−0.025 −16

−15

−14

−13 −12 Eb/N0 (in dB)

−11

−10

−9

Fig. 3. Iterative APP and BP decodings of the (980,49) code after 20 iterations.

6

Concluding Remarks

In this paper, characteristics of the BP based iterative decoding assuming very high noise have been considered. In particular, the expected noise value beyond which the iterative BP decoding is not feasible for the BSC model considered has been derived. The established results have been applied to improve a cryptanalytic technique: the fast correlation attack. We have showed that under the same conditions of realization for fast correlation attacks, BP based iterative decoding provides significant improvements upon APP based iterative decoding.

Acknowledgments The authors wish to thank one reviewer for pointing out the work of [7,17], as well as Tom Richardson and Rudiger Urbanke for several interesting communications which enlighted some interpretations of our results.

References 1. A.M. Chan and F.R. Kschischang, “A simple taboo-based soft-decision decoding algorithm for expander codes,” IEEE Commun. Lett., vol. 2, pp. 183-185, 1998.

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2. V. Chepyzhov and B. Smeets, “On fast correlation attack on certain stream ciphers,” Advances in Cryptology - EUROCRYPT ’91, Lecture Notes in Computer Science,vol. 547, pp. 176-185, 1991. 3. G. C. Clark, Jr. and J. B .Cain, Error-Correcting Coding for Digital Communications. New York: Plenum Press, 1982. 4. A. Clark, J. Dj. Goli´c, and E. Dawson, “A comparison of fast correlation attacks,” Fast Software Encryption - Cambridge ’96, Lecture Notes in Computer Science, vol. 1039, pp. 145-157, 1996. 5. M.P.C. Fossorier, M. Mihaljevi`c and H. Imai, “Reduced complexity iterative decoding of low density parity check codes based on belief propagation,” IEEE Trans. Commun., vol. 47, pp. 673-680, May 1999. 6. R. G. Gallager, “Low-density parity-check codes,” IRE Trans. Inform. Theory, vol. IT-8, pp. 21-28, Jan. 1962. 7. M.G. Luby, M. Mitzenmacher, M.A. Shokrollahi and D.A. Spielman, “Analysis of Low Density Codes and Improved Designs Using Irregular Graphs,” 30th ACM STOC, May 1998. 8. R. Lucas, M. Bossert, and M. Breitbach, “On Iterative soft-decision decoding of linear binary block codes and product codes,” IEEE Jour. Select. Areas Commun., vol. 16, pp. 276-298, Feb. 1998. 9. D.J.C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, pp. 399-431, Mar. 1999. 10. D.J.C. MacKay, “Free Energy Minimization Algorithm for Decoding and Cryptanalysis,” Electronics Lett., vol. 31, pp. 446-447, Mar. 1995. 11. J. L. Massey, Threshold Decoding. Cambridge, MA: MIT Press, 1963. 12. W. Meier and O. Staffelbach, “Fast correlation attacks on certain stream ciphers,” Journal of Cryptology, vol. 1, pp. 159-176, 1989. 13. A. Menezes, P.C. van Oorschot and S.A. Vanstone, Handbook of Applied Cryptography. Boca Raton: CRC Press, 1997. 14. M. Mihaljevi´c and J. Goli´c, “A comparison of cryptanalytic principles based on iterative error-correction,” Advances in Cryptology - EUROCRYPT ’91, Lecture Notes in Computer Science, vol. 547, pp. 527-531, 1991. 15. M. Mihaljevi´c and J. Goli´c, “A method for convergence analysis of iterative errorcorrection decoding,” ISITA 96 - 1996 IEEE Int. Symp. Inform. Theory and Appl., Canada, Victoria, B.C., Sep. 1996, Proc. pp. 802-805 and submitted to IEEE Trans. Inform. Theory, Apr. 1997. 16. J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. San Mateo, CA: Morgan Kaufmann, 1988. 17. T. Richardson and R. Urbanke, “The Capacity of Low-Density Parity Check Codes under Message-Passing Decoding,” submitted to IEEE Trans. Inform. Theory, Nov. 1998. 18. T. Siegenthaler, “Decrypting a class of stream ciphers using ciphertext only,” IEEE Trans. Comput., vol. C-34, pp. 81-85, 1985. 19. M. Sipser and D.A. Spielman, “Expander codes,” IEEE Trans. Inform. Theory, vol. 42, pp. 1710-1722, Nov. 1996. 20. K. Yamaguchi, H. Iizuka, E. Nomura and H. Imai, “Variable Threshold Soft Decision Decoding,” IEICE Trans. Elect. and Commun., vol. 72, pp. 65-74, Sep. 1989. 21. K. Zeng, C.H. Yang and T.R.N. Rao, “An improved linear syndrome method in cryptanalysis with applications,” Advances in Cryptology - CRYPTO ’90, Lecture Notes in Computer Science, vol. 537, pp. 34-47, 1991.

An Authentication Scheme over Non-authentic Public Channel in Information-Theoretic Secret-Key Agreement
Shengli Liu and Yumin Wang National Key Laboratory on ISN, Xidian University Xi’an, 710071, People’s Republic of China [email protected], [email protected]

Abstract. It is necessary to authenticate the messages over an insecure and non-authentic public channel in information-theoretic secretkey agreement. For a scenario where all three parties receive the output of a binary symmetric source over independent binary symmetric channels as their initial information, an authentication scheme is proposed based on coding theory, which uses the correlated strings between the two communicants to authenticate the messages over the public channel. How to select coding parameters to meet the safety requirements of the authentication scheme is described. This paper also illustrates with an example that when the adversary’s channel is noisier than the communicants’ channels during the initialization phase, such an authentication scheme always exists, and the lower bound of the length of authenticators is closely related to the safety parameters, code rate of the authentication scheme and bit error probabilities of independent noise channels in the initialization phase.

1

Introduction

In the past few years, the information-theoretic secret-key agreement protocols [1][2][4][5][6][7] secure against adversaries with infinite computing power have attracted much attention, for those protocols discard the unproven assumptions on the hardness of certain computational problems such as the discrete logarithm or the integer factoring problem which are essential to public-key cryptographic protocols. Information-theoretic secret-key agreement over authentic public channel takes place in a scenario where two parties Alice and Bob who want to generate a secret key, have access to random variables X and Y , respectively, whereas the adversary Eve knows a random variable Z. The three random variables X, Y , and Z are distributed according to distribution PXY Z , and a key agreement protocol consists of three phases: Advantage distillation, Alice and Bob use their advantage over Eve offered by the authenticity of the public channel to generate an advantage over Eve in terms of their knowledge about each other’s information; Information reconciliation, Alice and Bob agree on a


This work was supported by National Natural Science Foundation of China.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 294–301, 1999. c Springer-Verlag Berlin Heidelberg 1999


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mutual string S by using error-correction techniques; Privacy amplification, the
partially secret S is transformed into a shorter, highly secret string S . Bennett
et.al. [6] have shown that the length of S can be nearly the R´enyi entropy of S when given Eve’s complete knowledge Z = z and the entire communication held over the public channel. Most of the protocols assume the existence of an insecure and authentic channel[2][4][5][6][7], which means that Eve can eavesdrop on the communication between Alice and Bob, but she can not modify or introduce fraudulent messages over the channel without detection. However, the existing public channels are usually non-authentic as well as insecure. In other words, Eve can see every message and replace it by an arbitrary message of her choice, and she even may impersonate either party by fraudulently initiating a protocol execution. So it is necessary to authenticate the public discussions in a secret key agreement over a non-authentic channel. For a scenario where all three parties receive the output of a binary symmetric source over independent binary symmetric channels as their initial information, an authentication scheme is proposed based on coding theory, which uses the correlated strings between the two communicants for authentication and makes information-theoretic secret-key agreement against active adversaries possible.

2

Secret-Key Agreement and the Scenario Employed

Generally, a key-agreement protocol consists of three phases[1]: – An initialization phase in which Alice, Bob and Eve receive random variables X,Y and Z, respectively, which are jointly distributed according to some probability distribution PXY Z . – During the communication phase Alice and Bob alternate sending each other messages M1 , M2 , . . . , Mt , Where Alice sends M1 , M3 , . . . and Bob sends M2 , M4 , . . .. Each message depends possibly on the sender’s entire view of the protocol at the time it is sent and possibly on privately generated random bits. Let t be the total number of messages and let Mt = [M1 , . . . , Mt ] denote the set of exchanged messages. – Finally, Alice and Bob each either accepts or rejects the protocol execution, depending on whether they believe to be able to generate a secret key. If Alice accepts, she generates a key S depending on her view of their protocol.
Similarly, if Bob accepts, he generates a key S depending on his view of the protocol (maybe with the help of privacy amplification techniques). We consider the following scenario(see Figure 1) which is inspired by[1] in this paper. 1. Initialization phase: A source (maybe a satellite) broadcasts random bits U n = (U0 , U1 , . . . , Un−1 ) through three independent binary symmetric channels CA , CB and CE with bit error probabilities A , B and E to Alice, Bob and Eve, and they receive random variable X n = (X0 , X1 , . . . , Xn−1 ), Y n =

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(Y0 , Y1 , . . . , Yn−1 ), and Z n = (Z0 , Z1 , . . . , Zn−1 ) respectively, which means PUi (0) = PUi (1) = 0.5, PXi Yi Zi |Ui = PXi |Ui ·PYi |Ui ·PZi |Ui , i = 0, 1, . . . , n−1, n−1
PXY Z (xi , yi , zi ) and PX n Y n Z n (x0 , . . . , xn−1 , y0 , . . . , yn−1 , z0 , . . . , zn−1 ) = i=0

2. Communication phase: Alice and Bob interchange messages over a public channel. We assume that the public channel is an ideal noiseless channel, but it is insecure and non-authentic. 3. Decision phase: Alice and Bob each either accepts or rejects the protocol execution. If both of them accept the results, they generate the final secret key.

Xn (Alice) ✛

CA ✛

✻

Source Un

✲ CB

Y n✲

(Bob)

✻

❄ CE Zn

❄

(Eve)

✲

✻ ❄ Public Channel

✛

Fig. 1. The scenario in the information-theoretic secret-key agreement.

The above scenario is a special case of the general key agreement protocol, which is well motivated by models such as discrete memoryless sources and channels previously considered in information theory. Such a scenario is relatively easy to analyze, and its result will be helpful to ongoing research. It is shown in [2] that such a secret key agreement over authentic public channel is possible even when Eve’s channel is superior to the other two channels, i.e. A ≥ E or B ≥ E . But M.Maurer proved recently that such a secret key agreement over non-authentic public channel is only possible when Eve’s channel is inferior to the other two channels[1]. For this reason we assume that A < E and B < E in this paper.

3

The Authentication Scheme

Some concepts and facts about coding theory are introduced first.

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Definition 1. [1] 0-1 distance from a codeword c1 to a codeword c2 , denoted by d(c1 → c2 ), is defined as the number of transitions from 0 to 1 when going from c1 to c2 , not counting the transitions from 1 to 0. The 0-1 distance of two codewords is different from the Hamming distance and it is not symmetric, i.e. d(c1 → c2 ) = d(c2 → c1 ) in general. Definition 2. The minimum 0-1 distance of a code C, denoted by d0→1 (C), is defined as the smallest value among the distances of any two different codewords in C, i.e. d0→1 (C) = min d(ci → cj ), ci , cj ∈ C. i,j,i=j

The minimum 0-1 distance of any conventional linear code is 0 for the existence of zero-code. Lemma 1. Every conventional linear code of length n with minimum Hamming distance d can be converted to a code of length 2n with minimum 0-1 distance d by replacing every bit in the original code by pair of bits, namely by replacing 0 by 01 and 1 by 10. The authentication scheme is described as follows: Prerequisite: Alice, Bob and Eve obtain initial information (X0 , X1 , . . .), (Y0 , Y1 , . . .) and (Z0 , Z1 , . . .) over independent binary symmetric channels with bit error probabilities A , B and E respectively, where A < E and

B < E .
Safety parameters: s, s > 1. Authentication performance: The receiver accepts the sender’s legitimate messages with probability at least 1 − 1/s2 while rejects Eve’s fraudulent
messages with probability at least 1 − 1/s 2 . Protocol: 1. Let AB = A + B − 2 A B , BE = B + E − 2 B E , AE =

A + E − 2 A E . Choose N and d from a set (N, d) satisfying


N∆1 ≥ [s (1 − AB − min( BE , AE ))/2]2 
 s N∆1 + s (N − d)∆1 + d∆2 ≤ d( BE − AB ) and


 s N∆1 + s (N − d)∆1 + d∆3 ≤ d( AE − AB )

(1) (2) (3)

where ∆1 = AB (1 − AB ), ∆2 = BE (1 − BE ) and ∆3 = AE (1 − AE ), and make sure that a (N, K, d) linear code C exists for some integer K. It should be noted that if N is large enough, such a (N, K, d) linear code always exists, a fact that will be shown in the next section. Parameters N, K, d and corresponding coding rules are public.
2. Convert the (N, K, d) linear code C to the corresponding 0-1 code C of length 2N with minimum 0-1 distance d, using the method provided by lemma 1. 3. Every time the sender sends a K-bit message to the receiver, she appends a N -bit authenticator to the message. Without loss of generality, we assume that Alice wants to send a message M = (M0 , M1 , . . . , MK−1 ) to

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Bob. The authenticator must be chosen according to the following rules: First of all, Alice encodes the message and gets the linear codeword C = (C0 , C1 , . . . , CN −1 ) according to the coding rules of the (N, K, d)
linear code. Then she finds out the corresponding 0-1 codeword C =






(C0 , C1 , . . . , C2N −1 ) from C . If Ci1 = Ci2 = . . . = CiN = 1 in the codeword, then she selects (Xi1 , Xi2 , . . . , XiN ) from (X0 , X2 , . . . , X2N −1 ) as the authenticator. Finally she appends (Xi1 , Xi2 , . . . , XiN ) to (M0 , M1 , . . . , MK−1 ) and sends it to Bob. 4. When Bob receives the message, he also gets the corresponding 0-1 code



word C = (C0 , C1 , . . . , C2N −1 ) from (M0 , M1 , . . . , MK−1 ). Similarly, he


determines (Yi1 , Yi2 , . . . , YiN ) from (C0 , C1 , . . . , C2N −1). Then he compares (Yi1 , Yi2 , . . . , YiN ) with the authenticator (Xi1 , X i2 , . . . , XiN ), if the number of different bits, say x, is less than N AB + s N AB (1 − AB ), he accepts the message or he rejects it. 5. (Xi1 , Xi2 , . . . , XiN ) and (Yi1 , Yi2 , . . . , YiN ) should be discarded from the initial correlated strings after each message transmission, and never used again for any purpose. Remark. The required authentication performance can be accomplished by choosing proper safety parameters. Theorem 2. The above authentication scheme ensures that the receiver accepts sender’s legitimate messages with probability at least 1−1/s2 while rejects Eve’s
fraudulent messages with probability at least 1 − 1/s 2 . Proof. Let AB = A + B − 2 A B be the bit error probability between corresponding bits of Alice’s and Bob’s strings, and AE = A + E − 2 A E and

BE = B + E − 2 B E be the bit error probabilities between corresponding bits of Alice’s and Eve’s and between Bob’s and Eve’s strings, respectively. We still assume Alice is the sender(When Bob is the sender, BE should be changed into

AE in the following proof). If the message sent by Alice is not modified by Eve, the subscripts of (Yi1 , Yi2 , . . . , YiN ) determined by Bob will be consistent with those of (Xi1 , Xi2 , . . . , XiN ) received by Bob since the public channel is noiseless. Let x denote the number of different bits between them, the expected value and the standard deviation of x are µ = N AB and σ=



N AB (1 − AB ).

 Since Bob accepts message only when x < N AB + s N AB (1 − AB ), we get the following result from the Chebyshev inequality Pr {|x − µ| < sσ} > 1 − σ 2 /(sσ)2  =⇒ Pr {|x − N AB | < s N AB (1 − AB ) > 1 − 1/s2

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 =⇒ Pr {x < N AB + s N AB (1 − AB ) > 1 − 1/s2 , which means that Bob accepts legitimate messages with probability at least 1 − 1/s2 . When Eve has intercepted a message together with its authenticator (M0 , M1 , . . . , MK−1 )||(Xi1 , Xi2 , . . . , XiN ), her best strategy for creating a new



authenticator for a different message M = (M0 , M1 , . . . , MK−1 )(hoping that it will be accepted by Bob) is to copy those bits from the received authenticator that are also contained in the new authenticator and to take as guesses for the remaining bits her copies of the bits in (Z0 , Z2 , . . . , Z2N −1 ), introducing bit errors in those bits with probability BE . The maximal probability of successful deception is hence determined by the number l of bits that Eve must guess and the total number N of bits in the forged authenticator. When Eve tries to deceive Bob, the expected value and the standard deviation of the bits in the forged authenticator that disagree with Bob’s corresponding bits are


µ = (N − l) AB + l BE and


σ =

  (N − l) AB (1 − AB ) + l BE (1 − BE ) = (N − l)∆1 + l∆2 ,

where ∆1 = AB (1 − AB ) and ∆2 = BE (1 − BE ). In fact, the 0-1 distance from a codeword C1 to a codeword C2 is the number of bits that Eve must guess when trying to convert the authenticator corresponding to C1 into the authenticator
corresponding to C2 . Since the minimum 0-1 distance of the code C is d, we obtain l ≥ d, which means that Eve must guess at least d bits to forge the authenticator. It is easy to prove that if


N∆1 ≥ [s (1 − AB − BE )/2]2 , the derivative function of f(x),


 f(x) = ( BE − AB )x − s N∆1 − s (N − x)∆1 + x∆2 ,


is non-negative, i.e. f (x) ≥ 0, when x ≥ 0. So we get f(l) ≥ f(d). The scheme assumes that 
 d( BE − AB ) ≥ s N∆1 + s (N − d)∆1 + d∆2 holds, so


 l( BE − AB ) ≥ s N∆1 + s (N − l)∆1 + l∆2








also holds, which means µ − µ ≥ sσ + s σ . Let x still denote the number of different bits between the authenticator received by Eve and that forged by Eve. From the Chebyshev inequality we know





























Pr {|x − µ | < s σ } > 1 − σ 2 /(s σ )2 =⇒ Pr {x > µ − s σ } > 1 − 1/s 2 .











Since µ − s σ ≥ µ + sσ, we get Pr {x > µ + sσ} > 1 − 1/s 2 , i.e. 
Pr {x > N AB + s N AB (1 − AB ) > 1 − 1/s 2 .


Therefore Bob rejects fraudulent messages with probability at least 1 − 1/s 2 .

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Shengli Liu and Yumin Wang

The Existence of the Authentication Scheme

In this section, we shows with an example that as long as A < E and B < E , it is possible to find a proper linear code to implement the above scheme and accomplish the required authentication performance. We take extended Reed-Solomon codes over a finite field GF (2r ) [3] as an example to illustrate that it is possible to find a proper linear code to implement the authentication scheme and accomplish the required authentication performance as long as A < E and B < E . Let N = 2r be the code length, and let information digit K = c · N where 0 < c < 1, then the minimum Hamming distance is d = (1 − c) · N + 1. When the extended Reed-Solomon code is converted to the 0-1 code, we know that the minimum 0-1 distance is still d. Substituting d = (1 − c) · N + 1 to the inequality (2), we obtain ((1 − c)N + 1) · ( BE − AB ) ≥ 
 s N∆1 + s N [c∆1 + (1 − c)∆2 ] + ∆2 − ∆1

(4)

N 0 (bit)

It is obvious that there exists a N0 to make both (1) and (4) hold for all integers N , N ≥ N0 . Therefore, we can always find an extended RS code (N, cN, (1 − c)N + 1) to implement the authentication scheme, and the code rate is K/(K + N ) = c/(c + 1)(see Figure 2).

25000

K/(K+N)=1/3, εΑ=0.1, εΒ=0.02, εΞ=0.3 K/(K+N)=1/9, εΑ=0.1, εΒ=0.02, εΞ=0.3 K/(K+N)=1/3, εΑ=0.01, εΒ=0.02, εΞ=0.3 K/(K+N)=1/9, εΑ=0.01, εΒ=0.02, εΞ=0.3

Lower bound of authenticator length

20000

15000

10000

5000

0 2

3

4 '

5 '

Safety parameters: s ,s (s = s )

Fig. 2. The lower bound of length of authenticators N0 as function of safety
parameters s, s for code rate K/(K + N ) = 1/3 and 1/9 with the bit error rate

A , B , E = 0.01, 0.02, 0.3 and 0.1,0.02,0.3.


We assume that s = s for simplicity. The dashed lines denote the case when A = 0.01, B = 0.02 and E = 0.3 while the solid lines denote the case

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when A = 0.1, B = 0.02 and E = 0.3. It is evident that the lower bound of
authenticators N0 grows with increasing s, s , decreases with decreasing code rate K/(K + N ), and the larger E / A or E / B , the smaller N0 is.

5

Conclusion

When two communicants and an adversary obtain correlated information through independent binary symmetric channels from a random source, and the adversary’s channel is noisier than those of communicants, information-theoretic secret-key agreement secure against active adversaries is always possible since an authentication scheme based on coding theory can always be implemented at the required safety level with the help of correlated strings between the two communicants. The authentication scheme based on extended RS code is simulated and the result shows that the lower bound of the length of authenticator is closely related to safety parameters, code rate and the bit error rates of the channels. Although a linear code satisfying the requirements can always be found, the scheme proposed in this paper may be not very practical since the authenticator may be too long or the code rate may be too low. So how to design a practical authentication scheme with high code rate and moderate authenticator length remains an open problem.

Acknowledgments The authors thank Dr. Wenping Ma for interesting discussions on the subject, and the referees for their valuable comments.

References 1. Maurer, U.: Information-theoretically Secure Secret-key Agreement by NOTauthenticated Public Discussion. In: Walter Fumy(Ed.): Advances in CryptologyEUROCRYPT’97, Lecture Notes in Computer Science, Vol. 1233. Springer-Verlag, Berlin Heidelberg New York (1997) 209-225 2. Maurer, U.M.: Secret Key Agreement by Public Discussion from Common Information. IEEE Transaction on Information Theory. 39(3) (1993) 733-742 3. Blahut, R.E.: Theory and Practice of Error Control Codes, Reading, MA: AddisonWesley. (1983) 4. Cachin,C., Maurer,U.: Linking Information Reconciliation and Privacy Amplification. J. Cryptol. 10 (1997) 97-110 5. Benett, C.H., Bessette, F., Brassard, G., Salvail,L., Smolin, J.: Experimental Quantum Cryptography. J.Cryptol. 5(1) (1992) 3-28 6. Bennett, C.H., Brassard, G., Crepeau,C., Maurer, U.M.: Generalized privacy amplification. IEEE Transaction on Information Theory. 41(6) (1995) 1915-1923 7. Brassard, G., Salvail, L.: Secret-key reconciliation by public discussion. In: Tor Helleseth(Ed.): Advances in Cryptology-Eurocrypt’93. Lecture Notes in Computer Science, Vol. 765. Springer-Verlag, Berlin Heidelberg New York (1994) 410-423

A Systolic Array Architecture for Fast Decoding of One-Point AG Codes and Scheduling of Parallel Processing on It Shojiro Sakata and Masazumi Kurihara The University of Electro-Communications Department of Information and Communication Engineering 1-5-1 Chofugaoka Chofu-shi, Tokyo 182-8585, Japan

Abstract. Since before we have proposed a systolic array architecture for implementing fast decoding algorithm of one point AG codes. In this paper we propose a revised architecture which is as its main framework a one-dimensional systolic array, in details, composed of a threedimensional arrangement of processing units called cells, and present a method of complete scheduling on it, where not only our scheme has linear time complexity but also it satisfies restriction to local communication between nearest cells so that transmission delay is drastically reduced.

1

Introduction

One-point codes from algebraic curves are a class of algebraic geometry (AG) codes which are important because not only they have potentially better performance for longer code lengths than the conventional codes such as BCH codes and RS codes, but also they can be decoded efficiently. We have given several versions [1][2][3][4] of fast decoding methods for them based on the BerlekampMassey-Sakata (BMS) algorithm [5][6]. As the second step toward practical use of one-point AG codes, we must devise efficient hardware implementation of the decoding method. In the setting, the vector version [4] of decoding method is suitable for parallel processing by nature. In this paper, we propose a special kind of systolic array architecture for its parallel implementation. This is a revision of our papers [7][8] partly presented at ISIT-95 and at ISIT-97 as well as a result of these previous trials. K¨ otter [9][10] gave a parallel architecture of his fast decoding method, which has a form nearer to the one-dimensional Berlekamp-Massey (BM) algorithm in comparison with our decoding method relying deeply on the multidimensional BMS algorithm. His architecture consisting of a set of feedback shift registers is an extension of Blahut’s implementation [11] of the BM algorithm. The shift registers have nonlocal links among delay units, which are not desirable because they give rise to long delay in transmitting data necessary for computations of the BM algorithm. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 302–313, 1999. c Springer-Verlag Berlin Heidelberg 1999


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303

Instead of shift register, we propose a systolic array architecture having only local links between neighboring cells (processing units). The main framework of this systolic array is one-dimensional, and it consists of a series of pipelined processors along which all data are conveyed. But, since each processor is composed of a two-dimensional arrangement of cells, the detailed structure is threedimensional. We assume that an array of size m is given as a set of input data, which is usually a syndrome array obtained from a given received word, where the integer m is comparable to code length. The input data are fed component by component into the cells constituting the first or leftmost processor of our systolic array. Then, all the relevant data are processed and transmitted component by component from the cells constituting each processor to those of its right-neighboring processor, and so on. As the output from the cells of the last or the rightmost processor, we can get the necessary data for decoding, in particular, the coefficients of polynomials composing a basis of the module of error locator functions, whose common zeros coincide with the error locators. Our main concern is in how to give a complete schedule of all the operations of cells constituting our systolic array, where we assume that each cell can compute by itself a small piece of intermediate values necessary for decoding and communicate the relevant data with its neighboring cells. The important issues are synchronization of all the operations of cells and local communication of the data through the links existing only between nearest cells (without any nonlocal feedback link). This is an extension of our previous work [12] on a systolic array architecture for implementing the BM algorithm of decoding RS codes. The present situation is more complex because of the multidimensional character of our fast decoding algorithm for one-point AG codes. By this scheme we can get much reduction in time complexity of decoding with some amount of space complexity, i.e. a number of cells, compared with the serial scheme. Furthermore, our parallel scheme is more efficient than K¨otter’s, in particular, w.r.t. time complexity, where ours O(m) is much better than his O(m2 ) for the input data size m.

2

Preliminaries

In this paper we use the same terminologies as in the reference[4] except for some modifications and additions. As preliminaries, we give a brief sketch of concepts and symbols appearing in the subsequent sections together with short descriptions of correspondences between the present and the previous symbols. Some important symbols used in [4] and their correspondences to the present ones are explained in brackets [ ]. A one-point code from an algebraic curve X over a finite field K = F q is defined from a prescribed point Q on the curve X , the set P of K-rational points excluding Q, an affine ring K[X ] = K[φ1 , · · · , φN ]/I and a nonnegative integer m, where K[X ] is the set of all algebraic functions having no pole except at the point Q, and the functions φi , 1 ≤ i ≤ N , constitute its basis. The

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Shojiro Sakata and Masazumi Kurihara

ring K[X ] is a K-linear space denoted as K[Σ] which is spanned by a set of
N functions {xp (:= i=1 φpi i )|p = (pi )1≤i≤N ∈ Σ}, where Σ is a subset of the direct product ZN 0 of the set Z0 of nonnegative nintegers. Specifically, such a code is defined as C := {c = (cj )1≤j≤n ∈ K n | j=1 cj f(Pj ) = 0, f ∈ L(mQ)} for a subset L(mQ), which consists of algebraic functions f ∈ L(∞Q)(:= K[X ]) of pole order o(f) ≤ m, m ∈ Z0 , where P = {P1 , · · · , Pn }. The set O of all possible pole orders o(xp ) of functions xp corresponding to p ∈ Σ coincides with the set of all pole orders o(f) of functions f ∈ K[X ], which is a semigroup with identity 0 w.r.t. addition. Its elements are numbered in the increasing order, i.e. O = {ol |l ∈ Z0 }, where o0 = 0 < o1 < o2 < · · ·. The first nonzero pole order o1 is denoted as ρ, and a function x ∈ K[Σ] having o(x) = ρ is fixed. Then, the set K[Σ] becomes a K[x]-module over the univariate 0 polynomial ring K[x] provided x is regarded as an independent variable. [x = xb for a certain b0 ∈ Σ.] A basis {o(i) |1 ≤ i ≤ ρ} of the semigroup O is given by o(i) := min{ol ∈ O|ol ≡ i − 1 mod ρ}, 1 ≤ i ≤ ρ, in particular o(1) = 0. [The semigroup Σ (w.r.t. vector addition) called a cylinder is identical with the union i ∪ρi=1 {bi + kb0 |k ∈ Z0 } for a ρ-tuple bi , 1 ≤ i ≤ ρ, such that o(xb ) = o(i) , 1 ≤ i ≤ ρ.] In the context of decoding, an error syndrome array u = (ul ), l ∈ Z0 , is introduced. [On the cylinder Σ, the syndrome array is given as u = (up ), p ∈ Σ, where up = ul for o(xp ) = ol .] Furthermore, a syndrome array vector u = (i) (u(i)), 1 ≤ i ≤ ρ, accompanied with ρ component arrays u(i) = (ujk ), (j, k) ∈ Σ (i), is considered, where Σ (i) is a subset of the direct product P × Z0 for P := {1, · · · , ρ} ⊂ Z0 . [The array vector u corresponds to a syndrome array u = (up ), p ∈ 2Σ, defined on the double cylinder 2Σ := {p + q|p, q ∈ Σ} ⊂ ZN 0 , which is represented as an array vector u = (ui ), 1 ≤ i ≤ ρ, with component (i) array ui = (uip ), p ∈ Σ in [4], such that uip = ubi +p , p ∈ Σ, In fact, ujk =

).] As K[Σ] is a residue class ring modulo an ideal I, the ubi + bj + k b0 (= ui j b + k b0 (i) array components ujk having the same value of o¯(i) (j, k) := o(i) + o(j) + kρ ∈ O are pairwise dependent through K-linear dependence of functions xp , p ∈ 2Σ, having the same pole order o(xp ) = ol modulo the linear subspace xq |o(xq ) < ol , q ∈ Σ. Thus, we introduce a pair of mappings η(l, i): Z0 × P −→ P and κ(l, i): Z0 ×P −→ Z0 defined by j = η(l, i), k = κ(l, i) if and only if there exists a pair (j, k) such that ol = o¯(i) (j, k), i.e. (j, k) ∈ Σ (i) ; otherwise we denote η(l, i) = κ(l, i) = ∅, where η(l, i) = (ol −i+1 mod ρ)+1, and κ(l, i) = (ol −o(i) −oη(l,i)))/ρ (i) if η(l, i), κ(l, i) = ∅. In fact, we know only the values ujk from a given received word such that o(i) (j, k) ≤ m. Example 1 Throughout this paper we use the following code accompanied with a set of instance data to illustrate our method. We consider one-point codes C (of codelength n = 64) over K = F 24 from the Hermitian curve X : X 5 + Y 4 Z + Y Z 4 = 0 having genus g = 6. For the point Q = (0 : 1 : 0), the linear space L(∞Q) is spanned by {xi y j |0 ≤ i, 0 ≤ j ≤ 3}, where the functions x := Z/X

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and y := Y /X have pole order o(x) = 4 and o(y) = 5, respectively. Thus, the cylinder Σ = {(i, j) ∈ Z20 |j ≤ 3} one-to-one corresponds to the semigroup of pole orders O = {o(xi y j )|(i, j) ∈ Σ} = {0, 4, 5, 8, 9, 10, 12, 13, 14, 15, 16, 17, · · ·}, where the first pole order is ρ = 4(= o(x)), and o(1) = 0(= o(1)), o(2) = 5(= o(y)), o(3) = 10(= o(y 2 )), o(4) = 15(= o(y 3 )). For an Hermitian code over F 24 defined by L(23Q), the values of o¯(i) (j, k) := o(i) +o(j) +kρ are shown in Table 1, and the corresponding functions η(l, i) and κ(l, i) are shown in the lefthalf of Table 2, where the symbol − means the empty value ∅. (All the tables are in Appendix.) The information of error locations is contained in a set of locator functions f ∈ K[Σ]. As any function in the module K[Σ] can be expressed uniquely as j ρ sj a form f = j=1 fj xb with fj = k=0 fjk xk ∈ K[x] of degree deg(fj ) = sj (fj,sj = 0), 1 ≤ j ≤ ρ, it is represented by the corresponding polynomial vector f = (fj ), 1 ≤ j ≤ ρ. The head number HN(f ) (or HN(f)) and the head exponent HE(f ) (or HE(f)) are defined from the pole order o(f), w.r.t. a ρtuple of prescribed weight vectors w i , 1 ≤ i ≤ ρ. In this paper, we restrict ourselves to the case of w i = bi , 1 ≤ i ≤ ρ. The exact error locations, which constitute a subset E of the set P, are obtained as the zeros of a so-called error locator module M(E) := {f ∈ K[Σ]|f(P ) = 0, P ∈ E}. Thus, we inquire a (i) ρ-tuple of polynomial vectors f (i) = (fj )1≤j≤ρ , 1 ≤ i ≤ ρ, which constitute a basis of the submodule M(E), or equivalently the corresponding polynomial (i) (i) matrix F = [fj ], 1 ≤ i, j ≤ ρ, where HN(f (i) ) = i, 1 ≤ i ≤ ρ, and fj = (i) sj (i) k (i) ) is equal to k=0 fjk x ∈ K[x], 1 ≤ i, j ≤ ρ. (The head exponent HE(f (i)

(i)

(i)




(i)

sj = deg(fj ) for a certain j, 1 ≤ j ≤ ρ, s.t. o(j) + sj ρ ≥ o(j ) + sj
ρ for any j  , 1 ≤ j  ≤ ρ.) For the code defined by L(mQ), the polynomial matrix F can be obtained as a minimal polynomial matrix of the syndrome array vector up to the pole order m + 4g by the BMS algorithm including majority voting scheme. In the subsequent sections, we often dispense with the subscripts and (i) superscripts, and denote f (i), f(i, j), f(i, j, k), s(i, j), etc. instead of f (i), fj , (i)

(i)

fjk , sj , etc.

3

BMS Algorithm and Its Pipelining

The vector-version of BMS algorithm[4] can be given in a modified form, which finds not only a pair of minimal and auxiliary polynomial matrices F (l) = [f(l, i)]1≤i≤ρ , G(l) = [g(l, i)]1≤i≤ρ but also the corresponding pair of discrepancy and auxiliary array vectors v(l) = (v(l, i))1≤i≤ρ , w(l) = (w(l, i))1≤i≤ρ iteratively at the increasing pole orders ol ∈ O, where each (l-th) iteration is for the set of 3-tuples (i, j, k) having the same value o¯(i) (j, k)(= ol ), l ∈ Z0 .

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The discrepancy and auxiliary array vectors v(l), w(l) are defined by a kind of operation by the minimal and auxiliary polynomial matrices F (l), G(l), respec(i) tively, upon the syndrome array vector u as follows. Denoting u(i, j, k) := ujk , 1 ≤ i ≤ ρ, (j, k) ∈ Σ (i) , ρ s(l,i,µ)   v(l, i, j, k) := f(l, i, µ, ν)u(µ, j, ν + k), µ=1 ν=1

w(l, i, j, k) :=

ρ t(l,i,µ)  

g(l, i, µ, ν)u(µ, j, ν + k),

µ=1 ν=1

where the values f(l, i, j, k) (0 ≤ k ≤ s(l, i, j) := deg(f(l, i, j))), g(l, i, j, k) (0 ≤ k ≤ t(l, i, j) := deg(g(l, i, j))) are the coefficients of polynomials f(l, i, j), g(l, i, j), and the values v(l, i, j, k), w(l, i, j, k) ((j, k) ∈ Σ (i)) are the components of arrays v(l, i), w(l, i), respectively. ¯ = (η(l, i), κ(l, i)) coincides The value d(f (l, i)) := v(l, i, ¯j, k¯ −s(l, i)) for (¯j, k) with the discrepancy of the minimal polynomial vector f (l, i) w.r.t. the syndrome array vector u, provided the value d(f (l, i)) does not vanish, where s(l, i) = HE(f (l, i)). The following is a modified BMS algorithm (without majority logic scheme) for a syndrome array vector given up to the pole order m. Algorithm Step 1 (Initialization) l := 0; s(0, i) := 0, 1 ≤ i ≤ ρ; f (0, i) := ei (the i-th unit vector), 1 ≤ i ≤ ρ; c(0, i) := −1, 1 ≤ i ≤ ρ; g(0, i) := ∅, 1 ≤ i ≤ ρ; v(0, i, j, k) := u(i, j, k), (j, k) ∈ Σ (i) , 1 ≤ i, j ≤ ρ; w(0, i, j, k) := ∅; Step 2 (Discrepancy computation) FN := {f(l, i)|d(f (l, i)) = 0, 1 ≤ i ≤ ρ}; ¯ := κ(l, i); Step 3 (Updating) for each f (l, i), 1 ≤ i ≤ ρ, ¯j := η(l, i), k df ¯ := d(f (l, i))/d(g(l, j)); dg ¯ Case A: if f (l, i) ∈ FN , for (j, k) ∈ Σ (i) or Σ (j) f(l + 1, i, j, k) := f(l, i, j, k), v(l + 1, i, j, k) := v(l, i, j, k), g(l + 1, ¯ j, j, k) := g(l, ¯j, j, k), w(l + 1, ¯j, j, k) := w(l, ¯j, j, k); Case B: if f (l, i) ∈ FN and s(l, i) ≥ k¯ − c(l, ¯j), ¯ for (j, k) ∈ Σ (i) or Σ (j) df g(l, ¯j, j, k − s(l, i) + k¯ − c(l, ¯j)), dg df ¯ − c(l, ¯j)), w(l, ¯j, j, k − s(l, i) + k v(l + 1, i, j, k) := v(l, i, j, k) − dg g(l + 1, ¯ j, j, k) := g(l, ¯j, j, k), w(l + 1, ¯j, j, k) := w(l, ¯j, j, k); ¯ − c(l, j), ¯ Case C: if f(l, i) ∈ FN and s(l, i) < k (i) (¯ j) for (j, k) ∈ Σ or Σ f(l + 1, i, j, k) := f(l, i, j, k) −

¯ + c(l, ¯j) + s(l, i)) − df g(l, ¯j, j, k), f(l + 1, i, j, k) := f(l, i, j, k − k dg ¯ + c(l, j) ¯ + s(l, i)) − df w(l, j, ¯ j, k), v(l + 1, i, j, k) := v(l, i, j, k − k dg g(l + 1, ¯ j, j, k) := f(l, i, j, k), w(l + 1, ¯j, j, k) := v(l, i, j, k);

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Cace D: if f (l, i) ∈ FN and g(l, ¯j) = ∅, for (j, k) ∈ Σ (i) or Σ (j) ¯

¯ + s(l, i) − 1), f(l + 1, i, j, k) := f(l, i, j, k − k ¯ v(l + 1, i, j, k) := v(l, i, j, k − k + s(l, i) − 1), g(l + 1, ¯ j, j, k) := f(l, i, j, k), w(l + 1, ¯j, j, k) := v(l, i, j, k); Step 4 (Termination check) l := l + 1; if ol > m then stop else go to Step 2. ¯ j, k) are quite similar to The computations of v(l + 1, i, j, k) and w(l + 1, j, those of f(l+1, i, j, k) and g(l+1, ¯j, j, k), respectively, owing to their relationships so that there are involved many redundancies in their computations. However, these duplicate computations and the redundant structure of the discrepancy and auxiliary array vectors are indispensable for efficient parallel implementation of the BMS algorithm, in particular, in calculation of the discrepancy d(f (l, i)) through the array v(l, i, j, k). By introducing a pair of modifications of the mapping κ(l, i), we have a pipelined version of the above algorithm, which can be implemented easily on a kind of systolic array. First, putting κ(i) := (o(i) − i + 1)/ρ, 1 ≤ i ≤ ρ, we introduce oˇ(i) (j, k) := o(i) + kρ + j − 1, oˆ(i) (j, k) := oˇ(i) (j, k) − κ(i)ρ, where oˇ(i) (j, k) and oˆ(i) (j, k) are defined only for (j, k) ∈ P × Z0 s.t. oˇ(i) (j, k), ˇ (i) and Σ ˆ (i), respecoˆ(i) (j, k) ∈ O. We denote the sets of those points (j, k) as Σ tively. Then, it is easy to see that the mappings κ ˇ (l, i) := (ol − o(i) − η(l, i) + 1)/ρ, κ ˆ (l, i) := κ ˇ(l, i) + κ(i) satisfy the following relationships ˇ (l, i) = k, η(l, i) = j; oˆ(i) (j, k) = ol ⇔ κ ˆ (l, i) = k, η(l, i) = j; oˇ(i) (j, k) = ol ⇔ κ κ(l, i) = κ ˇ (l, i) − κ(η(l, i)) = κ ˆ (l, i) − κ(η(l, i)) − κ(i). Furthermore, the values κ ˇ (l, i), κ ˆ (l, i) are nondecreasing w.r.t. l ∈ Z0 , and in particular κ ˇ (l, i) ≤ κ ˇ (l + 1, i) ≤ κ ˇ (l, i) + 1, κ ˆ(l, i) ≤ κ ˆ(l + 1, i) ≤ κ ˆ (l, i) + 1. Example 2 (Continued) In our example of code, the functions κ ˇ (l, i) and κ ˆ (l, i) are shown in the righthalf of Table 2. Finally, we define modified polynomial vectors fˇ (l, i) and arrays vˆ(l, i) having the coefficients and components vˆ(l, i, j, k) := v(l, i, j, k − κ(i) − κ(j) − s(l, i)), fˇ(l, i, j, k) := f(l, i, j, k − κ ˇ (l, i) + s(l, i)),

308

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where HN (fˇ (l, i)) = i, HE(fˇ (l, i)) = κ ˇ (l, i). Then, we can show that the updatings of them and their auxiliary counterparts are given as follows: fˇ(l + 1, i, j, k) := fˇ(l, i, j, k + κ ˇ (l, i) − κ ˇ (l + 1, i)) ¯ −ˇ g (l, j, j, k + κ ˇ(l, i) − κ ˇ (l + 1, i)), 0 ≤ k ≤ κ ˇ (l + 1, i), ˇ ¯ ¯ gˇ(l + 1, j, j, k) := gˇ(l, j, j, k) or f (l, i, j, k), depending on case, df ˆ w(l, ˆ ¯j, j, k), (j, k) >T (¯j, k), dg w(l ˆ + 1, ¯ j, j, k) := w(l, ˆ ¯ j, j, k) or vˇ(l, i, j, k), depending on case. vˆ(l + 1, i, j, k) := vˆ(l, i, j, k) −

The components vˆ(l, i, j, k), w(l, ˆ i, j, k) of discrepancy and auxiliary arrays ˆ where the total ˆ (i) s.t. (j, k) ≥T (¯j, k), vˆ(l, i), w(l, ˆ i) are given for (j, k) ∈ Σ   (i) order ≥T is defined by (j, k) ≥T (j , k ) if and only if o (j, k) ≥ o(i) (j  , k  ). It is important to note that they satisfy for ¯j = η(l, i), k¯ = κ(l, i), kˇ = κ ˇ (l, i), kˆ = κ ˆ (l, i) ˇ = f(l, i, i, s(l, i)), vˆ(l, i, ¯j, k) ˆ = v(l, i, ¯j, k¯ − s(l, i)), fˇ(l, i, i, k) where the above righthand sides are equal to the head coefficient of the polynomial vector f (l, i) and the discrepancy d(f (l, i)), respectively. Furthermore, both polynomial coefficients fˇ(l, i, j, k) and gˇ(l, i, j, k) are synchronized in updating fˇ(l, i, j, k) to fˇ(l + 1, i, j, k), and similarly both vˆ(l, i, j, k) and w(l, ˆ i, j, k) are synchronized in updating vˆ(l, i, j, k) to vˆ(l + 1, i, j, k) as shown in the above formulae. These updating formulae are a generalization of the similar ones in parallelization of the BM algorithm[12].

4

Systolic Array and Scheduling

To implement parallelization of our algorithm, we introduce a special kind of systolic array architecture as follows, where instead of the subset {o ∈ O|0 ≤ o ≤ m}, we take {l ∈ Z0 |0 ≤ l ≤ m} containing gaps for the purpose of making the following discussions easy. (The functions η, κ, etc. are redefined appropriately so that they are assumed to be completely defined.) (1) It consists of a series of m + 1 processors: P (0), · · · , P (m), where the lth processor P (l) is connected to the (l − 1)-th (left-neighboring) and the (l + 1)-th (right-neighboring) processors P (l − 1) and P (l + 1), 1 ≤ l ≤ m − 1, except for the 0-th (leftmost) and the m-th (rightmost) ones. There are g trivial (dummy) processors working only as delayers as well as m + 1 − g effective processors by which the BMS computations are executed. (2) The leftmost processor P (0) receives as input data the components of a syndrome array vector, and the rightmost processor P (m) outputs the components of the minimal polynomial matrix, from which one can get the desired basis of the error locator module. The l-th processor P (l), 1 ≤ l ≤ m, receives as input the components of the modified arrays vˆ(l − 1, i), w(l ˆ − 1, i)

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ˇ (l − 1, i) from the and the coefficients of the polynomial vectors fˇ (l − 1, i), g (l − 1)-th processor P (l − 1) and (if it is effective) calculates the components of the modified arrays vˆ(l, i), w(l, ˆ i) and the coefficients of the polynomial ˇ (l, i) together with the accompanying control data such as vectors fˇ (l, i), g s(l, i), c(l, i), d(f (l, i)), d(g(l, i)), 1 ≤ i ≤ ρ. Each computation of values vˆ(l, i, j, k), etc. is a combination of one multiplication and one addition over the symbol field K. (3) Each processor P (l), 0 ≤ l ≤ m, has ρ subprocessors S(l, i), 1 ≤ i ≤ ρ, which contain ρ cells C(l, i, j), 1 ≤ j ≤ ρ. In total, it consists of ρ2 cells C(l, i, j), 1 ≤ i, j ≤ ρ. For k ∈ Z0 , cell C(l, i, j) manipulates the four values vˆ(l, i, j, k), w(l, ˆ ¯ j, j, k), fˇ(l, i, j, k), and gˇ(l, ¯j, j, k) at a certain clock n ∈ Z0 , where ¯j = η(l, i) and n depends on l, i, j, k as well as vˆ, w, ˆ fˇ, gˇ. (4) We assume an artificial 2-dimensional arrangement of ρ2 cells C(l, i, j), 1 ≤ i, j ≤ ρ. (We disregard realizability of such an effective structure by the current VLSI technology.) Consequently, our systolic array architecture has a three-dimensional structure having three perpendicular axes corresponding to the indices l, i, j. The arrangement of ρ2 cells in each processor P (l) is determined so that each cell C(l, i, j) is situated at the point of coordinates −1 (l, φ−1 is the inverse of l (i), j), 0 ≤ l ≤ m, 1 ≤ i, j ≤ ρ, where the mapping φl the following permutation φl of the integers 1, 2, · · · , ρ defined by induction w.r.t. l. (Base)  if k = 1  1, , if k=odd & k = 1 φ0 (k) := ρ − k−3 2 k + 1, if k=even 2 (Induction) for 1 ≤ l ≤ m  φl−1(ρ), if    φl−1(k + 1), if φl (k) := φl−1(1), if    φl−1(k − 1), if

k k k k

=ρ = ρ =1 = 1

& & & &

k + l=even k + l=even k + l=odd k + l=odd

(5) The main data such as components of discrepancy arrays and coefficients of polynomials, i.e. elements of the symbol field K, are transmitted from the cells in the processor P (l) to the cells in the processor P (l+1), 0 ≤ l ≤ m−1. More precisely, these communications are made through the links which connect each cell C(l, i, j), 1 ≤ i, j ≤ ρ, in P (l) to two cells C(l − 1, i, j) and C(l − 1, i− , j) in the left-neighboring processor P (l − 1) and also to two cells C(l + 1, i, j) and C(l + 1, i+ , j) in the right-neighboring processor P (l + 1), where the numbers i− , i+ are determined uniquely by the condition that η(l, i) = η(l − 1, i− ) = η(l + 1, i+ ). These links are only between nearest cells of neighboring processors (as shown in Lemma 1 below). ˆ = (η(l, i), κ (6) By the processor P (l), for (¯j, k) ˆ(l, i)), (if it is effective) the value ˆ is tested for 0 or not at cell C(l, i, ¯j). If d(f (l, i)) = vˆ(l, i, ¯j, k) ˆ vˆ(l, i, ¯ j, k) turns out to be not equal to zero, a flag F1 is set to be 1 at ρ2 cells C(l, i, j),

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1 ≤ i, j ≤ ρ, in the subprocessor S(l, i), and kept onwards (Otherwise F1 := 0). At cell C(l, i, ¯ j) with the flag F1 = 1 all the data vˆ(l, i, j, k), w(l, ˆ ¯j, j, k), ˇ ¯ f(l, i, j, k), gˇ(l, j, j, k) are processed for updating, so that the new values vˆ(l + 1, i, j, k) and fˇ(l + 1, i, j, k) are obtained at cell C(l + 1, i, j), and the ¯ new values w(l+1, ˆ j, j, k) and gˇ(l+1, ¯j, j, k) are obtained at cell C(l+1, i+, j), respectively. Provided that F1 = 1 at cell C(l, i, ¯j), another flag F2 is set to be, e.g. 0, 1 or −1 (for controlling the updating) according to cases B, C or D. (7) The clock for synchronization and all the control data (flags F1 , F2 , the integers s(l, i), c(l, j), etc.) are communicated between subprocessors or maintained in subprocessors. In particular, we assume that our architecture is equipped with a global clock to maintain synchronization of the operations of all the cells C(l, i, j), 0 ≤ l ≤ m, 1 ≤ i, j ≤ ρ. Except for a global link for the clock information exchange among all the processors and the control data exchange among ρ2 cells of each processor, there is nothing but local links between nearest cells of the neighboring processors as shown below. (8) Timing of the above-mentioned data transmission and computation is adjusted such that manipulations of the data vˆ(l, i, j, k), etc. by ρ2 cells of the processor P (l) are done by one or two clocks later than manipulations of the corresponding data vˆ(l − 1, i, j, k), etc. having the same index k by ρ2 cells of the left-neighboring processor P (l − 1), 1 ≤ l ≤ r. The links between cells C(l − 1, i, j), 1 ≤ i, j ≤ ρ, of processor P (l − 1) and cells C(l, i, j), 1 ≤ i, j ≤ ρ, of processor P (l) are local in the sense that the following conditions are satisfied: −1 (a) The differences between positions φ−1 l−1 (i) and φl (i) are not greater than 1, 1 ≤ i ≤ ρ; (b) For ψl (k) := η(l, φl (k)) = (l − φl (k) + 1 mod ρ) +1, the difference −1 between values ψl−1 (i) and ψl−1 (i) is not greater than 1, 1 ≤ i ≤ ρ,

in view of the following lemma, which can be proved by induction. (Remark: It holds that φl (k) + ψl (k) ≡ l + 2 mod ρ, 1 ≤ k ≤ ρ, 0 ≤ l ≤ m.) Lemma 1 In case of l + k = odd, φl (k) = φl+1 (k − 1), 2 ≤ k ≤ ρ; φl (1) = φl+1(1); ψl (k) = ψl+1 (k + 1), 1 ≤ k ≤ ρ − 1; ψl (ρ) = ψl+1 (ρ). In case of l + k = even, φl (k) = φl+1(k + 1), 1 ≤ k ≤ ρ − 1; φl (ρ) = φl+1(ρ); ψl (k) = ψl+1 (k − 1), 2 ≤ k ≤ ρ; ψl (1) = ψl+1 (1). Finally, we can give a complete scheduling of the BMS computations on the above architecture, where a scheduling is a mapping from the set of all operations

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311

(calculations and other manipulations of relevant data) of the algorithm into the direct product N ×C of the set N (:= Z0 ) of clocks and the set C := {C(l, i, j)|0 ≤ l ≤ m, 1 ≤ i, j ≤ ρ} of all cells. A possible scheduling is given by the mappings nv , nw , nf , ng which assign the data vˆ(l, i, j, k), w(l, ˆ i, j, k), fˇ(l, i, j, k), gˇ(l, i, j, k) to the clocks ˆ (l, i) + l + k, nw (l, i, j, k) := κ ˆ(l, i) + l + k, nv (l, i, j, k) := κ nf (l, i, j, k) := l + k, ng (l, i, j, k) := l + k, respectively, of the cell C(l, i, j). It means that the value vˆ(l, i, j, k) is manipulated, i.e. calculated and/or stored to be sent to the right-neighboring nearest cell(s) C(l + 1, i, j) (and C(l + 1, ¯j, j)) at clock n = nv (l, i, j, k), and so on. (Reˇ := κ ˇ i, i, k) ˇ of fˇ (l, i) mark: For k ˇ (l, i), computation of the head coefficient f(l, ˇ In view of the properties κ is done just at clock n = l + k.) ˆ (l, i) ≤ κ ˆ (l + 1, i) ≤ κ ˆ (l, i)+ 1 and κ ˇ (l, i) ≤ κ ˇ (l + 1, i) ≤ κ ˇ (l, i)+ 1, this scheduling ensures that the set of data {ˆ v (l, i, j, k), w(l, ˆ ¯ j, j, k), fˇ(l, i, j, k), gˇ(l, ¯j, j, k)} available at cell C(l, i, j) at a certain clock n ∈ N can be available at two cells C(l + 1, i, j), C(l + 1, i+ , j) successively to obtain {ˆ v (l + 1, i, j, k), fˇ(l + 1, i, j, k)} at cell C(l + 1, i, j) and {w(l ˆ + 1, ¯ j, j, k), gˇ(l + 1, ¯ j, j, k)} at cell C(l + 1, i+ , j), at clock n + 1 ∈ N and/or at clock n + 2 under the requirement of local communication, provided that each cell has a buffer storage in addition to the register for the current data vˆ(l, i, j, k), etc. ˇ i, j, k) and gˇ(l, ¯i, j, k) can start just after all the The computations of f(l, input data have been fed into the 0-th processor and the controlling data such as the flags F1 , F2 , and the integers s(l, i), c(l, i), etc. have been fixed at each subprocessor S(l, i) as a result of the computations of vˆ(l, i, j, k) and w(l, ˆ ¯i, j, k). The integer values F1 , F2 , s(l, i), c(l, i), etc. control the finite field computations of fˇ(l, i, j, k) and gˇ(l, ¯i, j, k) during the whole process of the algorithm. Consequently, the total number of clocks required is about 3m. Thus, the time complexity is O(m) and the space complexity as the number of cells is O(ρ2 m). K¨ otter [9] gave a parallel implementation of a fast decoding algorithm of onepoint AG codes. In his architecture, he uses a set of ρ feedback shift registers each with m storage elements. That scheme has time complexity of O(m2 ) and space complexity of O(ρm). Since ρ is usually much less than m, our scheme is better than his in regard to the total complexity as well as in regard to time 1 complexity. For instance, in case of Hermitian codes where O(ρ) = O(m 3 ), our 5 time and space complexities O(m), O(m 3 ) can be compared with his O(m2 ), 4 O(m 3 ).

5

Concluding Remarks

In this paper we have proposed a special kind of systolic array architecture composed of three-dimensional arrangement of processing units called cells for implementing fast decoding of one-point AG codes and presented a method of scheduling of all the operations done by cells, where each cell executes a

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Shojiro Sakata and Masazumi Kurihara

combination of multiplication and addition over the finite field (symbol field) by itself and transmits the result to his nearest-neighboring cells at each clock. This architecture not only satisfies restriction to local communication which is required usually for any efficient systolic array architecture, but also has linear time complexity. We omit discussions about incorporating majority logic scheme and some other subtle aspects for decoding into our parallel architecture.

References 1. S. Sakata, J. Junsten, Y. Madelung, H. Elbrønd Jensen, and T. Høholdt, “A fast decoding method of AG codes from Miura-Kamiya curves Cab up to half the FengRao bound,” Finite Fields and Their Applications, Vol.1, No.6, pp.83–101, 1995. 2. S. Sakata, J. Junsten, Y. Madelung, H. Elbrønd Jensen, and T. Høholdt, “Fast decoding of algebraic geometric codes up to the designed distance,” IEEE Transactions on Information Theory, Vol.41, No.6, Part I, pp.1672–1677, 1995. 3. S. Sakata, H. Elbrønd Jensen, and T. Høholdt, “Generalized Berlekamp-Massey decoding of algebraic geometric codes up to half the Feng-Rao bound,” IEEE Transactions on Information Theory, Vol.41, No.6, Part I, pp.1762–1768, 1995. 4. S. Sakata, “A vector version of the BMS algorithm for implementing fast erasureand-error decoding of any one-point AG-codes,” (Eds., T. Mora, H. Mattson) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: Proc. AAECC12, Toulouse, France, June 1997, Lecture Notes in Computer Science, Vol. 1255, pp.291–310, Springer Verlag. 5. S. Sakata, “Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array,” J. Symbolic Computation, Vol.5, pp.321–337, 1988. 6. S. Sakata, “Extension of the Berlekamp-Massey algorithm to N dimensions,” Information and Computation, Vol.84, pp.207–239, 1990 7. M. Kurihara and S. Sakata, “A fast parallel decoding algorithm for one-point AGcodes with a systolic array architecture,” presented at the 1995 IEEE International Symposium on Information Theory, Whistler, Canada, September 1995. 8. S. Sakata and M. Kurihara, “A systolic array architecture for implementing a fast parallel decoding algorithm of one-point AG-codes,” presented at the 1997 IEEE International Symposium on Information Theory, Ulm, Germany, June 1997. 9. R. K¨ otter, “A fast parallel implementation of a Berlekamp-Massey algorithm for algebraic geometric codes,” Link¨ oping, Sweden, January 1995; Dissertation, Link¨ oping University, §3, pp.721–737, 1996. 10. R. K¨ otter, “Fast generalized minimum distance decoding of algebraic geometric and Reed Solomon codes,” Link¨ oping, Sweden, August 1993; IEEE Transactions on Information Theory, Vol.42, No.3, pp.721–737, 1996. 11. R. Blahut, Theory and Practice of Error Correcting Codes, Reading, MA: Addison Wesley, 1983. 12. S. Sakata and M. Kurihara, “A fast parallel implementation of the BerlekampMassey algorithm with a 1D systolic array architecture,” (Eds., G. Cohen, M. Giusti, T. Mora) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: Proc. AAECC-11, Paris, July 1995, Lecture Notes in Computer Science, Vol. 948, pp.415–426, Springer Verlag.

A Systolic Array Architecture for Fast Decoding of One-Point AG Codes

Appendix Table 1: Values o¯(i) (j, k) for a code.

k 0 1 2 3 4 5 6

i 1 2 3 4 j 1 2 3 4 1 2 3 4 1 2 3 4 1 2 31 0 5 10 15 5 10 15 20 10 15 20 4 9 14 19 9 14 19 14 19 8 13 18 23 13 18 23 18 23 12 17 22 17 22 22 16 21 21 20

15 20 19 23

Table 2: Functions η(l, i), κ(l, i), κ ˇ (l, i), κ ˆ (l, i).

l 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

i

ol 1 0 1 4 1 5 2 8 1 9 2 10 3 12 1 13 2 14 3 15 4 16 1 17 2 18 3 19 4 20 1 21 2 22 3 23 4

η(l, i) κ(l, i) κ ˇ (l, i) κ ˆ (l, i) 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 − − 1 − 1 2 − 1 2 3 − 1 2 3 4 1 2 3

−−0 −−1 −−0 −−2 −−1 1 −0 −−3 −−2 1 −1 2 1 0 −−4 −−3 1 −2 2 1 1 3 2 5 −−4 1 −3 2 1 2

− − 0 − 1 0 − 2 1 0 − 3 2 1 0 4 3 2

−−0 −−1 −−1 −−2 −−2 0 −2 −−3 −−3 1 −3 0 0 3 −−4 −−4 2 −4 1 1 4 0 0 5 −−5 3 −5 2 2 5

− − 0 − 1 1 − 2 2 2 − 3 3 3 4 4 4 4

−−0 −−1 −−1 −−2 −−2 0 −2 −−3 −−3 1 −3 1 0 3 −−4 −−4 2 −4 2 1 4 3 2 5 −−5 3 −5 3 2 5

− − 1 − 2 2 − 3 3 3 − 4 4 4 4 5 5 5

−− −− −− −− −− 2 − −− −− 3 − 3 3 −− −− 4 − 4 4 4 4 −− 5 − 5 5

313

Computing Weight Distributions of Convolutional Codes via Shift Register Synthesis Mehul Motani and Chris Heegard
School of Electrical Engineering, Cornell University Ithaca, NY 14853 USA [email protected], [email protected]

Abstract. Weight distributions of convolutional codes are important because they permit computation of bounds on the error performance. In this paper, we present a novel approach to computing the complete weight distribution function (WDF) of a convolutional code. We compute the weight distribution series using the generalized Viterbi Algorithm (GVA) and then find the minimum linear recursion relation in this series using the shift register synthesis algorithm (SRSA). The WDF follows from the minimum recursion. In order to generalize the use of the SRSA over certain commutative rings, we prove the key result that the set of finite recursions forms a principal ideal.

1

Introduction

Weight distributions of convolutional codes are important because they can be used to compute error performance bounds for the code. Viterbi and Omura [2] derive upper bounds for the probability of word and bit errors for a linear code in terms of the Hamming weights of all error events. Traditionally, the weight distribution function is computed using signal flow graph analysis and Mason’s gain formula [3]. Other methods have been presented by Fitzpatrick and Norton [5], Onyszchuk [6], and McEliece [7]. In [8], we presented an algorithm to compute univariate weight distribution functions. This paper extends the technique to compute multivariate weight distribution functions. Our method is to use the Viterbi algorithm to generate a recursive array and use the BerlekampMassey algorithm (BMA) to find the minimum recursion in the array. Massey showed [1] that the BMA synthesizes the minimum length shift register capable of generating the recursive array and so we will refer to it as the shift register synthesis algorithm (SRSA). The weight distribution function follows easily from the minimum recursion. We introduce the key steps of the problem by an example. We will compute the weight distribution function of the code [1 +D + D2 1 + D2 ], whose encoder and state diagram are shown in Fig. 1. We are interested in enumerating the error events for the code, i.e. those paths in the state diagram which deviate from the zero-state and remerge with


Supported in part by NSF grant # CCR-9805885.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 314–323, 1999. c Springer-Verlag Berlin Heidelberg 1999


Computing Weight Distributions of Convolutional Codes

315

0/00

v1

00 1/11

u

0/11 1/00

D

D

01

10 0/10

1/01

v2

0/01 11

1/10

Fig. 1. Encoder and State Diagram for code with G(D) = [1 + D + D2

1 + D2 ].

the zero-state exactly once. It can be seen by inspection that the free distance of this code is 5 and that there is exactly 1 error event of output weight 5. There are 2 events of weight 6, 4 of weight 7, 8 of weight 8, and so on. The number of events forms the sequence S = {1, 2, 4, 8, · · · }. By inspection, the sequence satisfies the recursion: S0 = 1, Sn = 2Sn−1 for n ≥ 1. Associating the number of events of weight n with the coefficient of X n , we get the weight distribution series for the code as X 5 + 2X 6 + 4X 7 + 8X 8 + · · · = X 5 (1 + 2X + 4X 2 + · · · ) = X 5

∞


Si X i .

i=0

To get the weight distribution function, consider the sum ∞
i=0

Si X i = S0 +

∞


2Si−1 X i = S0 + 2X

i=1

∞


Si−1 X i−1 = S0 + 2X

i=1

∞


Si X i .

i=0

Substituting S0 = 1 and solving, we get the output weight distribution function X5

∞
i=0

Si X i =

X5 . 1 − 2X

In the simple example above, we were able to easily generate the recursive series and identify the recursion. It is not so easy in general. The next section defines weight distribution functions. Section 3 explains some of the theory of linear recursion relations and proves the key result that the set of finite recursions is a principal ideal. Section 4 indicates how to generalize shift register synthesis to certain rings. Section 5 describes the generalized Viterbi Algorithm (GVA). Sections 6 and 7 describe how to generate the weight distribution series and compute the weight distribution function respectively. The final section makes some concluding remarks.

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Mehul Motani and Chris Heegard

Weight Distributions

We first introduce some notation. Usually, F will denote a field and R a commutative ring with identity. R[X1 , · · · , Xn ] and R[[X1 , · · · , Xn ]] are respectively the polynomial and power series rings in n variables. We denote the set of integers as Z and the set of rational numbers as Q. An error event is a nonzero finite weight codeword for which the encoder starts in the zero state, departs, and returns to the zero state exactly once. In addition, we will call two error events equal if they are shifts of each other. The weight distribution of a convolutional code is defined as the weight distribution of the set of error events. The output weight distribution series (OWDS) of a code is a power series in Z[[X]], where the coefficient of X n , an , indicates the number of error events of weight n. A recursion relation can be found in this series and it can expressed as a rational function, called the output weight distribution function (OWDF): T (X) = a1 X 1 + a2 X 2 + · · · + an X n + · · · =

Ω(X) , Λ(X)

(1)

where Λ(X), Ω(X) are polynomials over Z with deg Ω(X) < deg Λ(X). The input-output weight distribution series (IOWDS) is a multivariate power series in Z[[X, Y ]], where the coefficient of X n Y k , an,k , indicates the number of error events of weight n which correspond to input sequences of weight k. The rational function representation is called the input-output weight distribution function (IOWDF): T (X, Y ) =

∞
∞
n=1 k=1

an,k X n Y k =

Ω(X, Y ) . Λ(X, Y )

(2)

Although we assumed the existence of an encoder for the convolutional code, the OWDS and OWDF are independent of the encoder and depend only on the structure of the code. Since the IOWDS and IOWDF indicate the input sequence associated with the output codeword, they do depend on the choice of encoder. Traditionally, the WDF is computed by signal flow graph analysis and Mason’s gain formula [3]. From [3], we see that the denominator of the WDF is a polynomial with constant term one. In the following section, we will see that this implies that the sequence of coefficients of the WDS is a shift register sequence.

3

Recursion Relations

d Let R be a ring. Let f(X) = i=0 fi X i be an element of R[X], where fi may be equal to zero for any i (e.g. f(X) = 1 − X − X 2 − 0X 3 ). The sequence {Si : i = 0, 1, 2, . . . , } with entries from R is said to satisfy the linear recursion relation specified by f if d
i=0

fi Sr−i = 0 ∀ r ≥ d.

(3)

Computing Weight Distributions of Convolutional Codes

317

P f1

f2

-

Sd;1

fd

-

-

Sd;2

...

S0

Fig. 2. Shift Register defined by f(X) = 1 +

d

i=1

fi X i .

We also call f a recursion for the sequence {Si }. Define a polynomial with constant term one to be motonic. If f ∈ R[X] is motonic and {Si } satisfies the recursion specified by f, then (3) implies Sr =

d


−fi Sr−i

∀ r ≥ d.

(4)

i=1

This means {Si } is a shift register sequence and it can be generated by the shift register shown in Fig. 2. of the We can also think of the sequence {Si : i = 0, 1, 2, . . . , } as an element ∞ power series ring R[[X]] by associating with it the series S(X) = i=0 Si X i . The series S(X) is often called the generating function associated with the sequence {Si }. The generating function has a nice form when S(X) comes from a shift register sequence. Suppose the sequence {Si } is a shift register sequence which satd isfies the recursion specified by the motonic polynomial f(X) = 1 + i=1 fi X i . Then by a similar derivation as in [9], we can write S(X) =

∞
i=0

Si X i =

d−1 i=0

d−i−1 fi X i k=0 Sk X k . f(X)

(5)

Note that the numerator depends on the d initial states of the shift register. The denominator, f(X), specifies the recursion and is independent of the initial conditions. If f(X) is the minimum degree polynomial which specifies the recursion, it is called the characteristic function of the series. Suppose S ∈ R[[X1 , . . . , Xn ]] and there exist f and g in R[X1 , . . . , Xn ] such that S = fg . Then f is called a finite recursion. We make the distinction between an arbitrary recursion and a finite recursion because, unlike one dimensional recursions, not all multidimensional recursions are finite. The next theorem shows that for certain commutative rings, the set of finite recursions forms a singly generated ideal in R[X1 , . . . , Xn ].

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Theorem 1. Let R be a commutative Noetherian Unique Factorization Domain (UFD). Let K = R[X1 , . . . , Xn ], and L = R[[X1 , . . . , Xn ]]. Let S ∈ L and let I = {h ∈ K : Sh = g for some g ∈ K}. Then I is a principal (singly generated) ideal of K. Proof. The basic idea behind the proof is a follows. It is clear that I is an ideal of K. Next we show that any two elements of the ideal are multiples of some element in the ideal. By the Hilbert Basis Theorem, K is finitely generated. The result then follows by collapsing the finite basis to a single generator. Since 0S = 0, I = ∅. Suppose h1 , h2 ∈ I, a ∈ K, Sh1 = g1 , and Sh2 = g2 . Then S(h1 + h2 ) = Sh1 + Sh2 = g1 + g2 ∈ K. Also S(h1 a) = (Sh1 )a = g1 a ∈ K. So I is an ideal of K. Next, let d1 = gcd(h1 , g1 ) and d2 = gcd(h2 , g2 ). Then ∃ h1 , h2 , g1 , g2 ∈ K suchthat Sh1 = g1 Sd1h1 = d1 g1 Sh1 = g1 ⇒ ⇒   Sh2 = g2 Sd2h2 = d2 g2 Sh2 = g2    where gcd(h1 , g1 ) = 1 and gcd(h2 , g2 ) = 1. Note that the above equations imply that h1 , h2 ∈ I.   Now, let d= gcd(h1 , h2 ). Then  ∃ h1 , h2 ∈ K such that     Sh1 = g1 Sdh1 = g1 =⇒ =⇒ g1 h2 = g2 h1 Sh2 = g2 Sdh2 = g2 Recall that sinceR is a UFD, K is a UFD. gcd(h1 , g1 ) = 1 gcd(h1 , g1 ) = 1 h |h ⇒ ⇒ 1 2     gcd(h2 , g2 ) = 1 gcd(h2 , g2 ) = 1 h2 |h1 So h1 = uh2 where u ∈ K is a unit.  h1 = d1 h1 = d1 uh2 where h2 ∈ I. Then h1 = uh2 ⇒ h1 = uh2 ⇒ h2 = d2 h2 Finally, recall that since R is Noetherian, the Hilbert Basis Theorem implies K is Noetherian. So there exists a finite generating set for I which can be collapsed into a single generator by the above argument. So I is a principal ideal of K. 

4

Generalized Shift Register Synthesis

The Shift Register Synthesis Algorithm (SRSA), as described in [1], takes an input sequence, whose elements come from a field F , and generates the shift register of minimum length, L, which generates the sequence. The characteristic function of the shift register is a polynomial with coefficients from F , degree ≤ L and constant term one. In other words, the SRSA finds the minimum degree motonic polynomial in F [X] which specifies the recursion. Recalling that the set of recursions forms a principal ideal in F [X], it is easy to see that the SRSA solution is also the generator of the ideal. We want to generalize the SRSA and apply it to a sequence whose elements lie in a ring. For example, if we input an integer sequence, which has an underlying

Computing Weight Distributions of Convolutional Codes

319

motonic integer recursion, it turns out that the SRSA yields that motonic integer recursion. In this section, we show why. Recall that while Z is not a field, it can be embedded in its field of fractions, Q. In this case, we can consider the integer sequence as a rational sequence, and so the SRSA should yield the minimum motonic rational recursion. So we need to show that the minimum motonic rational recursion lies in Z[X]. Let S ∈ Z[[X]] ⊂ Q[[X]]. Let I = {h ∈ Z[X] : hS ∈ Z[X]} and J = {h ∈ Q[X] : hS ∈ Q[X]}. By Theorem 1, I and J are singly generated ideals. Let I = f and J = g for some f ∈ I and g ∈ J . Since Z[X] ⊂ Q[X], I ⊂ J . Figure 3 depicts the various inclusions.

Q [X ] J

Z[X ]

I f




g




Fig. 3. Z[X] ⊂ Q[X], I ⊂ J .

A simple argument shows that any generator of I also generates J : J = g ⇒ J = ug for any unit u ∈ Q[X]. By clearing the denominators, it is possible to choose u such that ug ∈ Z[X]. So ugS ∈ Z[X] which implies ug ∈ I. This means ug is a Z[X]-multiple of f and so J = f. Solution via Mason’s gain formula [3] means there exists a motonic polynomial in I. This implies that there exists a motonic generator of I, which, by the preceding argument, generates J . In a singly generated ideal, generators are unique up to multiplication by a unit, and so the motonic generator, if it exists, is unique. Thus the motonic generator of J lies in Z[X] and this is precisely what the SRSA computes. The discussion above is easily generalized by replacing Z by a Noetherian UFD R and Q by the field of fractions of R.

5

The Generalized Viterbi Algorithm (GVA)

A trellis for a convolutional encoder is an extension in time of the encoder’s state diagram. Thus a trellis is a directed graph consisting of vertices, or states, interconnected by edges, or branches. In a recent paper, McEliece [4] presented a generalized version of the Viterbi Algorithm which operates on a trellis in which the branch labels come from a semiring. Examples of semirings include

320

Mehul Motani and Chris Heegard 1

1

00

00 2

x

x 01

2

x 01

1

01

. . .

x 10

x x x

2

1

x 10

11

00 2

x

10

x x 11

x

11

Fig. 4. Trellis for code with G(D) = [1 + D + D2

1 + D2 ].

the non-positive real numbers and the familiar ring of polynomials, Z[X]. Figure 4 depicts a trellis section of an encoder for the convolutional code which has a generator matrix given by G(D) = [1 + D + D2 1 + D2 ]. As previously mentioned, the branch labels come from the semiring. A path is a sequence of branches and the path metric of a path P is the product (“·”) of the labels of the branches in the path, taken in order. The flow between two states, S1 and S2 , is defined as the sum (“+”) of the path metrics of all paths starting at S1 and ending at S2 . For example, when we use the semiring of polynomials over Z described above, the flow between two states is the weight enumerator polynomial that counts the number of paths of each possible weight between the states. In this context, the generalized Viterbi Algorithm is an efficient algorithm for computing the flow between two states [4].

6

Using the GVA to Compute the WDS

We can use the GVA to enumerate the paths which diverge immediately from the zero state and remerge with the zero state exactly once. Let the semiring be Z[X] with the usual polynomial multiplication and addition. We label each branch in the trellis with a monomial X n , where the exponent n corresponds to the Hamming weight of the output associated with the branch. Now all error events correspond to paths which leave and end with the zero state once. Now consider a path, p1 , which stays at the zero state for a finite time, then diverges and remerges with the zero state. Then p1 corresponds to the same error event as the path, p2 , which immediately diverges from the zero state, imitates p1 and remerges with the zero state. We can modify the original trellis in the following manner to prevent overcounting paths corresponding to the same error event. In the original trellis, remove all nonzero branches which diverge from the zero state except the initial one. In addition we break the self-loop at the zero state by removing the initial branch connecting the zero states. Note that a branch can be effectively removed by setting the branch label to zero. Figure 5 shows the modified trellis for the trellis of Fig. 4. Now we apply the GVA to the modified trellis, setting the initial flow to 1 for the zero state and 0 for all nonzero states. After each iteration, the flow at the zero state consists of terms in the WDS. It will soon be clear that the lower order

Computing Weight Distributions of Convolutional Codes 0 00

1 00

2

x

x 01

2

2

x

10

11

x

0

11

10

x

0 01

10

x

. . .

x

x x 11

01

1

x

x x

00 2

x

1

x

x x

2

01

1

x

1 00

x

0 01

1

10

1 00

321

10

x x 11

x

11

Fig. 5. Modified Trellis for code with G(D) = [1 + D + D2

1 + D2 ].

terms will stabilize after a certain number of iterations. So after every iteration the stabilized terms form a truncated version of the WDS. Let us apply the GVA to the convolutional code of Fig. 5. Table 1 shows the computations. After seven and eight iterations of the GVA, we see that the partial WDS are x5 + 2x6 + 4x7 + 4x8 + x9 and x5 + 2x6 + 4x7 + 7x8 + 5x9 + x10 respectively. Note that the first three terms remain unchanged after the seventh iteration. From Fig. 5, we see that changes to the flow at the 00 state depend on the flow at all other states. After the seventh iteration, the minimum degree term at the 10 state is 3x6 . Since any path connecting 10 to 00 has path metric at least x2 , it can affect only the coefficients of the terms of degree eight and higher. The minimum degree term at 01 is x5 . Since any path from 01 to 00 has path metric at least x3 , it also can affect only the coefficients of the terms of degree eight and higher. A similar argument for 11 proves that the terms of degree seven and lower in the WDS will remain unchanged after every iteration thereafter. The above paragraphs describe how to compute the OWDS. To compute the IOWDS, we simply label each branch with a monomial from Z[X, Y ], i.e. X n Y k , where n and k correspond to the Hamming weights of the output and input respectively associated with the branch. Table 1. GVA Computations for G(D) = [1 + D + D2 State 0 00 1 01 0 10 0 11 0

1 + D2 ].

Iteration 3 4 5 6 x5 x5 + x6 x5 + 2x6 + x7 x5 + 2x6 + 3x7 + x8 x3 x4 x4 + x5 2x5 + x6 4 4 5 5 6 5 x x +x 2x + x x + 3x6 + x7 x4 x4 + x5 2x5 + x6 x5 + 3x6 + x7 7 8 00 x5 + 2x6 + 4x7 + 4x8 + x9 x5 + 2x6 + 4x7 + 7x8 + 5x9 + x10 01 x5 + 3x6 + x7 3x6 + 4x7 + x8 6 7 8 6 10 3x + 4x + x x + 6x7 + 5x8 + x9 11 3x6 + 4x7 + x8 x6 + 6x7 + 5x8 + x9 1 0 x2 0 0

2 0 0 x3 x3

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Mehul Motani and Chris Heegard

Using the SRSA to Compute the WDF

We have seen that the stable terms of the GVA computations yield a truncated version of the WDS. From this partial WDS, we want to find the recursion relation and thus compute the WDF whose series expansion yields the entire WDS. It can be shown that this is equivalent to solving the Key Equation. The Euclidean Algorithm can be used to solve the Key Equation but the SRSA provides an algorithmically appealing alternative. The SRSA synthesizes the minimum length linear feedback shift register (LFSR) which generates the WDS coefficients. Theorem 1 in [1] allows one to compute an upper bound on the number, M, of required stable terms of the WDS based on a bound on the Xdegree, D, of the denominator of the WDF. Letting A be the weighted adjacency matrix of the state diagram of the code, it can be shown that
D≤ max {deg(I − A)ij } and M ≤ 2D. i

j

We assume we have a non-catastrophic encoder (reasonable since catastrophic encoders are not very interesting). A non-catastrophic encoder for the code implies that the state diagram has no zero weight loops except for the self-loop at the zero state. This means there are a finite number of error events with a given Hamming weight. Let us first compute the OWDF (as in [8]). The non-catastrophic encoder assures that the OWDS, S(X), is an element of Z[[X]]. Theorem 1 implies that the set of recursions is a principal ideal and the SRSA can be used to find the generator of that ideal, which is the minimum motonic recursion in Z[X]. us compute the IOWDF. It is clear that the IOWDS, S(X, Y ) =  Now let i j i,j Sij X Y , is an element of Z[[X, Y ]]. The IOWDS can also be represented by an infinite two dimensional array over Z, and computing the IOWDF is equivalent to finding the “minimum” two-dimensional recursion in this array. Although it is not clear what “minimum” means for recursions in multiple variables, Theorem 1 implies that the set of finite recursions is a principal ideal. The “minimum” recursion we are searching for is the generator of that ideal. The noncatastrophic encoder implies that S(X, Y ) is actually an element of Z[Y ][[X]], i.e. it is a univariate series with polynomial coefficients. In other words, the IOWDS can be represented as a sequence of polynomials in Y . The SRSA can then be used to compute the minimum motonic recursion in this sequence. The algorithm used to compute the recursion relation using the GVA and SRSA is described in pseudocode below. Algorithm 1: label and modify trellis as described while(number of stable terms < bound) run one GVA iteration if(one more stable term) call shift register synthesis algorithm end if end while

Computing Weight Distributions of Convolutional Codes

8

323

Conclusion

In this paper, we presented a novel method to compute the WDF of a convolutional code. We used the generalized Viterbi Algorithm to compute a recursive series and shift register synthesis to find the minimum recursion in this series. Areas of further research include issues related to the implementation of the algorithm. Since the algorithm requires symbolic computation in multivariate polynomial rings, an efficient implementation is needed to compute the weight distribution of large constraint length convolutional codes. We conclude by giving the OWDF for the best rate 1/2, constraint length ν = 6 convolutional code, which has generator G = [133 171]. The coefficients of the numerator Ω(X) and the denominator Λ(X) are shown below: Ω = [0 · 0 · 0 · 0 · 0 · 0 · 0 · 0 · 0 · 0 · 11 · 0 · -6 · 0 · -25 · 0 · 1 · 0 · 93 · 0 · -15 · 0 · -176 · 0 · -76 · 0 · 243 · 0 · 417 · 0 · -228 · 0 · -1156 · 0 · -49 · 0 · 2795 · 0 · 611 · 0 · -5841 · 0 · -1094 · 0 · 9575 · 0 · 1097 · 0 · -11900 · 0 · -678 · 0 · 11218 · 0 · 235 · 0 · -8068 · 0 · -18 · 0 · 4429 · 0 · -20 · 0 · -1838 · 0 · 8 · 0 · 562 · 0 · -1 · 0 · -120 · 0 · 0 · 0 · 16 · 0 · 0 · 0 · -1] Λ = [1 · 0 · -4 · 0 · -6 · 0 · -30 · 0 · 40 · 0 · 85 · 0 · -81 · 0 · -345 · 0 · 262 · 0 · 844 · 0 · -403 · 0 · -1601 · 0 · 267 · 0 · 2509 · 0 · 389 · 0 · -3064 · 0 · -2751 · 0 · 2807 · 0 · 8344 · 0 · -1960 · 0 · -16133 · 0 · 1184 · 0 · 21746 · 0 · -782 · 0 · -21403 · 0 · 561 · 0 · 15763 · 0 · -331 · 0 · -8766 · 0 · 131 · 0 · 3662 · 0 · -30 · 0 · -1123 · 0 · 3 · 0 · 240 · 0 · 0 · 0 · -32 · 0 · 0 · 0 · 2]

References 1. J. L. Massey, “Shift-Register Synthesis and BCH Decoding”, IEEE Transactions of Information Theory, Vol. 15, pp. 122-127, January 1969. 2. A. J. Viterbi and J. K. Omura, Principles of Digital Communication and Coding, New York: McGraw-Hill Book Company, 1979. 3. S. Lin and D. J. Costello, Jr., Error Control Coding: Fundamentals and Applications, Englewood Cliffs: Prentice Hall, 1983. 4. R. J. McEliece, “On the BCJR Trellis for Linear Block Codes”, IEEE Transactions of Information Theory, Vol. 42, pp. 1072-1092, July 1996. 5. P. Fitzpatrick and G.H. Norton, “Linear recurring sequences and the path weight enumerator of a convolutional code” Elect. Lett., 27 (1991), pp. 98-99. 6. I. Onyszchuk, “Finding the Complete Path and Weight of Convolutional Codes”, JPL TDA Progress Report, 42-100, 1990. 7. R. J. McEliece, “How to Compute Weight Enumerators for Convolutional Codes”, Communications and Coding, M. Darnell and B. Honary, eds., Taunton, Somerset, England: Research Studies Press Ltd., pp. 121-141, June 1998. 8. M. Motani and C. Heegard, “The Viterbi Algorithm Meets The Key Equation”, Proceedings of International Symposium on Information Theory (Boston, MA), August 1998. 9. S. Golomb, Shift Register Sequences, Revised Edition, Aegean Park Press, May, 1982.

Properties of Finite Response Input Sequences of Recursive Convolutional Codes Didier Le Ruyet1 , Hong Sun2 , and Han Vu Thien1 1

2

Conservatoire National des Arts et M´etiers, Laboratoire Signaux et Syst`emes 75141 Paris Cedex 03, France [email protected] Huazhong University of Science and Technology, Departement of Electronic and Information Engineering, 430074 Wuhan, China [email protected]

Abstract. A recursive convolutional encoder can be regarded as an infinite impulse response system over the Galois Field of order 2. First, in this paper, we introduce finite response input sequences for recursive convolutional codes that give finite weight output sequences. In practice, we often need to describe the finite response sequence with a certain Hamming weight. Then, different properties of finite response input sequences are presented. It is shown that all finite response input sequences with a certain Hamming weight can be obtained in closed-form expressions from the so-called basic sequences. These basic sequences are presented for important recursive convolutional encoders and some possible applications are given.

1

Introduction

Recursive convolutional codes have seldom been employed in the past because their weight enumerating function is equivalent to that of the non recursive convolutional codes [1]. But they have been renewed since they have been used to construct serial and parallel concatenated convolutional codes (turbo codes) whose performances are near Shannon limit (see [2] and [3]). The works of Battail et al. [4] have shown that recursive convolutional codes mimic random coding if the denominator polynomial is chosen as a primitive polynomial. In comparison with non recursive convolutional codes, the input sequences with finite weight are associated with output sequences with infinite weight, except for a fraction of finite weight input sequences which generate finite weight output sequences. These input sequences are called finite response input sequences (FRISs). In [5], FRISs have been introduced ; the enumeration of FRISs for a Hamming weight w=2 is simple but however, no practical method to enumerate these sequences with a certain Hamming weight w greater than 2 has yet been given. The goal of this paper is to study the properties of finite response input sequences with weight w and to show how these sequences can be enumerated from one or more basic FRISs. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 324–333, 1999. c Springer-Verlag Berlin Heidelberg 1999


Properties of FRIS

325

In the next section, we recall some classical definitions of convolutional codes. The third section we give different properties of FRIS and introduce basic FRIS. An example is given to show how these properties can be used to enumerate all the FRIS in closed form. Then, the basic FRISs are presented for some important recursive convolutional encoders. Finally, we will show how these properties can be used to find the Hamming weight of the output sequence of any FRIS and to build interleavers for turbo codes.

2

Review of Basics

In order to keep the following expositions self-contained, we shall introduce recursive convolutional codes and some definitions to be used later in this section. A rate 1/r recursive convolutional encoder maps the input sequence of information bits u0 , u1 , u2 , . . . into the output sequence of r-dimensional code blocks y0 , y1 , y2 , . . . with yn = (y1n , y2n , ..., yrn ) . The encoder also goes through the internal state sequence s0 , s1 , s2 , ..., where each encoder state sn at time n is a M-tuple : sn = [s1n , s2n , ...sM n] . M is the number of delay cells of the encoder and sin is the state at time n of the i-th delay cell. The structure of a recursive systematic convolutional encoder of rate 1/2 is shown in Fig.1. A recursive encoder can also be regarded as an infinite impulse response (IIR) system over the finite field GF(2) with input u(D) and output y(D), where D is the unit-delay operator: y(D) = u(D)G(D) with G(D) =




P1 (D) P2 (D) Pr (D) , , ..., Q(D) Q(D) Q(D)

(1)

 and y(D) = (y1 (D), y2 (D), ..., yr (D)).

326

Didier Le Ruyet, Hong Sun, and Han Vu Thien

y1 n

un

D

p0

s1n

D

qM - 1

q2

q1

s2 n

p1

D

p2

qM

s Mn

pM - 1

pM

y2n Fig. 1. The structure of a recursive systematic convolutional encoder of rate 1/2. where Q(D) is a primitive polynomial of degree M: Q(D) = q0 + q1 D + ... + qM DM and Pi (D) is a polynomial of degree at most equal to M: Pi (D) = p0i + p1i D + ... + pM i DM . When the recursive convolutional encoder is systematic, we have y1n = un since P1 (D) = Q(D). Since Q(D) is a primitive polynomial, the encoder generates a pseudo noise (PN) sequence or a maximum length sequence. The period of the PN sequence is 2M − 1. The weight of the output sequence for one period of the PN sequence is 2M −1 [6]. An example of state diagram is shown in Fig.2 for the primitive polynomial Q(D) = 1 + D + D3 . Each edge is labeled by xwI y wO where wI and wO are respectively the weight of the corresponding input and output bit. As the edge drawn in dotted line corresponds to an input bit equal to 0, we can clearly observe the loop corresponding to the PN sequence of period 7 and that the output weight of the PN sequence is equal to 4. We say that the encoder with Q(D) is IIR, since the weight-one input sequence (impulse input) produces an infinite response, i.e. an infinite weight output sequence. Definition 1. A finite response input sequence (FRIS) is an input sequence whose first ”1” causes the encoder state to leave the zero state S0 = [0, 0, ..., 0] at time n0 and whose last ”1” brings it back to S0 at time n0 + L − 1 (L > 0). A FRIS will produce a finite weight output sequence. These FRISs are represented by F (D).

Properties of FRIS [110 ]

y

327

x

1 [111 ] [100 ] S 1 [ 000 ] SO

y

x xy

xy

xy 1

[ 011 ]

x

xy 1

S*

y

[ 001 ] x

y

[101 ] 1

[ 010 ]

Fig. 2. The state diagram for a primitive polynomial Q(D) = 1 + D + D3 .

3

Properties of Finite Response Input Sequences (FRIS)

We have the following theorems about F (D). Theorem 1. A FRIS of a recursive convolutional encoder satisfies the equation: F (D) ≡ 0

(mod Q(D))

(mod 2) .

(2)

Proof. From (1), if and only if Q(D)|u(D) (mod 2) , i. e. Q(D) is a factor of u(D) over the finite field GF(2), then yi (D) becomes a finite order polynomial or a finite weight output sequence. Since Q(D) is a primitive polynomial, the encoder generates a maximum length sequence of period 2M −1 . We then have: D0 ≡ D2

M

−1

≡1

(mod Q(D))

(mod 2) .

(3)

− 1) (mod 2) .

(4)

Then, (2) becomes: F (D) ≡ 0

(mod Q(D))

(mod D2

M

−1

Theorem 2. If we have a FRIS F(D) of weight w noted F (w) (D): F (w)(D) = Dn1 + Dn2 + ... + Dnw

(5)

where n1 = 0 and n2 , ..., nw are any positive integer, then there exists a family of weight w FRISs : (w) Fm (D) = Dm0 (Dn1 +m1 (2 0 m2 ...mw

M

−1)

+ Dn2 +m2 (2

M

−1)

+ ... + Dnw +mw (2

M

−1)

) (6)

where m1 = 0 and m0 , m2 , ..., mw can be any integer, positive, negative, or zero.

328

Didier Le Ruyet, Hong Sun, and Han Vu Thien

Proof. From (4),(5) and (6), we obtain: (w) (D) ≡ Dm0 F (w)(D) ≡ 0 Fm 0 m2 ...mw

(mod Q(D))

(7)

(mod D2

M

−1

− 1)

(mod 2) .

This theorem tells us that if we find any FRIS in a family, we can deduce all the FRISs of this family. We note that there are two different kinds of FRISs called simple and complex FRISs. Definition 2. A FRIS is simple if its last ”1” solely brings back the encoder state to S0 . Otherwise, the FRIS is complex since the encoder state returns to S0 more than once. We will now choose a unique representative for each family of simple FRISs, called basic FRIS. (w)

Definition 3. F0 (D) is called a basic FRIS for weight w if and only if the following three conditions are satisfied: (w)

F0

(D)is a FRIS with the form (5) 0 < ni − ni−1 < 2

M

−1

(8)

(∀i)

(9)

nw = min .

(10)

Condition (8) means that the first ”1” of a basic FRIS should occur at time 0; condition (9) means that after rearranging n1 , n2 , ...nw in ascendant form, the duration between two consecutive “1” should be less than 2M − 1; condition (10) means that we choose as the basic FRIS the sequence with the minimal length. The basic FRISs of a recursive convolutional encoder depend only on Q(D). We call F (w)(D) which satisfies conditions (8) and (9) a secondary basic (w) FRIS FS (D). The next theorem will show how to describe all the FRISs with weight w.  Theorem 3. Supposing w = i wi (wi > 1), all the FRISs can be obtained in (w) (w ) the form (6) from F0 (D) and from combinations of F0 i (D).  In particular for w=2 and w=3, since we have no combination by w = i wi (wi > 1), each FRIS is obtained from basic FRISs according to (6). The next theorem will give us the total number of basic FRISs for each weight w. Theorem 4. For w=2, there exists only one basic FRIS: 1 + D2 For w=3, there exists  M  2 −2 basic F RISs. A33

M

−1

(11)

Properties of FRIS

For 4 ≤ w < 2M − 1, there exists   M (2 − 2)(2M − 3)w−3 − Nw Aw w

basic F RISs.

329

(12)

Nw is the number of F (w) (D) which are constructed from secondary basic FRISs (w ) FS i (D) ; Anp is the number of ordered selections of p elements from a set of n elements, and c means c rounded to the nearest integer towards plus infinity. Proof. Since Q(D) is a primitive polynomial of degree M, (s1 |s0 = S0 ) = S1 when an input “1” occurs at time 0, where S1 = [1, 0, ...0]; and then, in the absence of an input, sn goes through all possible 2M − 1 nonzero encoder states and repeats with period 2M − 1; it returns to S0 if and only if an input “1” occurs and the current state is S ∗ = [0, ..., 0, 1]. So, if we exclude the first “1” and the last “1” of this FRIS, the w − 2 other “1”s can occur under any state sni |sni = S0 , sni = S ∗ , sni = sni−1 . Note that, for the second ”1” of the FRIS, sni−1 = s0 . Therefore, there are (2M − 2)(2M − 3)w−3 different secondary basic FRISs including those that are constructed from F (wi ) (D); on the other hand, from (6) each family includes Aw w secondary basic FRISs if ni − ni−1 = nj − nj−1 (i = j) and possibly less than Aw w otherwise. As a result, we conclude that there exist ((2M − 2)(2M − 3)w−3 − Nw )/Aw w  basic FRISs. For w = 2, there is only one basic FRIS which has the first “1” corresponding to the leaving of the zero state to S1 and the other “1” for the return from M (2) s2M −1 = S ∗ to the zero state S0 , that is, F0 (D) = 1 + D2 −1 . For w = 3, N3 = 0, then there are (2M − 2)/A33  basic FRISs. Example 1. Supposing M=3 and Q(D) = 1 + D + D3 . (2) For w=2, since 2M − 1 = 7, F0 (D) = 1 + D7 . All weight-2 FRISs can be written as follows according to (6) : (2) Fm0 m2 (D) = Dm0 (1 + D7m2 ), (3) For w=3, there exists (2M − 2)/A33  = 1 basic FRIS, F0 (D) = 1 + D + D3 . All weight-3 FRISs can be written as follows according to (6) : (3) Fm0 m2 m3 (D) = Dm0 (1 + D1+7m2 + D3+7m3 ), for example, (3) F6,−1,−1 (D) = 1 + D2 + D6 , (3)

F4,0,−1(D) = 1 + D4 + D5 . (2)

For w=4, since 4 = 2+2, and F0 (D) = 1 + D7 , we have F (4)(D) which are (2) (4) combinations of secondary basic FRISs FS (D) written by F∗ (D): (4) (2) (2) F∗ (D) = F0 (D) + Dli F0 (D), li = 1, 2, ..., 6. M Clearly, here N4 = 6, ((2 − 2)(2M − 3)4−3 − N4 )/A44  = 1 and we have (4) (4) one F0 (D) that is, F0 (D) = 1 + D2 + D3 + D4 . Therefore, the following two equations describe all simple weight-4 FRISs : F (4)(D) = Dm0 (1 + D2+7m2 + D3+7m3 + D4+7m4 ),

330

Didier Le Ruyet, Hong Sun, and Han Vu Thien

F∗ (D) = Dm0 [(1 + D7m1 ) + Dli (1 + D7m2 )], where m0 , mi can be any integer and li = 1, 2, ..., 6. And the following equation describe all complex weight-4 FRISs: (4) Fcom (D) = Dm0 (1 + D7m1 + D7m2 + D7m3 ) where m0 , mi can be any integer and mi = mj (i = j). (4)

4

Tables

In this section, we will give a list of basic FRISs for recursive convolutional encoders with M=2, 3, 4 and 5. The following basic FRISs have been obtained from an exhaustive search since there is no known method to find them. Table 1. Basic FRISs for M = 2 Q(D) = 1 + D + D2 . (w)

w

F0

2 3 4

1 + D3 1 + D + D2 1 + D + D3 + D4

Table 2. Basic FRISs for M = 3 Q(D) = 1 + D + D3 . (w)

w

F0

2 3 4

1 + D7 1 + D + D3 1 + D2 + D3 + D4

Table 3. Basic FRISs for M = 4 Q(D) = 1 + D + D4 . (w)

(w)

(w)

w

F0

w

F0

w

F0

2 3 3 3

1 + D15 1 + D + D4 1 + D2 + D8 1 + D5 + D10

4 4 4 4

1 + D2 + D4 + D5 1 + D5 + D6 + D7 1 + D + D3 + D7 1 + D + D5 + D8

4 4 4

1 + D3 + D6 + D8 1 + D3 + D4 + D9 1 + D4 + D8 + D10

Properties of FRIS

331

Table 4. Basic FRISs for M = 5 Q(D) = 1 + D + D2 + D3 + D5 . (w)

(w)

(w)

w

F0

w

F0

w

F0

2 3 3 3 3 3 4 4 4 4 4 4 4 4

1 + D31 1 + D3 + D8 1 + D7 + D9 1 + D + D12 1 + D6 + D16 1 + D4 + D17 1 + D4 + D5 + D6 1 + D + D4 + D7 1 + D + D3 + D10 1 + D2 + D7 + D11 1 + D6 + D8 + D11 1 + D4 + D9 + D12 1 + D8 + D10 + D12 1 + D3 + D5 + D13

4 4 4 4 4 4 4 4 4 4 4 4 4 4

1 + D2 + D12 + D13 1 + D + D5 + D14 1 + D6 + D7 + D14 1 + D2 + D8 + D14 1 + D10 + D13 + D14 1 + D4 + D8 + D15 1 + D9 + D10 + D15 1 + D5 + D11 + D15 1 + D3 + D11 + D16 1 + D9 + D14 + D16 1 + D13 + D15 + D16 1 + D + D9 + D17 1 + D7 + D12 + D17 1 + D11 + D13 + D17

4 4 4 4 4 4 4 4 4 4 4 4 4

1 + D3 + D15 + D17 1 + D5 + D16 + D17 1 + D6 + D9 + D18 1 + D5 + D12 + D18 1 + D7 + D16 + D18 1 + D3 + D7 + D19 1 + D8 + D9 + D19 1 + D4 + D10 + D19 1 + D11 + D18 + D19 1 + D10 + D17 + D20 1 + D3 + D9 + D20 1 + D6 + D13 + D21 1 + D8 + D17 + D21

5 5.1

Examples of Application Hamming Weight of the Output Sequences of Finite Input Response Sequences

In this section, we will show how to use the properties introduced above to compute the Hamming weight of the output sequence of any FRIS. Theorem 5. Consider an arbitrary FRIS of weight w F (w) (D): F (w)(D) = Dm0 (Dn1 + Dn2 + ... + Dnw ), where n1 = 0 and ni > ni−1 , (∀i). d[F (w)(D)] denotes the Hamming weight of the output sequence. We have (w)

d[F (w)(D)] = d[FS (D)] + d[P N ]

w 

bi

(13)

i=2

(w)

where FS (D) is the secondary basic FRIS FS (D) = Dl1 + Dl1 +l2 + ... + D (w)

with



i li

,

l1 = 0 li ≡ ni − ni−1 − 1 (mod 2M − 1) + 1   (ni − ni−1 − li ) bi = . 2M − 1

(14)

332

Didier Le Ruyet, Hong Sun, and Han Vu Thien

d[P N ] is the weight of the output sequence for one period of the PN sequence. d[P N ] = 2M −1 since Q(D) is a primitive polynomial. This theorem tells us that we can calculate the Hamming weight of the output sequence of any FRIS from its associated secondary basic FRIS. We will now give a method to find the Aw w secondary basic FRISs from the basic FRIS. (w) Consider a basic FRIS F0 (D) (w)

F0

(D) = Dn1 + Dn2 + ... + Dnw ,

(15)

where n1 = 0 , ni > ni−1 (∀i). From this basic FRIS, we can deduce all the simple FRIS of the family F (w) (D) = Dm0 (Dn1 +m1 (2 l1

l2

M

−1)

+ Dn2 +m2 (2

M

−1)

+ ... + Dnw +mw (2

M

−1)

) (16)

lw

= D + D + ... + D , where m1 = 0 , li = m0 + ni + mi (2M − 1)(∀i) . All the secondary basic FRISs can be obtained from the basic FRIS by permutation of n1 , n2 , ..., nw and then searching m0 , m2 , ..., mw to satisfy the inequality l1 < l2 < ... < lw and l1 = 0 , li − li−1 < 2M − 1. Example 2. Supposing M=3, w=3 and Q(D) = 1 + D + D3 . There is only one basic FRIS : (3) F0 (D) = D0 + D + D3 = 1 + D + D3 . We have (3) FS1 (D) = D0 + D3 + D1+7 = 1 + D3 + D8 with m0 =0,m2 =0, m3 =1 . (3) FS2 (D) = D−1 (D1 + D3 + D0+7 ) = 1 + D2 + D6 with m0 =-1,m2=0,m3 =1 . ... 5.2

Interleaver Construction for Turbo Codes

Turbo codes are a parallel concatenation of recursive systematic convolutional codes [2]. The turbo encoder consists of two recursive convolutional codes and an interleaver of size N. An example of a turbo encoder is shown in Fig.3. The N bits information sequence u(D) is encoded twice : firstly by C1 and secondly after interleaving by C2. A tail sequence composed of M bits is added after the information sequence in order to bring the internal state of the first encoder to the zero state. As a consequence only FRISs are allowed. So we can use the properties of FRISs for the construction of the interleaver. The interleaver should improve the weight distribution and the free distance of turbo codes. An optimal interleaver should map the input sequences u(D) which generate low weight output sequences y1 (D) with sequences v(D) which generate high weight output sequence y2 (D) and vice versa. For the construction of the interleaver, we can take into account only the input sequences u(D) which generate low weight output sequences. These sequences can be enumerated using the properties of FRISs introduced above. The

Properties of FRIS

333

u( D )

u( D)

C1

y 1( D )

C2

y 2( D )

E

v( D)

Fig. 3. The structure of a turbo encoder of rate 1/3. weight of the associated output sequence y1 (D) is calculated by using (13). The weight of the output sequence y2 (D) can also be obtained using a generalization of this principle. In [7], we have shown that these properties combined with a tree research method for construction of the interleaver can produce very good interleavers.

6

Conclusion

The finite response input sequences (FRISs) for a recursive convolutional encoder with a primitive polynomial can be defined by (4). In this paper, new practical properties of FRISs with a certain Hamming weight w are presented. We have introduced the basic FRIS and shown that we could write all FRISs with weight w in closed-form expressions from these basic FRISs. These properties can be employed in many applications, such as the computing of the weight enumerators of these codes and the construction of efficient interleavers for turbo codes.

References 1. Forney, G. D.: Convolutional codes I: Algebraic structure. IEEE Trans. Inform. Theory IT16 (1970) 720–738 2. Berrou, C., Glavieux, A.,Thitimajshima, P.: Near Shannon limit error correcting coding and decoding : Turbo-codes. Proc. of Int. Conf. on Comm., Geneva, Switzeland (1993) 1064–1070 3. Benedetto, S., Divsalar, D., Montorsi, G., Pollara, F.: Serial concatenation of interleaved codes : performance analysis, design and iterative decoding. IEEE Trans. Inform. Theory IT44 (1998) 909–926 4. Battail, C., Berrou, C.,Glavieux, A.: Pseudo-random recursive convolutional coding for near-capacity performance. Proc. of GLOBECOM’93, Houston, Texas, USA (1993) 23–27 5. Podemski, R.,Holubowicz, W.,Berrou, C.,Battail, G.: Hamming distance spectra of turbo-codes. Ann.Telecommun. 50 (1995) 790–797 6. Golomb, S. W.: Shift register sequences. revised version Aegean Park, Laguna Hills, CA (1982) 7. Le Ruyet, D., Sun, H., Vu Thien, H.: New method for the construction of interleavers. submit to Int. Symp. on Info. Theory, Sorrento, Italia (2000)

Lower Bounds for Group Covering Designs K.K.P. Chanduka, Mahesh C. Bhandari, and Arbind K. Lal Department of Mathematics Indian Institute of Technology, Kanpur, 208 016 India {mcb,arlal}@iitk.ac.in, [email protected]

Abstract. A group covering design (GCD) is a set of mn points in n disjoint groups of size m and a collection of b k−subsets, called blocks, such that every pairset not contained in the same group occurs in at least one block. For m = 1, a GCD is a covering design [5]. Particular cases of GCD’s, namely transversal covers, covering arrays, Sperner systems etc. have been extensively studied by Poljak and Tuza [22], Sloane [24], Stevens et al. [26] and others. Cohen et al. [8], [9] and Sloane [24] have also shown applications of these designs to software testing, switching networks etc.. Determining the group covering number, the minimum value of b, for given k, m and n, in general is a hard combinatorial problem. This paper determines a lower bound for b, analogous to Sch¨ onheim lower bound for covering designs [23]. It is shown that there exist two classes of GCD’s (Theorems 15 and 18) which meet these bound. Moreover, a construction of a minimum GCD from a covering design meeting the Sch¨ onheim lower bound is given. The lower bound is further improved by one for three different classes of GCD’s. In addition, construction of group divisible designs with consecutive block sizes (Theorems 20 and 21) using properties of GCD’s are given.

1

Introduction

Let K and M be sets of positive integers and let λ be a positive integer. A triple (X, G, B) is a group divisible design (GDD), denoted GD[K, λ, M; v], if (i) X is a finite set of v elements (points); (ii) G = {G1 , G2 , . . . , Gn }, n > 1, is a partition of X with |Gi | ∈ M. The elements of G are called groups; (iii) B is a collection of subsets, called blocks, of X such that |B| ∈ K and |B ∩ G| ≤ 1 for every B ∈ B and G ∈ G; (iv) every pairset {x, y} ⊂ X such that x and y belong to distinct groups is contained in exactly λ blocks. Observe that a GD[K, λ, {1}; v] is a pairwise balanced design, denoted P BD[K, λ; v]. If K = {k} and M = {m} then GD[K, λ, M; v] is often denoted by GD[k, λ, m; v]. A GD[k, λ, m; km] is called a transversal design and is denoted by T D[k, λ; m], while a GD[k, λ, 1; v] is known as a balanced incomplete block design, denoted B[k, λ; v]. If λ = 1 one usually represents GD[k, λ, m; v], T D[k, λ; m], respectively by GD[k, m; v], T D[k; m]. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 334–345, 1999. c Springer-Verlag Berlin Heidelberg 1999


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A pair (X, B) is said to be a (v, k, λ)−covering design, denoted AD[k, λ; v], if |X| = v and B is a collection of k−subsets of X such that every distinct pair
of elements of X is contained in at least λ blocks of B. Let C(k, λ; v) = min |B| :  (X, B) is an AD[k, λ; v] is called the covering number. In [23], Sch¨ onheim has shown that    v λ(v − 1) ≡ α(k, λ; v) (1) C(k, λ; v) ≥ k k−1 where x is the least integer such that x ≤ x. This has been further sharpened by Hanani[14] in the following theorem. Theorem 1. [14] Let (X, B) be a covering design AD[k, λ; v]. If λ(v − 1) ≡ 0 (mod k − 1) and λv(v − 1)/(k − 1) ≡ −1 (mod k) then C(k, λ; v) ≥ α(k, λ; v) + 1. In [4], Caro and Raphael has shown that if λ = 1 and the conditions of Theorem 1 are not satisfied then equality in (1) is attained for v ≥ v0 (k).    v v−1 Theorem 2. [4] For v ≥ v0 (k), C(k, 1; v) = k k−1 ≡ α(k, λ; v), unless k − 1 is even, (v − 1) ≡ 0 (mod k − 1) and v(v − 1)/(k − 1) ≡ −1 (mod k) in which case C(k, 1; v) = α(k, 1; v) + 1. Recently, the authors [2], Honkala [15] and Zhang [28] have used covering designs to improve lower bounds on binary covering codes. An attempt to extend the counting arguments of covering designs to study q−ary covering codes for q ≥ 3 has been made by Chanduka [5], Chen and Honkala [7]. It motivates a study of a new combinatorial design, called group covering design, which generalizes the concepts of covering designs and group divisible designs. Particular cases of group covering designs, namely, transversal covers, covering arrays, Sperner systems etc. have been extensively studied by Poljak and Tuza [22], Sloane [24], Stevens et al. [26] and others. Cohen et al. [8], [9] and Sloane [24] have also shown applications of these designs to software testing, switching networks etc. Definitions and basic properties of group covering design and group covering number are discussed in section 2 of this paper. Lower bounds for the group covering number are also obtained in this section. Section 3 contains a few lower bounds for group covering number obtained from known designs. Few constructions for a class of group covering designs are given in section 4. It is shown that for any k there exists two classes of GCD’s which meet the bound given by (5). The paper concludes by constructing some GDD’s with consecutive block sizes using properties of group covering designs. The following two theorems of Hanani [14] give necessary conditions for the existence of a GDD and a TD. Theorem 3. [14] : If a group divisible design GD[k, λ, m; v] exists then v ≥ km, v ≡ 0

(mod m), λ(v − m) ≡ 0

(mod k − 1)

(2)

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and λv(v − m) ≡ 0

(mod k(k − 1)).

(3)

Theorem 4. [14] : If v = km, m > 1, λ = 1 and m ∈ T D(k) then m ≥ k − 1. Thus, if 1 < m < k−1 then a GD[k, m; km] does not exist. Further, Tarry [27] has shown that a GD[4, 6; 24] does not exist. Therefore, the conditions of Theorem 3 are not sufficient for the existence of a GD[k, λ, m; v]. However, group divisible designs with block sizes 3 and 4 are known to exist (see e.g., [3] and [14]) for all λ, m and v satisfying the necessary conditions of Theorem 3 with the exceptions of GD[4, 2; 8] and GD[4, 6; 24]. Very little is known for the case k ≥ 5 except for some special cases, see Assaf [1]. If λ = 1 and v > km then the following theorem gives a necessary condition for the existence of a group divisible design. Theorem 5. If there exists a group divisible design GD[k, m; v] with v > km then v ≥ k(k − 1) + m. Proof. Let (X, G, B) be a group divisible design  GD[k, m; v] and let B ∈ B. Then there exists a group G ∈ G such that B G = φ. Let x0 ∈ G. For any point x ∈ B, let Bx be the block which contains the  pairset {x0 , x}. Then for any two distinct points x and y of B, (B \{x }) (By \{x0 }) = φ. For if x 0   y0 ∈ Bx By , y0 = x0 , then {y0 , x0 } ⊆ Bx By , contradicting the fact that  each pair is contained in exactly one block. Also (B \{x }) G = φ for every x 0  x ∈ B. Thus, the set G ( x∈B Bx \{x0 }) contains m + k(k − 1) points and this number cannot exceed v. Remark 1. If m = 1, the above theorem gives v ≥ k(k − 1) + 1, the famous Fisher s inequality for λ = 1. The above theorem also shows nonexistence of many group divisible designs. For example, it is easy to verify that GD[8, 2; 30], GD[9, 3; 51] and GD[10, 3; 48] do not exist. For other elementary results on GDD and covering designs the reader is referred to [10].

2

Group Covering Designs

It has been observed that a group divisible design may not always exist for certain values of v, k and m. To deal with such cases, the notion of group covering designs analogous to covering designs is introduced in this section. The study of group covering designs apart from being a generalization of covering designs finds applications in q-ary covering codes, q ≥ 3 (see, [5], [6] and [7]). Throughout this paper, unless otherwise stated m, n and k are assumed to be positive integers with n ≥ k ≥ 2 and the members of a pairset belong to distinct groups.

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Definition: A triple (X, G, B) is a group covering design (GCD), denoted GC[k, m; mn], if |X| = mn, G is a partition of X into m−subsets and B is a collection of k−subsets of X such that |G ∩ B| ≤ 1 for all G ∈ G and B ∈ B and every pairset  is contained in at least one block of B. In case | G B | = 1 for G ∈ G and B ∈ B, the group covering design GC[k, m; km] is called a transversal covering design, denoted T C[k; m] (also referred to as transversal covers [26]). It follows immediately that an AD[k, 1; n] is a
group covering design GC[k, 1; n] and  vice versa. The number G(k, m; mn) = min |B| : (X, G, B)is a GC[k, m; mn] is called the group covering number. In case n = k, G(k, m; km) will be denoted by tc(k; m). A trivial lower bound for G(k, m; mn) is given by G(k, m; mn) ≥

nm2 (n − 1) . k(k − 1)

(4)

Equality in (4) is attained if there exists a GD[k, m; mn]. Analogous to Sch¨ onheim lower bound[23] for covering designs, the following theorem gives a better lower bound for the group covering number. Theorem 6. Let (X, G, B) be a group covering design GC[k, m; mn]. Then    mn m(n − 1) ≡ β(k, m; mn) (say). (5) G(k, m; mn) ≥ k k−1 Proof. Let x0 ∈ X. Then the number of pairsets containing x0 is m(n − 1). If B ∈ B is a block containing x0 then B can cover (k − 1) of these pairsets. Hence the number of blocks in the group covering design containing x0 is at least m(n − 1)/(k − 1). A simple counting argument gives kG(k, m; mn) ≥ mn m(n − 1)/(k − 1) . As G(k, m; mn) is an integer, the result follows.    n n−1 Remark 2. If m = 1 then C(k, 1; n) = G(k, 1; n) ≥ k k−1 which is the best known lower bound for the covering number C(k, 1; n) [23]. A similar result was proved by Chen and Honkala [7] while estimating the minimum number of (R + 2)-weight codewords required to cover all 2-weight words of any q-ary covering code of length n and covering radius R. If a group divisible design GD[k, m; mn] exists then k, m and n must satisfy 2 (n−1) ( 2) and ( 3) and the number of blocks is nm k(k−1) . This observation gives the following theorem. Theorem 7. Let k, m and n satisfy ( 2) and ( 3). If a group divisible design 2 (n−1) GD[k, m; mn] does not exist then G(k, m; mn) ≥ nm k(k−1) + 1. In particular, when n = k and 1 < m < k − 1, by Theorem 4 tc(k; m) ≥ m2 + 1. In [26], Stevens et al have shown that if m ≥ 3 and k ≥ m + 2 then tc(k; m) ≥ m2 + 3 with the only exception being tc(5; 3) = 11.

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Let (X, G, B) be a group covering design GC[k, m; mn] and let x0 ∈ X. Following the arguments of Theorem 6, the number of blocks containing x0 , denoted f(x0 ), satisfies   m(n − 1) f(x0 ) ≥ ≡ m0 (say). k−1 Let g(x0 ) = f(x0 ) − m0 . If g(x0 ) > 0 and m(n − 1) ≡ 0 (mod k − 1) then there exists at least one pairset {x0 , y} which is contained in two or more blocks. This proves the following lemma. Lemma 1. Let (X, G, B) be a group covering design GC[k, m; mn] and let {x, y} be a pairset. If either g(x) = 0 or g(y) = 0 and m(n − 1) ≡ 0 (mod k − 1) then the pairset {x, y} is contained in exactly one block of the group covering design. The following theorem is analogous to the result obtained by Hanani for covering designs, see [14]. Theorem 8. Let (X, G, B) be a group covering design GC[k, m; mn]. If m(n − 1) ≡ 0 (mod k − 1) and nm2 (n − 1)/(k − 1) ≡ −1 (mod k) then G(k, m; mn) ≥ β(k, m; mn) + 1. Proof. If possible, suppose G(k, m; mn) = β(k, m; mn). By hypothesis  nm2 (n−1)+(k−1) and hence β(k, m; mn) = x∈X g(x) = 1. So there exists a k(k−1) unique x0 ∈ X such that g(x0 ) = 1 and g(x) = 0 for x = x0 . Hence by Lemma 1, there exists a pairset {x0 , u} which is contained in at least two blocks. Thus g(u) > 0 for u = x0 , a contradiction. The above theorem gives many improvements to (5), e.g., G(7, 2; 26) ≥ 16, G(7, 2; 32) ≥ 24, G(5, 4; 32) ≥ 46, G(5, 4; 52) ≥ 126, G(11, 4; 64) ≥ 36, G(5, 4; 72) ≥ 246. Lemma 2. Let (X, G, B) be a group covering design GC[k, m; mn] and let m(n− 1) ≡ 0 (mod k − 1) and nm2 (n − 1)/(k − 1) ≡ −2 (mod k). If G(k, m; mn) = β(k, m; mn) then there exists a unique pairset {x0 , y0 } ⊆ X which is contained in exactly k blocks and every other pairset is contained in exactly one block. 2  Proof. Note that, G(k, m; mn) = nm (n−1)+2(k−1) and x∈X g(x) = 2. k(k−1) Thus, there exists x0 ∈ X with g(x0 ) > 0. Hence there exists y0 ∈ X such that {x0 , y0 } is contained in at least two blocks. Thus g(y0 ) ≥ 1. Hence g(x0 ) = g(y0 ) = 1 and g(x) = 0 for all other x ∈ X. Hence by Lemma 1, every pairset having x0 or y0 will be contained in exactly one block. Since each block contains k 2 pairsets and k(k−1) β(k, m; mn) = nm (n−1) + (k − 1), the pairset {x0 , y0 } 2 2 2 will be contained in exactly k blocks and the result follows.

Theorem 9. Let (X, G, B) be a group covering design GC[k, m; mn] and let m(n − 1) ≡ 0 (mod k − 1) and nm2 (n − 1)/(k − 1) ≡ −2 (mod k). If m(n − 1)/(k − 1) < k + m − 2 then G(k, m; mn) ≥ β(k, m; mn) + 1.

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Proof. Suppose G(k, m; mn) = β(k, m; mn). By Lemma 2, there exists a pairset {x0 , y0 } contained in exactly k blocks of B and f(x0 ) = f(y0 ) = m0 + 1. Let y0 ∈ G. Then for every y ∈ G, y = y0 , the pairset {x0 , y} must be contained in exactly one block of B. Thus f(x0 ) ≥ k + (m − 1), a contradiction. As a consequence of the above theorem, there are many improvements to (5), e.g., G(6, 2; 22) ≥ 16, G(8, 3; 45) ≥ 35, G(5, 4; 28) ≥ 35, G(10, 4; 76) ≥ 62. Lemma 3. Let n ≥ k > 3 and let (X, G, B) be a group covering design GC[k, m; mn]. If m(n − 1) ≡ 0 (mod k − 1), nm2 (n − 1)/(k − 1) ≡ −3 (mod k) and G(k, m; mn) = β(k, m; mn) then there exists three elements x0 , y0 and z0 of X belonging to distinct groups such that each of the pairsets {x0 , y0 }, {y0 , z0 } and {z0 , x0 } is contained in 12 (k + 1) blocks of B and all other pairsets are contained in exactly one block.  nm2 (n−1)+3(k−1) and x∈X g(x) = 3. Proof. Observe that G(k, m; mn) = k(k−1) Thus, there exists a point x0 ∈ X with g(x0 ) > 0. Following the arguments given in Lemma 2, there exists y0 ∈ X with g(y0 ) > 0 and x0 and y0 are in 2 two distinct groups. Note that k(k−1) β(k, m; mn) = nm (n−1) + 32 (k − 1). Now 2 2 if g(x0 ) = 2 and g(y0 ) = 1 then g(x) = 0 for all other x ∈ X and the pairset {x0 , y0 } will be contained in 32 (k − 1) + 1 blocks of B. Hence g(x0 ) = g(y0 ), a contradiction. Therefore, g(x0 ) = g(y0 ) = 1 and there exists z0 ∈ X such that g(z0 ) = 1. Hence for each x ∈ {x0 , y0 , z0 }, the number of pairsets containing x ( counting multiplicity ) that are contained in more than one block is k − 1. By Lemma 1 these (k − 1) extra pairsets containing x0 must contain either y0 or z0 . Similar statements hold for y0 and z0 . If x0 and z0 belong to the same group then since the pairset {x0 , y0 } occurs in exactly k blocks, there will be no extra pairset containing z0 , i.e., g(z0 ) = 0, a contradiction. Hence x0 , y0 and z0 must belong to distinct groups, each of the pairsets {x0 , y0 }, {y0 , z0 } and {z0 , x0 } must occur in (k + 1)/2 blocks and all other pairsets occur exactly once. The proof of the following theorem, being similar to the proof of Theorem 9 is omitted. Theorem 10. Let (X, G, B) be a group covering design GC[k, m; mn] and let m(n − 1) ≡ 0 (mod k − 1) and nm2 (n − 1)/(k − 1) ≡ −3 (mod k). If m(n − 1)/(k − 1) < (k − 3)/2 + m, then G(k, m; mn) ≥ β(k, m; mn) + 1. The above theorem gives improvements to (5) for many values of k, m and n; e.g., G(7, 2; 20) ≥ 10, G(9, 2; 26) ≥ 10, G(11, 2; 32) ≥ 10, G(9, 4; 52) ≥ 36. Remark 3. If m = 1, Lemma 2 and 3 give analogous results for covering designs. Theorem 11. G(5, 2; 14) ≥ 10. Proof. Suppose (X, G, B) is a group covering design GC[5, 2; 14] with 9 blocks. Let G = {G1 , . . . , G7 }. For simplicity, let Gi = {1, 2} meaning thereby that 1(2) represents the first (second) element of the group Gi . Moreover for B ∈ B (pairset

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P ) it is convenient to write B = x1 . . . x7 (P = x1 . . . x7 ) with xi ∈ Gi ∪{θ}. Here xj = θ will mean B(P ) does not contain any element of Gj . Since β(5, 2; 14) = 9, by Lemma 3 there exists 3 pairsets, say P1 = 11θ . . . θ, P2 = θ11θ . . . θ and P3 = 1θ1θ . . . θ, that are contained in exactly 3 blocks of B and every other pairset is contained in exactly 1 block. Following the argument given in Lemma 3, it is easy to see that for each i = 1, 2, 3 there exists exactly 4 blocks having xi = 1. Since each of the pairsets 12θ . . . θ, 21θ . . . θ and 22θ . . . θ occurs in exactly 1 block, there exists 6 blocks B1 , . . . , B6 whose first 2 entries are 11, 11, 11, 12, 21 and 22 respectively. Since each of the pairsets P2 and P3 occurs in exactly 3 blocks and x3 = 1 for only 4 blocks, the following two possibilities arise. Case 1: B1 = 111 ∗ ∗ ∗ ∗ B2 = 111 ∗ ∗ ∗ ∗ B3 = 111 ∗ ∗ ∗ ∗ B4 = 122 ∗ ∗ ∗ ∗ B5 = 212 ∗ ∗ ∗ ∗ B6 = 221 ∗ ∗ ∗ ∗ Case 2: B1 = 111 ∗ ∗ ∗ ∗ B2 = 111 ∗ ∗ ∗ ∗ B3 = 112 ∗ ∗ ∗ ∗ B4 = 121 ∗ ∗ ∗ ∗ B5 = 211 ∗ ∗ ∗ ∗ B6 = 22γ ∗ ∗ ∗ ∗ where ∗ denotes an element of {θ, 1, 2} and γ = 1. Suppose that Bi ’s have the configuration given in Case 1. Then each pairset with x1 = 1, x2 = x3 = θ must be contained in exactly one Bi , 1 ≤ i ≤ 4. Without loss of generality, let B5 = 212αβθθ and let P = 1θθαθθθ. If P is contained in B1 or B2 or B3 , say B ∗ , then the pairset Q = θ1θαθθθ will be contained in B ∗ and B5 , a contradiction. Otherwise P is contained in B4 and hence the pairset R = θθ2αθθθ is contained in B4 and B5 , a contradiction. Thus β(5, 2; 14) > 9. The proof in Case 2 follows similarly.

3

Lower Bounds for G(k, m; mn) from Known Combinatorial Designs

A relation between covering numbers and group covering numbers is given by the following theorem. Theorem 12. If n ≥ m + 1 then C(m + 1, 1; mn + 1) ≤ n + G(m + 1, m; mn). Proof. Let (Y, G, P) be a minimum group covering design GC[m+1, m; mn]  and let G = {G , G , . . . , G }. Let y ∈  Y. Consider X = Y {y } and B = 1 2 n 0 0     P P  , where P  = {G1 {y0 }, G2 {y0 }, . . . , Gn {y0 }}. It is easy to verify that (X, B) is a covering design AD[m + 1, 1; mn + 1]. Thus C(m + 1, 1; mn +1) ≤ n + G(m + 1, m; mn). As an immediate consequence of the above theorem, we have, Corollary 1. If n ≥ m + 1 then G(m + 1, m; mn) ≥ β(m + 1, m; mn) + ∆(mn + 1; m + 1) where ∆(mn + 1; m + 1) = C(m + 1, 1; mn + 1) − α(m + 1, 1; mn + 1). Proof. Observe that α(m + 1, 1; mn + 1) − n = β(m + 1, m; mn). If ∆(mn + 1; m + 1) > 0 then the above corollary gives an improvement to (5), e.g., G(4, 3; 18) ≥ β(4, 3; 18) + 2 = 25, and G(5, 4; 28) ≥ β(5, 4; 28) + 1 = 35. Remark 4. Let n(mn + 1) ≡ −1 (mod m + 1). Then Hanani [14] has shown that C(m + 1, 1; mn + 1) ≥ α(m + 1, 1; mn + 1) + 1 and hence by Corollary 1 G(m + 1, m; mn) ≥ β(m + 1, m; mn) + 1.

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Theorem 13. Let t ≥ n ≥ 3. If there exists a transversal design T D[n; m] then G(n, m; m(t(n − 1) + 1)) ≤ tm2 + G(n, m(n − 1); m(n − 1)t). Proof. Let {(Xj , Gj , Bj ) : j = 1, . . . , t} be a collection of t transversal designs T D[n; m] satisfying Xi ∩ Xj = G0 for all i = j and Gj = {G0 = Gj1, Gj2 , . . . ,  Gjn }. Let Y = j,l,l=1 Gjl , and let G = {G1 , G2 , . . . , Gt } where Gi = 2≤l≤n Gil . Let P be a collection of n−subsets of Y such that (Y, G, P) is a minimal group covering design GC[n, m(n − 1); m(n − 1)t]. Then (X, H, B) where X = Y ∪  G0 , H = j Gj and B = P ∪ B1 ∪ · · · ∪ Bt is a GC[n, m; m((n − 1)t + 1)]. If m is a prime power then a T D[m + 1; m] exists [14]. Hence we have Corollary 2. Let m be a prime power with m = n − 1 ≥ 2 and t ≥ n. If G(n, m(n − 1); t) = β(n, m(n − 1); t) then G(n, m; t(n − 1) + 1) = β(n, m; t(n − 1) + 1). Proof. Note that β(n, m(n − 1); t) = tm2 (n − 1)(t − 1)/n. Hence by the above theorem, G(n, m; t(n − 1) + 1) ≤ β(n, m; t(n − 1) + 1) and now the result follows from (5).

4

Exact Bounds for G(k, m; mn)

Constructions of minimum group covering designs can be quite challenging. In this section we use the structure of covering designs, meeting certain conditions, to construct few classes of minimum group covering designs. In [4], Caro and Raphael have shown that for each k, the size of the block, there exists a minimum covering design. It is used to establish the existence of a minimum group covering design for any k. Recall that tc(k; m) = G(k, m; km). Using extremal set theory, Kleitman and Spencer [16] have proved the following theorem.  s−1 Theorem 14. [16] Let s be a positive integer and let g(s) = s/2 −1 . Then tc(k; 2) = min{s : g(s) ≥ k}. If a covering design meeting the Sch¨ onheim bound exists then the following theorem gives a construction of a minimum group covering design. Theorem 15. Let (v − 1) ≡ 0 (mod k − 1), n = (v − 1)/(k − 1), nv ≡ −2 (mod k), and let v > k(k − 2) + 2. If there exists a minimum covering design AD[k, 1; v] with C(k, 1; v) = α(k, 1; v) then there also exists a minimum group covering design GC[k, k − 1; v − 1] with G(k, k − 1; v − 1) =

(v−1)(v−k) k(k−1)

.

Proof. Let (Y, P) be a covering design AD[k, 1; v] with C(k, 1; v) = α(k, 1; v). Then by Remark 3 there exists a unique pairset {u, v} ⊆ Y which is contained in exactly k blocks, say P1 , P2 , .  . . , Pk and all other pairsets are contained in exactly one block. Observe that Pi Pj = {u, v} for all 1 ≤ i < k j ≤ k. Let S = i=1 Pi . Clearly | S |= k(k − 2) + 2 and Y \S = φ. Let x0 ∈ Y \S. Since any pairset containing x0 must be contained in exactly one block of P,

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the number of blocks containing x0 is n. Let T1 , . . . , Tn be the blocks containing x0 . Clearly G = {Ti \{x0 } : 1 ≤ i ≤ n} is a partition of X = Y \{x0 } and {u, v} ⊆ Ti  for 1 ≤ i ≤ n. Let B = P\{T1 , T2 , . . . , Tn }. Note that for each B ∈ B, | Ti B |≤ 1 for 1 ≤ i ≤ n. Since (Y, P) is a covering design, every pairset {s, t} ⊆ X such that s and t are from distinct Ti \{x0 } will be contained in a block of B. Hence (X, G, B) is a group covering design GC[k, k − 1; v − 1]

(v−1)(v−k)

with G(k, k − 1; v − 1) ≤| B |= α(k, 1; v) − n = . k(k−1) 

v(v−k) Therefore by (5), G(k, k − 1; v − 1) ≥ k(k−1) . Hence the theorem follows.

If n ≡ 0 or 1 (mod 3) and k = 3 then Hanani [14] has shown that a GD[3, 2; 2n] exists and hence G(3, 2; 2n) = 2n(n − 1)/3 . If n ≡ 2 (mod 3) and v = 2n + 1 then Fort and Hedlund [13] have shown that there exists a covering design AD[3, 1; v] with C(3, 1; v) = α(k, 1; v). Hence by Theorem 15 the following corollary is immediate. Corollary 3. If n ≥ 3 then G(3, 2; 2n) = 2n(n − 1)/3 . If n ≡ 0 or 1 (mod 4) and k = 4 then Brouwer et al. [3] have shown that a GD[4, 3; 3n] exists and hence G(4, 3; 3n) = 3n(n − 1)/4 . If n ≡ 2 or 3 (mod 4) and v = 3n + 1 then Mills [19],[20] has shown that there exists a covering design AD[4, 1; v] with C(4, 1; v) = α(4, 1; v) with the exception of v = 19. Hence by Theorem 15 we have the following result. Corollary 4. If n ≥ 4 and n = 6 then G(4, 3; 3n) = 3n(n − 1)/4 . Remark 5. If n = 6, Corollary 1 gives G(4, 3; 18) ≥ 25 > 3n(n − 1)/4. If k = 5 and v = 4n + 1, then Mills and Mullin [21] have shown that C(5, 1; v) = α(5, 1; v) whenever n ≡ 2 (mod 5), n > 787 or n ≡ 4 (mod 5), n > 189. In each of these cases, the hypothesis of Theorem 15 are satisfied. Hence we have the following corollary. Corollary 5. If n ≡ 2 (mod 5), n > 787 or n ≡ 4 (mod 5), n > 189 then G(5, 4; 4n) = 4n(n − 1)/5 . Let v0 (k) be as in Theorem 2 and let v2 (k) = max{v0 (k), 3(k − 2)(k + 1)/2 + 4}. If v ≥ v2 (k), the following theorem follows from Theorems 2 and 15.

Theorem 16. Let v ≥ v2 (k), (v − 1) ≡ 0 (mod k − 1) and let v(v − 1)/(k − 1) ≡ −2  (mod k).  Then there exists a GC[k, k − 1; v − 1] with (v−1)(v−k) G(k, k − 1; v − 1) = . k(k−1) Theorem 17. Let k ≥ 3, (v−1) ≡ 0 (mod k −1), n = v−1 k−1 , nv ≡ −3 (mod k), and let v > 3(k + 1)(k − 2)/2 + 3. If there exists a minimum covering design AD[k, 1; v] with C(k, 1; v)= α(k, 1; v)then there also exists a GC[k, k − 1; v − 1] with G(k, k − 1; v − 1) =

(v−1)(v−k) k(k−1)

.

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Proof. Let (Y, P) be a covering design AD[k, 1; v] with C(k, 1; v) = α(k, 1; v). Then by Remark 3 there exists three pairsets, say {x, y}, {y, z} and {z, x}, each of which is contained in exactly (k + 1)/2 blocks of P (not necessarily distinct) and all other pairsets are contained in exactly one block. Let B1 , B2 , . . . , Bs be the blocks containing the pairsets {x, y}, {y, z} and {z, x} and let S = ∪si=1 Bi . Note that (k + 1)(k − 3)/2 + 3 ≤ |S| ≤ 3(k + 1)(k − 2)/2 + 3. Since v > 3(k + 1)(k − 2)/2 + 3, X\S = φ. Let x0 ∈ X\S and let T1 , T2 , . . . , Tn be the blocks containing x0 . Let X = Y \{x0 }, G = {Ti \{x0 } : 1 ≤ i ≤ n} and let B = P\{T1 , T2 , . . . , Tn }. Following the argument given in the proof of Theorem 15 it  is easy to verify that (X, G, B) is a minimum group covering design  containing

(v−1)(v−k) k(k−1)

blocks.

Let v0 (k) be as in Theorem 2 and let v3 (k) = max{v0 (k), 3(k − 2)(k + 1)/2 + 4}. If v ≥ v3 (k), by Theorems 2 and 17 we have the following theorem. Theorem 18. Let k ≥ 3, v ≥ v3 (k), (v − 1) ≡ 0 (mod k − 1) and let v(v − 1)/(k − 1) ≡ −3  (mod k).  Then there exists a GC[k, k − 1; v − 1] with (v−1)(v−k) G(k, k − 1; v − 1) = . k(k−1) Theorem 19. (i) G(3, 2; 8) = 8, (ii) (iii) G(5, 2; 12) = 8, (iv) G(6, 2; 14) = 7.

G(4, 2; 10)

=

8,

Proof. We use the notation for representing a block as described in Theorem 11. (i) By (5), G(3, 2; 8) ≥ 8. The following eight blocks gives the desired result. 111θ 12θ1 21θ2 222θ 1 θ 2 2 2 θ 1 1 θ 1 2 1 θ 2 1 2. (ii) By (5), G(4, 2; 10) ≥ 8. The following eight blocks gives the desired result. 2111θ 112θ1 121θ2 2222θ 1 1 θ 2 2 1 2 θ 1 1 2 θ 1 2 1 2 θ 2 1 2. (iii) By (5), G(5, 2; 12) ≥ 8. The following eight blocks gives the desired result. 12111θ 1112θ1 1121θ2 12222θ 2 1 1 θ 2 2 2 1 2 θ 1 1 2 2 θ 1 2 1 2 2 θ 2 1 2. (iv) By (5), G(6, 2; 14) ≥ 7. The following seven blocks gives the desired result. 12θ1221 21122θ1 222θ212 1122θ22 2 θ 1 1 1 2 2 θ 1 2 1 1 1 1 1 2 1 2 1 1 θ.

5

Group Divisible Designs with Consecutive Block Sizes

Recently Colbourn, Lenz, Ling, Rosa, Stinson and others have studied PBD’s with consecutive block sizes (see, [17], [11], [12] and [18]). However very little is known about GDD’s with consecutive block sizes. In this section we use Lemmas 2 and 3 to construct some GDD’s with two, three or four consecutive block sizes.

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Theorem 20. Let (X, G, B) be a minimum group covering design GC[k, m; mn] satisfying m(n − 1) ≡ 0 (mod k − 1) and nm2 (n − 1)/(k − 1) ≡ −2 (mod k) and |B| = β(k, m; mn). Then there exist group divisible designs GD[{k − 1, k}, {m − 1, m}; mn − 1] and GD[{k − 2, k − 1, k}, {m − 1, m}; mn − 2]. Proof. By Lemma 2 there exists a unique pairset P0 = {x0 , y0 } which is contained in exactly k blocks. Let X1 = X\{x0 }, X2 = X\P0 , G1 = {G\{x0 } : G ∈ G}, G2 = {G\P0 : G ∈ G}, B1 = {B\{x0 } : B ∈ B} and let B2 = {B\P0 : B ∈ B}. Then it is easy to verify that (X1 , G1 , B1 ) is a GD[{k − 1, k}, {m − 1, m}; mn − 1] and (X2 , G2 , B2 ) is a GD[{k − 2, k − 1, k}, {m − 1, m}; mn − 2]. Theorem 21. Let n ≥ k > 3 and let (X, G, B) be a minimum group covering design GC[k, m; mn] with |B| = β(k, m; mn). If m(n − 1) ≡ 0 (mod k − 1) and nm2 (n − 1)/(k − 1) ≡ −3 (mod k) then there exist group divisible designs GD[{k − 2, k − 1, k}, {m − 1, m}; mn − 2] and GD[{k − 3, k − 2, k − 1, k}, {m − 1, m}; mn − 3]. Proof. By Lemma 3 there exists three pairsets, say {x, y}, {y, z} and {z, x}, each of which is contained in exactly (k + 1)/2 blocks. Let X1 = X\{x, y}, X2 = X\{x, y, z}, G1 = {G\{x, y} : G ∈ G}, G2 = {G\{x, y, z} : G ∈ G}, B1 = {B\{x, y} : B ∈ B} and let B2 = {B\{x, y, z} : B ∈ B}. Observe that B1 contains blocks of sizes k − 2, k − 1 or k while B2 contains blocks of sizes k − 3, k − 2, k − 1 or k. A block of size k − 3 in B2 will exist only when there exist a block containing x, y and z. It can be easily verified that (X1 , G1 , B1 ) and (X2 , G2 , B2 ) are GD[{k − 2, k − 1, k}, {m − 1, m}; mn − 2] and GD[{k − 3, k − 2, k − 1, k}, {m − 1, m}; mn − 3], respectively. It may be observed that if mn ≥ v2 (k) (v3 (k)), m(n − 1) ≡ 0 (mod k − 1) and nm2 (n − 1)/(k − 1) ≡ −2 (mod k) (≡ −3 (mod k)) then Theorem 16 ( Theorem 18 ) guarantees the existence of a minimum group covering design with β(k, m; mn) blocks.

References 1. A. M. Assaf, An application of modified group divisible designs, J. Combin. Theory Ser. A 68 (1994), 152-168. 2. M. C. Bhandari, K. K. P. Chanduka and A. K. Lal, On lower bounds for covering codes, Designs, Codes and Cryptography, 15 (1998), 237-243. 3. A. E. Brouwer, A. Schrijver and H. Hanani, Group divisible designs with block size four, Discrete Math. 20 (1977), 1-10. 4. Y. Caro and Y. Raphael, Covering graphs: The covering problem solved, J. Comb. Theory, Ser. A, 83 (1998), 273-282. 5. K. K. P. Chanduka, ”Combinatorial Designs and Covering Codes”, Ph.D. thesis, Indian Institute of Technology, Kanpur, India (January 1998). 6. K. K. P. Chanduka, M. C. Bhandari and A. K. Lal, Further results on q−ary covering codes, manuscript under preparation. 7. W. Chen and I. S. Honkala, Lower bounds for q-ary covering codes, IEEE Trans. Inform. Theory 36 (1990), 664-671.

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8. D. M. Cohen, S. R. Dalal, J. Parelius and G. C. Patton, The combinatorial design approach to automatic test generation, IEEE software, 13 (1996), 83-88. 9. D. M. Cohen, S. R. Dalal, M. L. Fredman and G. C. Patton, The AETG systems: An approach to testing based on combinatorial design, IEEE Trans. Soft. Eng., 23 (1997), 437-444. 10. C. J. Colbourn and J. H. Dinitz, editors, ”The CRC Handbook of Combinatorial Designs”, CRC Press, Boca Raton, first edition, 1996. 11. C. J. Colbourn and A. C. H. Ling, Pairwise balanced designs with block sizes 8, 9 and 10, J. Combin. Theory Ser. A 77 (1997), 228-245. 12. C. J. Colbourn, A. Rosa and D. R. Stinson, Pairwise balanced designs with block sizes three and four, Can. J. Math. 43 (1991), 673-704. 13. M. K. Fort and G. A. Hedlund, Minimal covering of pairs by triples, Pacific J. Math. 8 (1958), 709-719. 14. H. Hanani, Balanced incomplete block designs and related designs, Discrete Math. 11 (1975), 255-369. 15. I. S. Honkala, Modified bounds for covering codes, IEEE Trans. Inform. Theory 37 (1991), 351-365. 16. D. J. Kleitman and J. Spencer, Families of K-independent sets, Discrete Math. 6 (1973), 255-262. 17. H. Lenz, Some remarks on Pairwise balanced designs, Mitt. Math. Sem. Giessen, 165 (1984), 49-62. 18. A. C. H. Ling, X. Zhu, C. J. Colbourn and R. C. Mullin, Pairwise balanced designs with consecutive block sizes, Designs, Codes and Cryptography, 10 (1997), 203-222. 19. W. H. Mills, On the covering of pairs by quadruples I, J. Combin. Theory Ser. A 13 (1972), 55-78. 20. W. H. Mills, On the covering of pairs by quadruple II, J. Combin. Theory Ser. A 15 (1973), 138-166. 21. W. H. Mills and R. C. Mullin, Covering pairs by quintuples: The case v congruent to 3 (mod 4), J. Combin. Theory Ser. A 49 (1988), 308-322. 22. S. Poljak and Z. Tuza, On the maximum number of qualitatively independent partitions, J. Combin. Theory Ser. A 51 (1989), 111-116. 23. J. Sch¨ onheim, On coverings, Pacific J. Math. 14 (1964), 1405-1411. 24. N. J. A. Sloane, Covering arrays and intersecting codes, J. Combin. Designs, 1 (1993), 51-63. 25. B. Stevens and E. Mendelsohn, New recursive methods for transversal covers, preprint. 26. B. Stevens, L. Moura and E. Mendelsohn, Lower bounds for transversal covers, Designs, Codes and Cryptography, 15 (1998), 279-299. 27. G. Tarry, Le probl`eme des 36 officiers, Compt. Rend. Assoc. Fr. Av. Sci. 1 (1900), 122-123; 2 (1901), 170-203. 28. Z. Zhang, Linear inequalities for covering codes: Part I-Pair covering inequalities, IEEE Trans. Inform. Theory 37 (1991), 573-582.

Characteristic Functions of Relative Difference Sets, Correlated Sequences and Hadamard Matrices Garry Hughes Royal Melbourne Institute of Technology, Melbourne, VIC 3001, Australia [email protected]

Abstract. Given a cocycle, α, the concept of a sequence being α- correlated provides a link between the cohomology of finite groups and various combinatorial objects: auto-correlated sequences, relative difference sets and generalised Hadamard matrices. The cohomology enables us to lift a map φ, defined on a group, to a map Φ, defined on an extension group, in such a way that Φ inherits some of its combinatorial properties from those of φ. For example, if φ is α-correlated, Φ will be the characteristic function of a relative difference set in the extension group determined by α. Many well-known results follow from choosing the appropriate extension groups and cocycles, α.

1

Introduction

It is well known that many objects of combinatorial interest are intimately connected; relative difference sets, sequences with certain auto-correlation properties and generalised Hadamard matrices for example. Recently several authors have related these objects to the cohomological theory of finite groups (see [6] and the references therein). That there is a relationship is, in hindsight, no surprise because combinatorial objects such as those mentioned are often defined and studied in terms of a group and its extensions. Cohomology is a natural language to discuss group extensions. In this spirit, the work presented here will use ideas from cohomology to show the equivalence of the combinatorial objects mentioned, under certain circumstances. The equivalence is constructive showing how to pass easily from one object to another (see Theorem 3). One of the advantages of this approach is that the theory covers, in a single framework, both splitting and non-splitting extensions (see the corollaries to Theorem 3). Another is that construction techniques for any of the objects can be proved using the most convenient equivalence (see Section 4). In Section 2 we introduce definitions that express certain distribution properties of the values of a map between groups. These definitions are fundamentally connected to each other and to objects from cohomology, namely “cocycles”. In Section 3 we review the results and definitions we need concerning cocycles and group extensions. We then show how, when given a map between groups, φ, with Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 346–354, 1999. c Springer-Verlag Berlin Heidelberg 1999


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a certain distribution property, we can lift this map to an extension group in such a way that the lifted map has a similar distribution property. The lifted map will be the characteristic function for a relative difference set in the extension group. We apply this to the split extension and the natural extension. In Section 4 we give some methods that enable us, when given such φ, to construct others. Finally we show that a Generalised Perfect Binary Array (see [2]) corresponds to a special example of the lifting of such a map φ. Throughout this paper we adopt the following notation: A will be a finite abelian group; ZA the group ring of A over the integers; if w is a vector, wi will denote its i-th component.

2

Relative Orthogonality and Relative Correlation

We give two definitions that describe the distribution properties of maps between groups. The definitions are then connected to cohomology. Definition 1. Let ψ : L × L → A be a map of finite groups with |A| dividing |L|. Let H be a subgroup of L. Then we say ψ is orthogonal relative to H if, in ZA, b ∈ L − H implies

ψ(b, j) = |L|/|A| a. j∈L

a∈A

Here L − H refers to the set of elements of L that are not in H. Furthermore, ψ is called orthogonal if it is orthogonal relative to the trivial subgroup.   H is called a forbidden subgroup and we may alternatively express relative orthogonality by saying that, when b is not in the forbidden subgroup, the sequence {ψ(b, j) : j ∈ L} has each element of A an equal number of times. Important in what follows is the effect of onto group homomorphisms on relative orthogonality. We have the following result, the first part of which we will use later to lift orthogonality on a group to relative orthogonality on an extension of that group. Lemma 1. Let β : R → S and θ : A → A be onto homomorphisms of groups. Let ψ : S × S → A be a map of groups and H a subgroup of S. Then i) ψ ◦ (β × β) : R × R → A is orthogonal relative to ker β if and only if ψ is orthogonal. ii) If ψ is orthogonal relative to H then so is θ ◦ ψ. Proof. We give an outline of the proof of i). Let T be a transversal for the cosets of ker β in R. For b ∈ R we have, upon collecting by cosets,

ψ(β(b), β(r)) = | ker β| ψ(β(b), β(t)). r∈R

t∈T

Now, because β is onto, as t runs over T then β(t) runs over S, so this last sum is equal to | ker β| s∈S ψ(β(b), s). The result now follows.  

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The concept of relative orthogonality is closely connected to that of the “twisted relative correlation” of sequences. The nature of this connection, which we prove in the next lemma, enables results concerning cohomology theory and relative difference sets to be interpreted in terms of correlation of sequences. We firstly need some definitions. Definition 2. Let φ : L → A and α : L × L → A be maps of groups with |A| dividing |L|. Let H be a subgroup of L. Then we say φ is α-correlated relative to H if, in the group ring ZA, b ∈ L − H implies

α(b, j)φ(j)(φ(bj))−1 = |L|/|A| a. j∈L

a∈A

Further, if H is the trivial subgroup we omit the phrase “relative to H” and if α is identically 1, we replace “1-correlated” by “correlated”.   So, in the definition above, we think of φ as being auto-correlated when the auto-correlation function has been “twisted” by α. We will use the following notation from cohomology theory (see [4]). If φ : L → A is a map of groups, the coboundary ∂φ : L × L → A is defined by ∂φ(m, n) = φ(m)φ(n)(φ(mn))−1 , ∀m, n ∈ L. We can now establish the connection mentioned above. Lemma 2. With the notation of the above definition, φ is α-correlated relative to H if and only if α∂φ is orthogonal relative to H. Proof. For b ∈ L, we have

α(b, j)∂φ(b, j) = φ(b) α(b, j)φ(j)(φ(bj))−1 . j∈L

Noting that φ(b)

3



a∈A a

=



j∈L a∈A a

= (φ(b))−1



a∈A a,

we obtain the result.  

Cohomological Aspects of Relative Orthogonality

For the remainder of this paper let G be a group of order v and we suppose w | v, where w denotes the order of A. When the “twisting” function α in the definition of α-correlation is of a special form there is an equivalence between the existence of α-correlated maps, generalised Hadamard matrices and relative difference sets in extension groups determined by α. The special form needed is that α is a cocycle (strictly a two dimensional cocycle with trivial action). We will define this shortly, but the following result of Perera and Horadam indicates the sort of relationship between cohomological and combinatorial objects that we are interested in. A v×v matrix, C, of elements of A, indexed by the elements of G (in some fixed order), is a (v/w)-generalised Hadamard matrix if, whenever 1 ≤ i = k ≤ v, the list cij c−1 kj , 1 ≤ j ≤ v has each element of A exactly v/w times (see [6] or [1]).

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Theorem 1. [6, Lemma 2.2] Let ψ : G × G → A be a cocycle and form the matrix Mψ = [ψ(x, y)] indexed by the elements x, y ∈ G (in some fixed order). Then ψ is orthogonal if and only if Mψ is a (v/w)-generalised Hadamard matrix.   3.1

Cocycles and Central Extensions

We summarise the definitions and results we need on cocycles and central extensions (for proofs see [4, Chapter 2]). We call the map α : G × G → A a cocycle if α(1, 1) = 1 and ∀x, y, z ∈ G it satisfies the equation α(x, y)α(xy, z) = α(y, z)α(x, yz). A consequence of this equation is that ∀x ∈ G we have α(x, 1) = α(1, x) = 1. The abelian group of all such cocycles under the multiplication (αα )(x, y) = α(x, y)α (x, y) is denoted Z 2 (G, A). If φ : G → A is such that −1 φ(1) = 1, the coboundary defined by ∂φ(x, y) = φ(x)φ(y)(φ(xy)) ∀x, y ∈ G is in Z 2 (G, A). If α, α ∈ Z 2 (G, A) and α = α ∂φ for some φ we say α and α are cohomologous and write α ∼ α . This is an equivalence relation and the group of equivalence (cohomology) classes, α, is denoted H 2 (G, A). Cocycles determine and are determined by central extensions. Consider a central extension, R, of G by A; that is consider a short exact sequence, ι

β

1 → A → R → G → 1,

(1)

where ι(A) = ker β is a subgroup of the centre of R. We take a section, λ, of β, that is a map λ : G → R such that β(λ(x)) = x, ∀x ∈ G and λ(1) = 1. Then λ defines a cocycle fλ ∈ Z 2 (G, A) by ι(fλ (x, y)) = λ(x)λ(y)(λ(xy))−1 . We note that λ(G) is a transversal for the cosets of ι(A) in R. Equivalently we could define λ given such transversal. Different choices of λ lead to cohomologous cocycles. Conversely, given a cocycle α ∈ Z 2(G, A) we have the central extension ι


β


1 → A → Eα → G → 1,

(2)

where Eα is the group {(a, x) : a ∈ A, x ∈ G} with a “twisted” multiplication defined by (a, x)(b, y) = (abα(x, y), xy) and ι (a) = (a, 1), β  (a, x) = x. We refer to this as the standard extension for α. We also refer to the section of β  given by λ (x) = (1, x) as the standard section for α since it determines the cocycle fλ
= α. 3.2

Extensions of Set Maps

Consider the central extension (1) and a fixed section, λ, of β. In this part we take a map φ : G → A, with φ(1) = 1, and define an extension to a map from R to A. This extension will preserve some of the relative correlation properties of the original map. When φ has certain correlation properties the extension will prove to be a characteristic function of a relative difference set in R. We will establish this in the next part.

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Any r ∈ R may be written in the form r = ι(ar )λ(xr ) where xr = β(r) and ar ∈ A is unique. Therefore, given φ : G → A we define the extension, Φλ : R → A, of φ by Φλ (r) = ar −1 φ(xr ).

(3)

If the central extension and section are the standard ones for α ∈ Z 2 (G, A) we write Φα for Φλ
. We have the following properties for the extension function. Lemma 3. i) {r ∈ R : Φλ (r) = 1} = {ι(φ(x))λ(x) : x ∈ G}; ii) ∂Φλ = (fλ ∂φ) ◦ (β × β). Proof. Only ii) needs any work. For r, s ∈ R we see xrs = xr xs and, because ι(A) is in the centre of R, we also have ars = ar as fλ (xr , xs ). The result now follows from the definitions of Φλ and ∂Φλ .   3.3

Relative Difference Sets and Cocycles

We firsly define a relative difference set (rds). Suppose M is a group of order vw and N a subgroup of order w where w | v. A v element subset, D, of M is called a (v, w, v, v/w)-rds in M relative to N if the list d1 d−1 2 , d1 , d2 ∈ D contains no elements of N and each element of M − N exactly v/w times. The fundamental result connecting such relative difference sets and cocycles is also due to Perera and Horadam. Theorem 2. [6, Theorem 4.1] Let ψ ∈ Z 2(G, A). Then ψ is orthogonal if and only if {(1, x) : x ∈ G} is a (v, w, v, v/w)-rds in Eψ relative to A × 1.   We are finally in a position to prove our main result which links the relative orthogonality and correlation properties of a map and its extension with a characteristic function for a relative difference set. Recall that |G| = v, |A| = w and w | v. Theorem 3. Consider the central extension, (1), and a section, λ, of β. Let φ : G → A be such that φ(1) = 1 and let Φλ be defined by (3). Then the following are all equivalent: i) fλ ∂φ is orthogonal; ii) D = {r ∈ R : Φλ (r) = 1} is a (v, w, v, v/w)-rds in R relative to ι(A) = ker β; iii) ∂Φλ is orthogonal relative to ι(A); iv) Φλ is correlated relative to ι(A); v) φ is fλ -correlated. Proof. The equivalence of i) and iii) follows from Lemmas 1 and 3; that of iii) and iv) and also that of i) and v) from Lemma 2. We need only show that i) and ii) are equivalent. Let ψ = fλ ∂φ. There is an isomorphism Γ : Eψ → R given by Γ (a, x) = ι(aφ(x))λ(x) (see [4] for this). We see that Γ (a, 1) = ι(a). Let D∗ = {(1, x) : x ∈ G} ⊆ Eψ . Therefore Γ (D∗ ) = {ι(φ(x))λ(x) : x ∈ G} and, by Lemma 3, D = Γ (D∗ ). Applying the isomorphism Γ , Theorem 2 tells us that the orthogonality of ψ is equivalent to D = Γ (D∗ ) being a (v, w, v, v/w)-rds in Γ (Eψ ) = R relative to Γ (A × 1) = ι(A) = ker β.  

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In view of part ii) of the above equivalence, we can regard Φλ as a characteristic function for the relative difference set {ι(φ(x))λ(x) : x ∈ G} in R. If we begin with a cocycle and use the standard extension, (2), and standard section for that cocycle we obtain the following corollary. Corollary 1. Let α ∈ Z 2 (G, A) and φ : G → A with φ(1) = 1. Then the following are equivalent: i) α∂φ is orthogonal; ii) Φα (a, x) = a−1 φ(x) is correlated relative to A × 1; iii) φ is α-correlated; iv) {(φ(x), x) : x ∈ G} is a (v, w, v, v/w)-rds in Eα relative to A × 1.   We note that instead of starting with a map φ we could equally well begin with a (v, w, v, v/w)-rds D, in Eα , and define φ(x) = a for (a, x) ∈ D. The map φ is well-defined on G and would be α-correlated. When we choose α ≡ 1 in the above corollary we obtain some well known results on splitting relative difference sets, because E1 = A × G. Corollary 2. i) φ is correlated if and only if the set {(φ(x), x) : x ∈ G} is a (v, w, v, v/w)-rds in A × G relative to A × 1. ii) If v = w, let b ∈ G and ∆b,φ : G → A be the difference operator ∆b,φ(x) = φ(x)(φ(bx))−1 . Then ∆b,φ is onto (equivalently one-to-one) for every b = 1 if and only if {(φ(x), x) : x ∈ G} is a (v, v, v, 1)-rds in A × G relative to A × 1.   As an example of the last result; if G is the additive group of the field GF(q), q odd, then for any e = 0, g, h ∈ GF(q) there is a “quadratic” (v, v, v, 1)-rds in G × G relative to G × 0, namely {(ex2 + gx + h, x) : x ∈ G}. This is also proved, basically, in [7]. As we have seen the split extension provides certain examples. So, too, does the natural extension. Let M, N be groups of order, respectively, vw, w with N in the centre of M. Let D be a transversal for the cosets of N in M with 1 ∈ D. The natual short exact sequence id

π

1 → N → M → M/N → 1, with π(m) = mN and section λ(dN ) = d, d ∈ D, defines a cocycle fD ∈ Z 2 (M/N, N ) as follows. For d, d ∈ D we have dd N = d∗ N for unique d∗ ∈ D. So we define fD (dN, d N ) = dd (d∗ )−1 . We take the map φ on M/N to be identically 1 and extend it to M as before. Write r ∈ M uniquely as r = dr nr , dr ∈ D, nr ∈ N and define Φλ (r) = dr r−1 = n−1 r . We now have the following by using Theorems 1 and 3. Corollary 3. For N central in M and a transversal D, with 1 ∈ D, the following are equivalent: i) D is a (v, w, v, v/w)-rds in M relative to N ; ii)Φλ (r) = dr r−1 is correlated relative to N ; iii) fD is a (v/w)-generalised Hadamard matrix.   The construction of a generalised Hadamard matrix from a relative difference set appears in [3]. We have proved the equivalence of these objects in the case of the parameters and groups above.

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Base Sequences and Generalised Perfect Binary Arrays

We have seen that maps that are auto-correlated when the correlation function is “twisted” by a cocycle can be lifted to produce relative difference sets in extension groups. In view of this we call φ : G → A, where φ(1) = 1, a base sequence with respect to α ∈ Z 2 (G, A) if it satisfies any of the equivalent conditions of Corollary 1. In the special case of α being symmetric (that is α(x, y) = α(y, x), ∀x, y ∈ G), G being an abelian group and A having order 2, base sequences have been studied under the name Generalised Perfect Binary Arrays, or GPBAs, by Jedwab and others (see [2]). We will discuss this correspondence later. 4.1

Construction of Base Sequences

Jedwab [2] gives many construction techniques for GPBAs. Some of the techniques seem specific to the situation he is studying, but many work in more general circumstances. We present some of these. By virtue of Theorem 3 these techniques give constructions for rds in extension groups. Theorem 4. For α ∈ Z 2 (G, A) and φ : G → A, with φ(1) = 1, we have: i)If α is cohomologous to α ∈ Z 2(G, A), say α = α ∂µ, then φ is a base sequence wrt α if and only if µφ is a base sequence wrt α ; ii) If Θ : G → G is an isomorphism, φ is a base sequence wrt α if and only if φ ◦ Θ is a base sequence wrt α ◦ (Θ × Θ) ∈ Z 2 (G , A); iii) Let φ1 : G1 → A and φ2 : G2 → A. Define the tensor product, φ1 ⊗ φ2 , to be φ1 ⊗ φ2 (x1 , x2 ) = φ1 (x1 )φ2 (x2 ). Similarly given α1 ∈ Z 2 (G1 , A) and α2 ∈ Z 2 (G2 , A) define α1 ⊗ α2 . Then φ1 ⊗ φ2 is a base sequences wrt α1 ⊗ α2 if and only if φ1 , φ2 are base sequences wrt α1 , α2 respectively. iv) Let Ω : A → A be an onto homomorphism. If φ is a base sequence wrt α then Ω ◦ φ is a base sequence wrt Ω ◦ α ∈ Z 2 (G, A ). Proof. These are most easily seen using the definition: φ is a base sequence wrt α if and only if α∂φ is orthogonal. All the parts except iii) follow from Lemma 1 with the following observations: in i) α∂φ = α ∂(µφ); in ii) (α∂φ) ◦ (Θ × Θ) = α ◦ (Θ × Θ)∂(φ ◦ Θ); in iv) Ω ◦ (α∂φ) = (Ω ◦ α)∂(Ω ◦ φ). It remains only to prove iii). Suppose ψ = (α1 ⊗ α2 )∂(φ1 ⊗ φ2 ) is orthogonal. Let ψ1 = α1 ∂φ1 and 1 = x ∈ G1 . Then
(x,y)∈G1×G2

  ψ (x , 1), (x, y) =


(x,y)∈G1×G2

ψ1 (x , x).1 = | G2 |




ψ1 (x , x).

x∈G1

From the definition of orthogonality we deduce ψ1 is orthogonal. An equivalent proof works for φ2 . The converse can be proven by a similar argument but is also a consequence of a result on the Kronecker product of generalised Hadamard matrices (see [6, Theorem 5.1]).  

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Part i) of the preceeding theorem tells us that if we want a base sequence wrt some cocycle, we may as well assume the cocycle is a representative of a cohomology class in H 2 (G, A). If G ∼ = Zs1 × · · · × Zsr is abelian, representatives for cohomology classes of symmetric cocycles are easy to describe.Indeed, let Ext(G, A) = {α ∈ H 2 (G, A) : α symmetric}, then Ext(G, A) ∼ = i,j Z(si ,tj) , where A ∼ = Zt1 × · · · × Ztk and (si , tj ) refers to the greatest common divisor of si , tj (see [4] for this). In view of this fact, and the theorem above, if we seek a base sequence wrt some symmetric cocycle when G is abelian, we may as well assume G = Zps , A = Zqt for primes p, q. If p = q then we may take the cocycle to be a coboundary and we are in the situation of Corollary 2. 4.2

GPBAs

We will now outline how we may regard a GPBA as a base sequence wrt to a very specific cocycle. Full details of proofs are given in [5]. We take the definition of GPBA from [2]. Let G = Zs1 × · · · × Zsr , A = {±1} be a group of order two, z = (z1 , . . . , zr ) where zi = 0 or 1, and s = (s1 , . . . , sr ). Also, let φ : G → A be a map of groups with φ(1) = 1. If z = 0 then φ is called a GPBA(s) of type 0, or a PBA(s), if it is correlated. When z = 0 a more involved definition is needed. We define yet more groups: G∗ = Z(z1 +1)s1 × · · · × Z(zr +1)sr . Thus, the arithmetic in the i-th coordinate of G∗ is mod 2si or mod si according as zi = 1 or zi = 0. Further, define the following subgroups of G∗ , H = {h ∈ G∗ : hi = 0 if zi = 0; hi = 0 or si if zi = 1}, K = {k ∈ H : k has even weight}. We now define an extension of φ to G∗ . We may write any g ∈ G∗ uniquely in the form g = x + h where x ∈ G and h ∈ H by taking x = g mod s = (g1 mod s1 , . . . , gr mod sr ) and h = g − x. Here gi mod si refers to the unique residue in the range 0, . . . , si − 1. Now put  a(x) if h ∈ K 7(g) = −a(x) if h ∈ / K. Finally, then, φ is called a GPBA(s) of type z = 0 if 7 is correlated relative to H. The concept of GPBA relates to the ideas in the earlier part of this paper by taking the short exact sequence ι

β

1 → A → G∗ /K → G → 0, where ι is the homomorphism ι(A) = H/K and β the homomorphism g + K → g mod s. Using λ(x) = x + K as a section of β we have, in the notation of the previous section, 7(g) = Φλ (g + K). We may prove from the results earlier that 7 is correlated relative to H if and only if Φλ is correlated relative to H/K. The section λ defines a cocycle in Z 2 (G, A), which we call fJ , in the usual way and we have the following result.

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Theorem 5. [5, Theorem 5.6] For any z, φ is a GPBA(s) of type z if and only if φ is a base sequence wrt fJ .   Finally we note that, because of the isomorphism of Ext(G, A) mentioned above, fJ depends only on s and z, and in a particularly simple manner (for details see [5]).

References 1. D. L. Drake, Partial λ-geometries and generalised Hadamard matrices, Canad. J. Math. 31 (1979), 617-627. 2. J. Jedwab, Generalized Perfect Arrays and Menon Difference Sets, Des. Codes and Cryptogr. 2 (1992), 19-68. 3. D. Jungnickel, On automorphism groups of divisible designs, Canad. J. Math. 34 (1982), 257-297. 4. G. Karpilovsky, Projective Representations of Finite Groups, Marcel Dekker, New York, 1985. 5. G. Hughes, Cocyclic Theory of Generalized Perfect Binary Arrays, Royal Melbourne Institute of Technology, Department of Mathematics, Research Report 6, May 1998. 6. A. A. I. Perera and K. J. Horadam, Cocyclic Generalised Hadamard Matrices and Central Relative Difference Sets, Des. Codes and Cryptogr. 15 (1998), 187-200. 7. A. Pott, Finite Geometry and Character Theory, Lecture Notes in Mathematics 1601, Springer-Verlag, Berlin, 1995.

Double Circulant Self-Dual Codes Using Finite-Field Wavelet Transforms F. Fekri, S.W. McLaughlin, R.M. Mersereau, and R.W. Schafer Center for Signal & Image Processing Georgia Institute of Technology, Atlanta, GA 30332-0250 {fekri,swm,rmm,rws}@ee.gatech.edu WWW home page: http://www.ee.gatech.edu/users/fekri/ Abstract. This paper presents an example of integrating recently developed finite-field wavelet transforms into the study of error correcting codes. The primary goal of the paper is to demonstrate a straightforward approach to analyzing double circulant self-dual codes over any arbitrary finite-field using orthogonal filter bank structures. First, we discuss the proper combining of the cyclic mother wavelet and scaling sequence to satisfy the requirement of self-dual codes. Then as an example, we describe the encoder and decoder of a (12,6,4) self-dual code, and we demonstrate the simplicity and the computation reduction that the wavelet method offers for the encoding and decoding of this code. Finally, we give the mother wavelet and scaling sequence that generate the (24,12,8) Golay code.

1

Introduction

Although wavelets and filter banks over real or complex fields have been studied extensively for years, the finite-field wavelet transform has received little attention because of the very limited application of this transform in the signal and image processing area. In the past there was some interest in extending wavelet transforms to the situation in which the complex field is replaced with a finite field. In [1] the authors show that unlike the real field case, there is no complete factorization technique for paraunitary filter banks (FB) over GF (p), for p a prime, using degree-one and degree-two building blocks. Relying on the Fourier transform defined over GF (pr ), the authors of [2] construct a wavelet transform for finite dimensional sequences over fields with a characteristic other than 2, p
= 2. In [3] an alias cancellation approach was used to design two-band filter banks over finite fields. The main problem with this approach was excluding fields of characteristic two, GF (2r ) for r ≥ 1, in which an element of order two does not exist. An extensive review of finite field transforms can be found in [4]. Recently, a formulation of the wavelet decomposition of a vector space over any finite-field has been derived in [5]. Since that formulation relies on a basis decomposition in the time domain rather than in the frequency domain, it does not require the existence of the number theoretic Fourier transform. Thus it becomes more attractive, particularly for the fields GF (2r ) having characteristic 2. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 355–363, 1999. c Springer-Verlag Berlin Heidelberg 1999


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The objective of this paper is to present an example of our attempt to bring together finite-field wavelet transforms and error correcting codes and to study error control coding in the signal processing context. This unified view uncovers a rich set of signal processing techniques that can be exploited to investigate new error correcting codes, and to simplify the encoding and decoding techniques of some existing codes. Self-dual codes have been widely studied [6]. In this paper, first we discuss a model to generate double circulant self-dual codes over any arbitrary finitefield using the notion of finite-field wavelet transform. Then as an example, we present an encoding and decoding method for the (12,6,4) binary self-dual code. More in-depth study of double circulant self-dual codes using cyclic orthogonal and biorthogonal wavelet transforms can be found in [7]. Let FN be the vector space of N -tuples over the finite-field F. Then a selfdual code of length N is a subspace C of FN such that C = C ⊥ , where C ⊥ is defined as: C ⊥ = {v ∈ FN : v, c = 0 ∀ c ∈ C} (1) Clearly, C has dimension M = N/2 and is called a (N, M, d) code where d is the minimum distance of the code. Since the primary focus of this paper is on linear block codes, before talking explicitly about self-dual codes, we briefly summarize some results on cyclic wavelet transforms.

2

Cyclic Wavelet Transforms over the Field F

It is well known that wavelet decomposition and reconstruction can be implemented as the analysis and synthesis components of a perfect reconstruction filter bank, respectively. Figure 1 shows the analysis and synthesis bank of a two-channel perfect reconstruction filter bank in which the synthesis filters g0 (n) and g1 (n) are the scaling sequence and mother wavelet, respectively. In [5] the authors show how to decompose a vector space V over a finite-field F onto two orthogonal subspaces V0 and W0 . Particularly, to have a two channel perfect reconstruction orthogonal filter bank, a design methodology is presented to obtain the analysis and synthesis filters over the fields of characteristic. Since the codewords c(n) of self-dual codes have finite even length, the vector space V is considered to be a vector space of finite dimension N = 2M. It can also be regarded as a space of periodic sequences of period N . Now, consider a two-channel perfect reconstruction orthogonal filter bank with the scaling sequence g0 (n) = {g0 (0), g0 (1), · · · , g0 (N − 1)} and the mother wavelet g1 (n) = {g1 (0), g1 (1), · · · , g1 (N − 1)}, where the mother wavelet is the time reverse of the scaling function g1 ((n))N = g0 ((−n − 1))N . It is worth noting that throughout the paper ((·))N denotes a modulo-N operation, or equivalently an N -point circular shift. Furthermore, the analysis filters are related to the synthesis filters by: hj ((n))N = gj ((−n))N

j = 0, 1

n = 0, . . . , N − 1.

(2)

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In the study of block codes, we will frequently use the fact that the algebra of M × M one-circulant matrices over the field F is isomorphic to the algebra of polynomials in the ring F[z −1 ]/(z −M − 1). This isomorphism allows us to simplify the proofs of the relations using matrix notation instead of z-transforms. Therefore, we introduce a matrix representation to express the relations of the cyclic filter bank. In the analysis bank, the filtering operation of periodic signals followed by decimation by a factor of two can be described using 2-circulant matrices. In other words, let H0 : V → V0 and H1 : V → W0 be two linear transformations (M × N matrices in matrix notation) that project a codeword c(n) ∈ V onto two orthogonal spaces V0 and W0 with wavelet coefficients x0 (n) and x1 (n), respectively. Then we have: x0 (n) = x1 (n) =

N −1
i=0 N −1


c(i)h0 ((2n − i))N = (H0 c)(n) (3) c(i)h1 ((2n − i))N = (H1 c)(n),

i=0

in which H0 and H1 are 2-circulant matrices. By using (2), these matrices can be written as:   gj (0) gj (1) gj (2) · · · gj (N − 1)  gj (N − 2) gj (N − 1) gj (0) · · · gj (N − 3)    j = 0, 1. (4) Hj =   .. .. .. .. ..   . . . . . gj (3) gj (4) · · · gj (1) gj (2) Similarly, in the synthesis bank, the upsampling of periodic signals by a factor of two followed by the filtering operation can be described by 2-circulant matrices G0 and G1 : c(n) =

M −1


x0 (i)g0 ((n − 2i))N +

i=0

= (G0 x0 )(n) + (G1 x1 )(n).

M −1


x1 (i)g1 ((n − 2i))N

i=0

(5)

Because of the relation (2), the synthesis matrices turn out to be the transposes of the analysis matrices: Gj = HjT

j = 0, 1.

From the perfect reconstruction constraint we deduce that:  H0 = IN ×N . [H0T H1T ] H1

(6)

(7)

Furthermore, since the operator T = [H0T H1T ]T is a 1-to-1 mapping V → V0 ×W0 in the finite-dimensional vector space, consequently it is onto as well. Hence T T T = I implies that T T T = I (i.e, T is unitary). Therefore: Hj HjT = IM ×M

j = 0, 1

and

H0 H1T = 0M ×M .

(8)

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h 0(n)

2

h1(n)

2

x0(n)

x 1(n)

x0(n)

x 1(n)

Analysis Bank

2

g 0(n)

2

g 1(n)

+

c(n)

Synthesis Bank

Fig. 1. Diagram of the two-band filter bank.

3

Double Circulant Self-Dual Codes

Assume that C ∈ FN is a double circulant self-dual code. The double circulant property requires that if c(n) = {c(0), c(1), . . . , c(N − 1)} is a codeword in C, then c((n − 2))N is also in C. Let c(n) be a codeword that is decomposed by the analysis bank of Fig. 1 to its wavelet coefficients x0 (n) and x1 (n). Although we do not need the result of the following theorem in this paper, we quote the following theorem from [7] to maintain the generality of the discussion. Theorem 1. Suppose the codeword c(n) ∈ C corresponds to the message block m(n), then for any double circulant self-dual code C there exists a cyclic orthogonal wavelet transform that maps the codeword c(n) to the message block, i.e, x0 (n) = x1 (n) = m(n). 3.1

Encoder Structure

Figure 2a shows the encoder of a (N, M, d) code that maps the message block m(n) of size M = N/2 to the codeword c(n). The encoder is realized by the synthesis portion of the two-band filter bank in which go (n) and g1 (n) are an orthonormal wavelet basis over F. The parameter α ∈ F introduced in this structure will be specified later to meet the self-dual constraint. The delay z −l , 0 ≤ l ≤ M − 1 controls the minimum distance of the generated code, and will also be discussed later. From the linearity of the wavelet transform, it is obvious that the generated code is linear. Furthermore, if c(n) is a codeword associated with the message m(n), then by the property of the multirate filters, there exists a message datum m((n − 1))M that is mapped to the codeword c((n − 2))N . Therefore, the code is double circulant. It is also worth noting that using the combined scaling sequence and the mother wavelet (synthesis bank) as the encoder, guarantees that the mapping of the message m(n) onto the codeword c(n) is a one-to-one mapping. This is true because the message block can be extracted by filtering c(n) through h0 (n) and downsampling it by a factor of two. In the following, we show that regardless of the amount of the delay l, the encoder generates a double circulant self-dual code in F. Using the matrix notation that we developed for the cyclic filter bank in Section 2, the N × M generator matrix G of this code can be written as: G = G0 + G1 Πl ,

(9)

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in which Πl is an M × M monomial matrix, which is a permutation of the identity matrix if the field is GF (2r ). In fact Πl is a one-circulant matrix defined by its first row that is zero everywhere except the (l + 1) position. Due to the isomorphism of the algebra of one-circulant matrices and the algebra of polynomials, the following statements can be readily proved. First, one can show that ΠlT is also a one-circulant matrix whose first row is zero everywhere except at the (M − l + 1) position. Furthermore, it can be proved that ΠlT Πl = α2 IM ×M .

(10)

Now, recalling the necessary and sufficient condition for self-dual codes, it is deduced that the M columns of the generator matrix G that are linearly independent must also lie in the dual space. Since dim C = dim C ⊥ = M, then the columns of G specify a basis set for the dual space as well. Consequently the generator matrix and the parity-check matrix of the code are the same. From the above argument we conclude that the if and only if condition of the self-dual codes is equivalent to GT G = 0. Using (9), (6), (8), and (10), we derive the following equality for the generator matrix of the encoder (Fig. 2a): GT G = I + α2 I.

(11)

Consequently, to meet the self-dual condition, we require that α2 + 1 = 0 for α ∈ F. channel error: e(n) x0(n)

m(n)

αz

2

g 0(n)

+

c(n)

c(n)

+

h0(n)

2

-l

+

s(n)

α z -(M-l) x1(n) 2

g 1(n)

h1(n)

2

(b) Syndrome Generator

(a) Encoder Structure

Fig. 2. Filter bank structure of the encoder and syndrome generator.

Our previous discussion of self-dual codes is valid over any arbitrary finitefield. Now, let us study the (12,6,4) code as an example in the binary field. As explained in [5], we choose the mother wavelet to be g1 (n) = {100010010101}. Therefore, the scaling sequence and the analysis filters are obtained by the relation g0 ((n))N = g1 ((−n − 1))N and (2) as: g0 (n) = {101010010001} h0 (n) = {110001001010} h1 (n) = {110101001000}.

(12)

The parameter α in GF (2) is equal to one, and the choice of the delay l determines the minimum distance of the generated code. Choosing the delay l from

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the set {1, 2, 4, 5} generates a code with a minimum distance four, while the other choices reduce the minimum distance of the code to two. It is worth noting that the codes generated by different values of the delay from the set {1, 2, 4, 5} are all equivalent to each other. By using any delay value from this set, all of the codeword weights are a multiple of two and the weight numerator of the code is: A4 = 15 , A6 = 32 , A8 = 15 , A12 = 1,

(13)

in which Ai denotes the number of codewords of weight i. In the rest of the paper, we choose delay l to be one. 3.2

Syndrome Generator and Decoder Structure

In the following we show that the structure in Fig. 2b constructs the syndrome of the code. Again, by using the relations that we developed for cyclic filter banks, we write: s(n) = (H0 + ΠlT H1 )(c + e)(n), (14) in which e(n) is the error pattern due to the communication channel. Given the self-dual requirement α2 + 1 = 0, the above equality is simplified further by combining the equations (6), (8), and (10) as: s(n) = (H0 + ΠlT H1 )e(n).

(15)

Therefore, the output of the system in Fig. 2b depends only on the error pattern. This structure can be simplified as in Fig. 3 in which h2 (n) = h0 (n) + h1 ((n + 2l))N . Now, the remaining problem is to interpolate the low (M) dimensional syndrome s(n) into the higher (N ) dimensional error pattern e(n). This is a conditional minimum distance estimation problem in the signal processing context. It is conditional because more than one error pattern is mapped into the same low dimensional syndrome. Therefore, the interpolator should choose (out of those possible valid choices) the error pattern that is most likely (has minimum weight) to achieve the maximum likelihood ML decoder performance.

channel error: e(n) c(n)

+

s (n)

e(n) s(n) h2(n)

2

00

2

u (n)

2

u (n)

+

00

s(n)

z -1 01

s (n) 01

(a) Syndrome Generator

(b) Polyphase Structure of the Syndrome Generator

Fig. 3. Polyphase representation of the syndrome generator.

Since the estimator is a signal dependent operator, there is no single operator solution for this estimation problem. Hence, to keep the decoder as simple as

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possible, we design the interpolator to be exact for all the weight one error patterns. Therefore, like the ML decoder, this decoder is guaranteed to correct all errors of weight one. Our approach to design the decoder is based on inverting the polyphase filters of the syndrome generator filter h2 (n). Figure 3b shows the polyphase structure of the syndrome generator in which u00 (n) = h2 (2n) and u01 (n) = h2 (2n + 1) are the polyphase components of h2 (n). Furthermore, let r00 (n) and r01 (n) be two filters with the z-transform satisfying: R0i (z)U0i (z) = 1 mod (z −M − 1)

i = 0, 1.

(16)

In other words, these two filters are the circular inverses of u00 (n) and u01(n), respectively. Now, let us define two sets E0 and E1 by distinguishing those errors that occur only in the even time indexes from those occur in the odd time indexes, respectively: E0 = {e(n) = {ζ0 , 0, ζ1 , 0, · · · , ζM −1 , 0} where: ζi ∈ {0, 1} ∀ i} E1 = {e(n) = {0, η0 , 0, η1 , · · · , 0, ηM −1 } where: ηi ∈ {0, 1} ∀ i} .

(17)

If e(n) ∈ E0 , then s01 (n) = 0 and s(n) = s00 (n). Therefore, we are required to invert the filter u00 (n) to estimate the error. By this argument, we realize that whenever e(n) ∈ E0 , the output at position 1 in Fig. 4 is the exact interpolation of the syndrome to e(n).

r (n) 00

~

s(n)

e(n)

Weight Computation r (n) 01

c(n)+e(n)

1 2

2

z -(N-1)

+

m(n) h0(n)

2

2

Conditional Interpolator

Fig. 4. Filter structure to reconstruct the message sequence. Taking into the account that half of the weight one errors belong to the set E1 , we need to identify these cases and be able to correct these errors as well. To do that, let us define the set E11 as : E11 = {e(n) : e(n) ∈ E1 & wt(e) = 1}

(18)

in which wt(e) means the weight of the error. It is clear that if e(n) ∈ E11 , then s(n) = u01 ((n − n0 ))M (n0 depends on the location of the ’1’ in e(n)) which generates a weight five output at node 1. Hence, whenever the weight of the output at node 1 is five, we select the output at node 2 as a correct estimate of the error. It is worth noting that weight five outputs at node 1 are also produced by errors from the set E05 = {e(n) : e(n) ∈ E0 & wt(e) = 5}. Consequently, the correctable errors by the decoder of Fig. 4 are from the set E = {E0 ∪ E11 } − E05 .

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Table 1. The number of correctable errors (by error weight) by the filter decoder in Fig. 4 and the ML decoder. Weight of the Error Filter Method ML Decoder 1 12 12 2 15 31 3 20 20 4 15 6 1 -

Table 1 gives the correctable errors by the filtering technique and ML decoder. In the following we investigate the amount of extra signal-to-noise ratio (SNR) that is required by the filter method to achieve the same performance as the ML decoder. Suppose pb is the probability of a bit error and perr is the word error probability. Then

Nwe can determine the word error rate by applying the formula perr = 1 − i=0 βi pib (1 − pb )N −i in which βi is the total number of correctable errors of weight i (given in Table 1). To plot the word error rate as a function of SNR, we choose DPSK and noncoherent FSK detection methods in which pb is (1/2)e−SN R and (1/2)e−SN R/2 , respectively. These graphs show that a very subtle extra SNR is required by the filter decoding method to achieve the performance of the syndrome table lookup decoder. It is important to note that the (12,6,4) self-dual code has been studied only as an example, and the wavelet method described in this paper can be used to generate self-dual codes of any length. As a final remark we give the generator filters of the two double circulant even self-dual codes (8,4,4), and (24,12,8) [7]. The (8,4,4) code is generated by a cyclic orthogonal filter bank with the scaling sequence and the mother wavelet equal to g0 = {9D} and g1 = {4C}, respectively. Similarly, the cyclic orthogonal filter bank that generates the Golay code (24,12,8) is constructed by g0 = {A80011} and g1 = {40DD55} (with no need for delay element in Fig. 2a). Note that the filter coefficients in GF (2) are represented in hexadecimal form. Furthermore, there exist several other pairs of scaling and mother wavelets that generate equivalent codes [7].

4

Conclusion

We report a new approach to study and implement self-dual codes by using finitefield wavelet transforms. This method allows us to construct double circulant selfdual codes of arbitrary length over any finite-field in a straightforward manner. We also introduce a decoder based on a polyphase filter inversion methodology. This decoder achieves nearly the performance of the syndrome table lookup decoder. Our approach, in addition to being a powerful tool to investigate error correcting codes, reduces the complexity and computation costs in the encoder and decoder by the polyphase implementation of the multirate filters.

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Noncoherent FSK Method

0

word error rate

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−5

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−10

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8

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−5

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−10

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−15

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4

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6

7

8

9

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Fig. 5. Comparison of the word error rate of the Filter decoding technique with that of the ML decoder method in DPSK and noncoherent FSK.

References 1. S. Phoong, and P. P. Vaidyanathan, “Paraunitary Filter Banks Over Finite Fields”, IEEE Trans. Signal Proc., vol. 45, pp. 1443–1457, June 1997. 2. G. Caire, and R. L. Grossman, and H. V. Poor, “Wavelet Transforms Associated with Finite Cyclic Groups”, IEEE Trans. on Information Theory, vol. 39, pp. 1157–1166, July 1993. 3. T. Cooklev, and A. Nishihara, and M. Sablatash, “Theory of Filter Banks over Finite Fields”, Proc. Asia Pacific conf. Circuits Syst., pp. 260–265, Taipei Taiwan, 1994. 4. F. Fekri, “Transform Representation of Finite Field Signals”, A qualifying examination report available at http://www.ee.gatech.edu/users/fekri, Georgia Institute of Technology, June, 1998. 5. F. Fekri, R. M. Mersereau, and R. W. Schafer,“Theory of Wavelet Transforms over Finite Fields”, Proc. Int. Conf. Acoust. Speech, and Signal proc., 605–608, March 1999. 6. F. J. MacWilliams, and N. J. A. Sloane, “The Theory of Error Correcting Codes, North-Holland publishing company, 1977. 7. F. Fekri, S. W. McLaughlin, R. M. Mersereau, and R. W. Schafer, “Error Control Coding Using Finite Field Wavelet Transforms, Part III: Double Circulant Codes”, in preparation.

Linear Codes and Polylinear Recurrences over Finite Rings and Modules (A Survey) V.L. Kurakin, A.S. Kuzmin, V.T. Markov, A.V. Mikhalev, and A.A. Nechaev Department of Mechanics and Mathematics and Center of New Information Technologies of Moscow State University

Abstract. We give a short survey of the results obtained in the last several decades that develop the theory of linear codes and polylinear recurrences over finite rings and modules following the well-known results on codes and polylinear recurrences over finite fields. The first direction contains the general results of theory of linear codes, including: the concepts of a reciprocal code and the MacWilliams identity; comparison of linear code properties over fields and over modules; study of weight functions on finite modules, that generalize in some natural way the Hamming weight on a finite field; the ways of representation of codes over fields by linear codes over modules. The second one develops the general theory of polylinear recurrences; describes the algebraic relations between the families of linear recurrent sequences and their periodic properties; studies the ways of obtaining “good” pseudorandom sequences from them. The interaction of these two directions leads to the results on the representation of linear codes by polylinear recurrences and to the constructions of recursive MDS-codes. The common algebraic foundation for the effective development of both directions is the Morita duality theory based on the concept of a quasi-Frobenius module.

Introduction The theory of linear codes over finite fields is a well-established part of discrete mathematics. The highlights of this theory include concepts of dimension, duality, weight enumerators and their generalizations, MacWilliams identities, cyclic codes, etc... [1,21,43,63,73]. The theory of linear recurrences (LR) over fields is also well developed; it has rather more ancient historical roots and important applications in various areas of mathematics (theory of algorithms, Monte Carlo method, cryptography). In particular, the analytical formulae for the general member of an LR were deduced, as well as the algebraic relations between the families of LR, cyclic types of such families over finite fields, the distribution laws for elements on the cycles of some LR and the ways of obtaining from them pseudorandom sequences (see [79,41,33,36] and the sources cited there). We point out also some papers on the properties of polylinear sequences over Galois fields: [44,61,65,66,67,68]. The attempts to extend the mentioned results to some class of finite rings more general than that of fields for long time dealt mostly with integral residue Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 365–390, 1999. c Springer-Verlag Berlin Heidelberg 1999


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rings [2,3,4,6,71,72,74,75], and in the last years especially with the ring Z4 (see e.g. [46,47,20,7,30,57,76]). The last works finally proved the actuality and importance of the investigations of linear codes and polylinear recurrences over arbitrary finite rings and modules that became so active in ’70-’90th [14,22,23,24,26,27,28,32,33,36,39,45], [48,49,52,53,54,55,58,59,60,70,77,78,62]. It turned out that sufficiently deep generalization requires not the absence of zero divisors in the base ring but the so called double annihilator relations in the module (see below p.1.1). This leads to the concepts of quasi-Frobenius and reciprocal module, providing a nice way to introduce for linear codes over an arbitrary finite module the main concepts of coding theory over fields so that its fundamental results remain valid. For instance, the parity check matrix and the dual code are defined over the reciprocal module (having the same number of elements as the given one) and the MacWiliams identities for complete weight enumerators remain true. We apologize for the brevity of our further comments caused by the space limits which were rather severe for a paper of such kind. Due to the same cause, the reference list is not exhaustive. We only tried to cite all the specialists that worked in the area during the last several years. We apologize in advance to the authors whose names may be missed: please consider it not as a malicious intention but as the sign of our insufficient competence. More complete reference lists could be found in e.g. [33,36,45,60]. We consider the following topics which are comparatively new: 1. General theory of linear codes over finite modules. 2. The comparison of linear codes over finite modules, linear spaces and fields. 3. Weight-functions on finite modules as a generalization of Hamming weight. 4. Presentations of codes over fields by linear codes over modules. 5. General theory of polylinear recurrences over finite modules. 6. Presentations of linear codes over modules by polylinear recurrences. 7. Recursive and linear recursive MDS-codes. 8. Periodic properties of polylinear recurrences over finite modules. 9. Pseudorandom sequences generated by polylinear recurrences.

1

General Theory of Linear Codes over Modules

Let R be a finite commutative ring with identity e, and R M a finite faithful module. Any submodule K < R M n is called a linear n-code over R M, its Hamming distance defined as d(K) = min{α : α ∈ K \ 0}, where α is Hamming weight of α. We choose as the main criterion of correctness for the theory of such codes the presence of notions of the parity-check matrix and the code Ko dual to the given code K, defined in such a way that in particular the equality Koo = K and the MacWilliams identity for complete weight enumerators of codes K and Ko should be valid. Consider the following fundamental example. Suppose we study linear codes L < R Rn over a ring R, and we define the dual code Lo in the usual way as Lo = {β ∈ Rn : βL = 0}. Then Loo ⊇ L, but the equality Loo = L is

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guaranteed if and only if R is a quasi-Frobenius ring [52,54] (for example principal ideal rings and in particular R = Zm are quasi-Frobenius). In such cases the codes over R can be studied without codes over modules, as it was done in [3,4,7,27,69,71,72]. On the other hand, if the ring R is not quasi-Frobenius then for deriving deep enough results the dual code should be built not over R but over the corresponding quasi-Frobenius module R Q (see below) that has the same number of elements as R. In its turn, the code dual to a linear code over R Q is built over the ring R. In the most general case, while studying the linear codes over arbitrary finite module R M, to obtain the results close enough to that of the theory of linear codes over fields, the code dual to the given one must be defined over the module ∗ R M , which is Morita dual to R M. Now we pass to the exact statements. 1.1. Quasi-Frobenius Modules and the Reciprocal Modules. For any ideal I R and any submodule K < R M we define their annihilators, correspondingly in M and R, by the equalities AnnM (I) = {α ∈ M : Iα = 0} < R M; AnnR (K) = {r ∈ R : rK = 0} R. A module R M is called quasi-Frobenius (a QF -module), if AnnR (AnnM (I)) = I and AnnM (AnnR (K)) = K for all I R and K < R M. For any finite commutative ring R there exists a unique (up to isomorphism) QF -module R Q [18,52]. It might be described as the character group Q = Hom(R, Q/Z) of the group (R, +), where the product rω ∈ Q of an element ω ∈ Q by an element r ∈ R is defined by the condition rω(a) = ω(ra) for any a ∈ R. We have (Q, +) ∼ = (R, +) and |Q| = |R|. A ring R is called quasi-Frobenius, if R R is a QF-module. Now, as we continue to discuss the example given in the Introduction, define the product of the row a = (a1 , . . . , an ) ∈ Rn by the row α = (α1 , . . . , αn ) ∈ Qn as aα = a1 α1 + . . . + an αn ∈ Q, and say the the code L0 = {α ∈ Qn : Lα = 0} over R Q is dual to the linear code L < Rn , while the code K0 = {a ∈ Rn : aL = 0} over R is dual to the linear code K < R Qn . Then, in particular, we have L00 = L, K00 = K, |L||L0 | = |K||Ko| = |R|n = |Q|n [52,54]. This construction is generalized to the linear codes over an arbitrary finite module R M using the following concept. We call the module R M ∗ = HomR (M, Q) of all homomorphisms R M → Q reciprocal to R M (or Morita-dual to R M). It may be presented also as R ˙ + α ˙ t be a direct sum Hom(M, Q/Z) in the following way. Let M = α1 +... of cyclic groups. Then any ϕ ∈ M ∗ generates t characters ϕ(αi ) = ωi ∈ Q =
∈ Hom(M, Q/Z) such that Hom(R, Q/Z), i ∈ 1, t, and unique character ϕ
induces an isomorphism of Rϕ(α
i ) = ωi (e), i ∈ 1, t. The correspondence ϕ → ϕ modules M ∗ → Hom(M, Q/Z). It is important to note that (M ∗ , +) ∼ = (M, +), so in particular M and M ∗ are equivalent alphabets. Let us define the product of α ∈ M by ϕ ∈ M ∗ as ϕα = ϕ(α) ∈ Q. Then for a fixed α ∈ M the correspondence ϕ → ϕα induces a homomorphism R M ∗ → Q belonging to M ∗∗ = HomR (M ∗ , Q). We identify this homomorphism with α, obtaining equality M ∗∗ = M. Note that if R M = R Q is a QF-module over R then the R-module M ∗ = Q∗ = HomR (Q, Q) is isomorphic to R [18,52]. In particular if R is a QF-ring and M = R, then there are natural identifications: Q = R and M ∗ = M = R.

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1.2. The Dual Code and the Parity-Check Matrix [55]. Let now K 1 then there exist linear codes over L which are better than linear codes over GF (q). Let BL (n, 3) (Bq (n, 3)) be the maximum of the cardinalities of linear n-codes over (L, +) (over GF (q)) with the distance 3. r

r

−1 −1 < n ≤ pδ qq−1 − (pδ − 1) for some k ≥ 2 and Proposition 7 ([60]) If pδ−1 qq−1 t−δ δ ∈ 1, t − 1, then BL (n, 3) = p Bq (n, 3).

It is a generalization of earlier result of [25] for some cases when p = 2. The attempts to build linear codes over modules which are better than linear codes over linear spaces unexpectedly failed. We say that a n-code K over M is majored by a n-code L over some alphabet L if |L| = |M|, |L| ≥ |K| and d(L) ≥ d(K). Theorem 8 ([55]) Let R M be a module over a local ring R, and let R¯ L be a ¯ = R/N. Then any linear space of cardinality |L| = |M| over the residue field R linear code K < R M n is majored by some linear code L < R¯ Ln . If M is a finite abelian group and L is a direct sum of elementary abelian groups of cardinality |L| = |M|, then any linear n-code over M is majored by some linear n-code over L. See also p. 7.2 below about the Asturian code.

3

Weight Functions on Finite Modules

The investigation of linear codes over modules is not so important for construction of codes which are better than codes over fields as for description of new linear representations of these codes. In [46,47] it was discovered that certain nonlinear binary codes (Kerdock codes) can be represented as linear codes over Z4 . Later in [20] a variant of this representation (the so called Gray map) was found which gives an isometry between the Lee-metric space Zn4 and the Hammingmetric space F2n 2 . Meanwhile in [37,38,56] a generalized Kerdock code over Fq , q = 2l , was built as a representation of some linear code over the Galois ring GR(q 2 , 4). Independently, in [8,9] the so-called homogeneous weight on the residue class ring Zm was introduced, and the resulting metric was characterized by algebraic-information-theoretic properties. With a suitable generalization of this weight to the ring GR(q 2 , p2 ) the Reed–Solomon map [56] isometrically embeds GR(q 2 , p2 )n into the space (Fnq q , wHam ). So the notion of homogeneous weight is closely related with the representations of codes over fields by linear codes over modules. Here we present the results of [22,24], where general notion of homogeneous weight on finite modules M over arbitrary finite (possibly noncommutative) rings R was introduced and investigated. A function w : M → R is called a weight if (W1) ∀x ∈ M : w(x) ≥ 0, w(x) = 0 ⇔ x = 0; (W2) ∀x ∈ M : w(x) = w(−x); (W3) ∀x, y ∈ M : w(x + y) ≤ w(x) + w(y).

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For any weight w : M → R the function ρw (x, y) = w(x−y) defines a translationinvariant metric on M, and every translation-invariant metric ρ on M arises in this way from the weight wρ (x) = ρ(x, 0). We call a function w : M → R egalitarian,  if (H1) there exists ζ ∈ R such that x∈U w(x) = ζ · |U| for any nonzero submodule U ≤ M. This function is called homogeneous if in addition (H2) ∀x ∈ M, ∀u ∈ R∗ : w(x) = w(ux). A module R M is called weighted if it admits an egalitarian weight w. In this case it admits also a homogeneous weight: w∗ (x) = |R∗ |−1 · u∈R∗ w(ux). Note that Hamming weight w = wHam on R M is homogeneous if and only if the module R M is simple. In [9] the following motivation for introducing the homogeneity axiom (H1) was given. For an arbitrary weight w on R M and n ∈ N the weight wn : R M n → R defined by wn (x) = w(x1 ) + · · · + w(xn ) for x = (x1 , . . . , xn ) ∈ M n turns M n into a translation-invariant metric space. Let now K be a linear code over R M, i. e. a submodule of R M n . Then the projection Ki of K onto the i-th coordinate is a submodule of R M. For information-theoretic purposes it is natural to require that  Ki = 0 for every i (i. e., K is a full-length code) and that the numbers Wi = x∈K w(xi ) satisfy the condition W1 = W2 = · · · = Wn The second condition holds for full-length linear codes over fields. It is satisfied by every full-length linear code over R M if and only if w satisfies (H1). We give the full description of weighted modules. Let N(R) be the nilradical of the ring R. The quotient ring R = R/N(R) has an Artin-Wedderburn-Molien decomposition R = R1 ⊕ R2 ⊕ · · · ⊕ Rt ,

(3.1)

where each ideal Rj is a simple subring of R. Hence there exist positive integers mj and prime powers qj (j ∈ 1, t) such that Rj is isomorphic to the matrix ring Mmj (Fqj ). The socle S(R M) of R M (p. 2.1), is an R-module, and S(M) = S1 ⊕ S2 ⊕ · · · ⊕ St ,

Sj = Rj M.

(3.2)

Theorem 9 A module R M is weighted if and only if S(M) is a cyclic R-module (i.e. nj ≤ mj for j ∈ 1, t) and the decomposition (3.2) does not contain F2 ⊕ F2 or F2 ⊕ F3 as a direct summand. Corollary 10 (a) A finite abelian group of order m is weighted if and only if it is cyclic and m ≡ 0( mod 6). (b) A faithful module R M over a finite commutative local ring is weighted if and only if it is a QF-module. The Corollary 10(a) implies that the Constantinescu—Heise criterion [9] is true not only for cyclic groups but for all finite abelian groups. A finite ring R is called a Frobenius ring if R R ∼ = S(R R) and RR ∼ = S(RR ).

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Theorem 11 For a finite ring R both modules R R and RR are weighted if and only if R is a Frobenius ring and the decomposition (3.1) does not contain F2 ⊕F2 or F2 ⊕ F3 . In this case the left and right homogeneous weights on R coincide. We denote by FR the class of all finite R-modules and define the Euler obius function function R : FR → N as R (M) = |{x ∈ M | M = Rx}|and the M¨ µR : FR → Z by the recursion formulae: µR (0) = 1, U ≤M µR (U) = 0 if M ∈ FR \ 0. Theorem 12 For a weighted module R M there exists the unique homogeneous  weight w = wh (x) such that x∈U wh (x) = |U| for any nonzero submodule U ≤ M. This weight has the form wh (x) = 1 −

4

µR (Rx) R (Rx)

for all

x ∈ M.

(3.3)

Scaled Isometries and Presentation of Codes

We describe here a general enough technique, based on the concept of scaled isometry, providing construction of presentations of linear codes over weighted modules, and give some examples of efficient applications of this technique. For a weighted module R M ∈ FR we fix someegalitarian weight wR . It is extended n n n to wR : M n → R by setting wR (x) = i=1 wR (xi ), and generates a metric n n n ρR (x, y) = wR (x − y) on M . Let now S N be another weighted module over some ring S with an egalitarian weight wS . Suppose that for some d ∈ N and ζ ∈ R \ 0 there exists a mapping σ : M → N d satisfying   ∀a, b ∈ M : ρdS σ(a), σ(b) = ζ · ρR (a, b). (4.1) We call σ an isometry with scale factor ζ or, for short, a scaled isometry. It induces for every n ∈ N a scaled isometry σ n : (M n , ρnR ) → (N dn , ρnd S ) with the same scale factor. With every code C ⊆ M n we associate the code C = σ n (C) ⊆ N nd and call C a σ-representation of the code C. Note that if C is distance invariant (relative to the metric ρnR ) then such is C (relative to the metric ρnd S ). If C is a linear code over R M, we call C a σ-linear code (and sometimes briefly an R M-linear code). An R M-linear code C is distance invariant but may be nonlinear. This approach allows to construct some new good codes and to find new compact representations of some well-known codes. In [20] an isometry between (Z4 , ρZ4 ) and (F22 , ρHam ) was rediscovered (the so-called Gray mapping Φ : Z4 → F22 , 0 → 00, 1 → 01, 2 → 11, 3 → 10), and the term Z4 -linear code was introduced for what we call a Φ-linear code. This approach allows to repeat the proof of Z4 -linearity of binary Kerdock code [46,47] and to prove the Z4 -linearity of Preparata, Delsarte-Goethals and some other codes. The more general form of this mapping for Galois rings in [56,34] gives 4.1. A Generalized Kerdock Code over Fq , q = 2l . Let R = GR(q 2 , 4) be the Galois ring of characteristic 4 and cardinality q 2 , q = 2l , l ≥ 1. A generalized

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Kerdock code Kq (m + 1) over Fq (m is odd) is a Reed-Solomon presentation of the so called base linear code KR (m) over the ring R. Let S = GR(q 2m , 4) be an extension of degree m of the ring R. The set m Γ (S) = {β ∈ S : β q = β} is closed under multiplication and consists of q m elements. Any element β ∈ S can be written uniquely as the sum β = β0 + 2β1 , where βt = γt (β) ∈ Γ (S), t = 0, 1. If we define ⊕ on Γ (S) by the rule u ⊕ v = γ0 (u + v) then (Γ (S), ⊕, ·) is Fqm and Γ (R) = {β ∈ R : β q = β} is the subfield Fq of Γ (S). Let θ be a primitive element of the field Γ (S). The base code KR (m) is defined as a linear code of length h = q m over R consisting of all words v = (v(0) . . . v(h − 1)) such that for some ξ ∈ S, c ∈ R S (ξθi ) + c, i = 0, h − 2, v(i) = T rR

v(h − 1) = c, (4.2)  S S (x) is the trace-function from S onto R: T rR (x) = σ σ(x), σ spans where T rR the group of automorphisms of S over R. Let now Γ (R) = {ω0 = 0, ω1 = e,. . . , ωq−1 }. Define γ∗ : R → Γ (R)q on elements r = r0 +2r1 ∈ R as γ∗ (r) = (r1 , r1 ⊕ω1 r0 , . . . , r1 ⊕ωq−1 r0 ). Then γ∗ (R) is a Reed–Solomon [q, 2, q − 1]q -code over Fq and therefore the mapping γ∗ (R) is called the RS-mapping [56]. It is easy to see that γ∗ (R) is a scaled isometry of the space R with the homogeneous weight into the Hamming space Γ (R)q . The code Kq (m + 1) is a concatenation of the code KR (m) (linear over R) and the code γ∗ (R) (linear over Γ (R)). It is the code of length n = q m+1 , consisting of all words γ∗h (u) = (γ∗ (u(0)), . . . , γ∗ (u(h − 1)), u ∈ KR (m). If q = 2, i.e. R = Z4 , this code is the original binary Kerdock code [1-5,7,15]. Theorem 13 ([34]) Let m = 2λ + 1 ≥ 3. Then the code Kq (m + 1) is √ an R-linear (n, n2 , q−1 n))q -code with the complete weight enumerator q (n − WKq (m+1) (x0 , ..., xq−1 ) = q−1 q−1 q−1   n/q 1  n −qλ q−1  qλ+1 xnj + (q m+2 − q) xj + q(q m − 1)(q m + q λ+1 ) xjq xj + 2 j=0

j=0

q−1 

λ n q +q

1 + q(q m − 1)(q m − q λ+1 ) xj 2 j=0

q−1  j=0

j=0

x−q j

λ+1

j=0

.

4.2. Presentations of the Extended Binary Golay Code [24,22]. It can be presented as a linear code over the ring R = F2 ⊕ F4 . Note that smaller rings of such form (F2 ⊕ F2 , F2 ⊕ F3 ) are not weighted. Proposition 14 Let e = e1 + e2 be the corresponding decomposition of the identity of R, and F4 = F2 (α). Let σ : R → F32 be the F2 -linear map defined by e1 → 111, e2 → 110, α → 011 and K ≤ R M 8 be the R-linear code with parity-check matrix   e1 e2 e e e 0 0 0  e e1 e2 e 0 e 0 0   (4.3)  e e e1 e2 0 0 e 0 . e2 e e e1 0 0 0 e

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Then the mapping σ is a scaled isometry from (R, ρR ) onto (F32 , ρ3Ham ) with the scale factor 32 , and the code σ 8 (K) is a linear binary (Golay) [24, 12, 8]-code. Another isometric representation of the same code is based on some egalitarian, but not homogeneous weight. Let R = F2 [x]/(x4 ) = F2 [z], where z = x+(x4 ) is the image of x in F2 [x]/(x4 ). Every a ∈ R has the unique representation a = a0 + a1 z + a2 z 2 + a3 z 3 with aj ∈ F2 . Define τ : R → F42 and w : R →  R by  setting τ (a) = (a0 + a1 + a2 + a3 , a1 + a3 , a2 + a3 , a3 ) and w(a) = wHam τ (a) . The function w is an egalitarian weight on R R. Proposition 15 Let K ≤ R R6 be the linear code with parity-check matrix   100vzz 0 1 0 z v z , (4.4) 001zzv where v = 1 + z 3 . The code τ 6 (K) is the linear (Golay) [24, 12, 8]-code over F2 . 4.3. Presentation of Ternary Golay Code [22]. Let R = F3 [x]/(x3 ) = F3 [z] with z = x + (x3 ). Then any a ∈ R has the unique representation a = a0 + a1 z + a2 z 2 with aj ∈ F3 . Let now σ : R → F33 be defined by σ(a) = 3 (a0 − a1 + a2 , a1 + a2 , a2 ). Then w : R → R, defined by w(a) = wHam σ(a) , is an egalitarian weight on R R. Proposition 16 Let K ≤ R R4 be the linear code with parity-check matrix   1 0 v v2 , (4.5) 0 1 v 2 −v where v = 1 + z 2 . The code σ 4 (K) is a linear (Golay) [12, 6, 6]-code over F3 . 4.4. Scaled Isometry over a Commutative QF-Ring [24,22]. Let now R be a finite commutative ring with the unique minimal ideal S, i. e. R is a local quasi-Frobenius ring with soc R = S. We construct a scaled isometry from the weighted R-module R R into a suitable Hamming space Fnq . Let J = rad R and R = R/J ∼ = Fq . The set Γ (R) = {a ∈ R | aq = a} is a full system of representatives for R, and thus has a natural field structure Γ (R), ⊕, ·). There exists a system of elements π0 , . . . , πl ∈ R such that πl is a generator of S and every x ∈ R has the unique representation x = a0 π0 + · · · + al πl

with ai ∈ Γ (R) for i ∈ 0, l.

(4.6)

We fix such system (π0 , . . . , πl ) and define the functions γi : R → Γ (R) by setting l x = i=0 γi (x)πi . Consider ((l + 1) × q l )-matrix G(l, q) with entries from Γ (R) whose columns are the vectors (a0 , . . . , al−1 , 1), (a0 , . . . , al−1 ) ∈ Γ (R)l , in some fixed order. The q-ary linear [q l , l + 1]-code with generator matrix G(l, q) is the generalized Reed–Muller code C = GRM(l, 1, q), cf. [21, Ch. 9.5]. It is a twoweight code with nonzero weights q l − q l−1 and q l .

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l

Proposition 17 The mapping σ : R → Γ (R)q , defined by   σ(x) = γ0 (x), γ1 (x), . . . , γl (x) · G(l, q),

(4.7)

a generalized is a scaled isometry with scale factor q l − q l−1 from (R, ρR ) onto   Reed–Muller [q l , l, q l − q l−1 ]-code GRM(l, 1, q), over the field Γ (R), ⊕, · . Some particular cases of this result can be found in [33,56,9]. For a generalization to arbitrary finite commutative local rings (using the notion of a quasi-Frobenius module) we refer to [24]. About linearly representable codes over chain rings see also [23]. We have also to mention here the works [70,78] concerning the extendibility of code isometries.

5

General Theory of Polylinear Recurring Sequences

As it was already pointed out in the Introduction, the theory of (poly)linear recurrences over modules was actively developing recently due to various applications, in particular, in coding theory. It is worth noting that, as in previous sections, quasi-Frobenius modules play special role. Here we use the results of [33,36] and of works cited therein, in particular [50,54,61,65,66,67,68,74,75,79]. 5.1. Main Concepts. For k ∈ N, we call any function µ : Nk0 → M a k-sequence over a module R M. We write µ = µ(z), where z = (z1 , . . . , zk ) is the row of free variables over N0 . The set M k of all k-sequences over M is an R-module relative to the usual operations on functions. Let Pk = R[x1 , . . . , xk ] = R[x] be the polynomial ring of k variables. For any s = (s1 , . . . , sk ) ∈ Nk0 denote the monomial xs11 . . . xskk by xs . Then any F (x) ∈ Pk has the form F (x) =  s the structure of a Pk -module on M k by F (x)µ = ν, ν∈ s fs x . We define 

k M , ν(z) = s fs µ(z + s). An ideal I of Pk is called monic if there exist monic polynomials F1 (x), . . . , Fk (x) ∈ R[x] (of one variable) such that F1 (x1 ), . . . , Fk (xk ) ∈ I.

(5.1)

If I is a monic ideal, then the quotient ring Pk /I is finite, and vice versa. The annihilator Ann(M) = AnnPk (M) = {F (x) ∈ Pk : F (x)M = 0} of any subset M ⊂ M k in the ring Pk is an ideal of Pk . We say that a sequence µ ∈ M k is a k-linear recurring sequence (k-LRS) over a module M if I = Ann(µ) is a monic ideal. In this case, polynomials (5.1) are called elementary characteristic polynomials of the k-LRS µ. Proposition 18 The set LM k of all k-LRS µ ∈ M k is a submodule of the Pk -module M k . For any subset I ⊂ Pk the set LM (I) = {µ ∈ M k | Iµ = 0}

k is also a submodule of this module. Moreover, LM (I) ⊂ LM if and only if Pk (I) is a monic ideal of Pk .

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For a monic ideal I Pk the set LM (I) is called a k-LRS-family over R M. Since I · LM (I) = 0 the Pk -module LM (I) may be considered as a module over the ring S = Pk /I = R[θ1 , . . . , θk ], where θs = xs + I. This ring will be called the operator ring of the ideal I (of the family LM (I)). If I = Ann(µ) for some µ ∈ M k , then S is said to be the operator ring of the k-sequence µ. 5.2. Relations between LRS-Families and their Annihilators. The following relations between 1-LRS families over a field P are well-known. For any monic polynomials F (x), G(x) ∈ P = P [x] LP (F ) + LP (G) = LP ([F, G]); LP (F ) ∩ LP (G) = LP ((F, G)). Any 1-LRS family M over the field P has the form M = LP (F ) for some monic polynomial F (x) ∈ P and is a cyclic P-module: M = PeF . For any u, v ∈ LP 1 v ∈ Pu ⇔ Mv (x)|Mu (x), where Mu (x) is the minimal (characteristic) polynomial of the LRS u. Any monic ideal I ⊆ P is the annihilator of some LRS over P . The attempts to generalized these results on k-LRS families over finite module R M gives the following results. Let Ak = Ak (R) be the set of all monic ideals of the ring Pk = R[x1 , . . . , xk ] and let Mk = Mk (M) be the set of all finite Pk -submodules of the module M k . In this case any element M ∈ Mk is a submodule of LM k and we have a pair of the Galois correspondences Ann : Mk → Ak ,

LM : Ak → Mk .

(5.2)

This means that for any M ∈ Mk , I ∈ Ak we get M ⊆ LM (Ann(M)),

I ⊆ Ann(LM (I)).

(5.3)

These inclusions are strict in general. Moreover we may state that Ann(M1 + M2 ) = Ann(M1 ) ∩ Ann(M2 ); LM (I1 + I2 ) = LM (I1 ) ∩ LM (I2 ); Ann(M1 ∩M2 ) ⊇ Ann(M1 )+Ann(M2 ); LM (I1 ∩I2 ) ⊇ LM (I1 )+LM (I2 ); (5.4) for any ideals I1 , I2 ∈ Ak and modules M1 , M2 ∈ Mk . The inclusions (5.4) are also strict in general, but in particular, we get Proposition 19 Let I1 , I2 be comaximal ideals of Pk . Then LM (I1 ∩ I2 ) = ·

LM (I1 ) + LM (I2 ) (a direct sum), and it is a cyclic Pk -module if and only if the modules LM (Is ), s = 1, 2, are cyclic. If M = R is a field and k = 1 then the correspondences (5.2) are bijections, and the inclusions (5.3),(5.4) are equalities. The finite modules satisfying these conditions (i.e. admitting the theory of LRS analogous to the theory of LRS over a field) are described as in section 1.

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Theorem 20 ([50,52,54]) The following conditions are equivalent: (a) R M is a QF-module; (b) the Galois correspondences (5.2) are bijective; (c) for any monic ideal I ⊆ Pk the family LM (I) is a QF-module over the ring S = Pk /I, and any module M ∈ Mk is a QF-module over the ring S = Pk / Ann(M); (d) the inclusions (5.4) are equalities; (e) for any recurrences µ, ν ∈ LM k we have ν ∈ Pk µ ⇔ Ann(µ) ⊂ Ann(ν). The following essential supplement to the last Theorem shows an interesting connection between cyclic LRS-families and QF-rings. Theorem 21 ([50,52,54]) Let R Q be a quasi-Frobenius module. Then for any monic ideal I ⊆ Pk the following conditions are equivalent: (a) I = Ann(µ) for some recurrence µ ∈ LQ k ; (b) M = LQ (I) is a cyclic Pk -module; (c) S = Pk /I is a quasi-Frobenius ring.

Corollary 22 Let F1 (x1 ), . . . , Fk (xk ) be monic polynomials from Pk and S = Pk /(F1 (x1 ), . . . , Fk (xk )). Then S is a QF-ring if and only if R is a QF-ring. Theorem 20(c) allows us to build the QF-module over any finite commutative ring S as a k-LRS family over some principal ideal ring. Really the ring S is an extension S = R[π1 , . . . , πk ] of a principal ideal subring R ≤ S [29]. Then S = Pk /I for some monic ideal I ≤ R[x1 , . . . , xk ]. Since R is a QF-ring, Theorem 20(c) implies that LR (I) is the required QF-module over S. 5.3. The Berlekamp—Massey Algorithm. Let R M be a finite left module over a finite (not necessary commutative) ring R. We say that a polynomial F (x) = xs − cs−1 xs−1 − ... − c1x − c0 ∈ R[x] generates (from the left) a sequence u(0, l − 1) = (u(0), u(1), ..., u(l − 1)) ∈ M l of length l if s ≥ l or s < l and u(i + s) = cs−1 u(i + s − 1) + ... + c1 u(i + 1) + c0 u(i),

i ∈ 0, l − s − 1.

A monic polynomial of the smallest degree which generates u(0, l − 1) is called its (left) minimal polynomial. The Berlekamp—Massey algorithm finds a minimal polynomial of a sequence of lengths l with complexity O(l2 ) operations of R and M. For the first time the Berlekamp—Massey algorithm was described for sequences over fields in [1] and [42]. Since then many versions of this algorithm over fields and rings were proposed (see, e.g., [19,64,68], and bibliography in [32]). In the presented general form the Berlekamp—Massey algorithm was built in [32].

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Presentations of Linear Codes by Polylinear Recurrences

Some linear codes over a finite module R M (and all linear codes over any QF module R Q !) may be described in terms of polylinear recurrences over R M. Any finite subset F = {i1 , ..., in } ⊆ Nk0

(6.1)

F

is called a polyhedron. Let M be the R-module of all functions δ : F → M. Any such function is uniquely determined by its valuation diagram δ[F ] = (δ(i1 ), ..., δ(in )) ∈ M n . It is clear that the module R M F is isomorphic to R M n . Of course for any k-sequence µ ∈ M we may also consider the valuation diagram µ[F ] = (µ(i1 ), ..., µ(iN )). Let now I be a monic ideal of Pk and K = LF M (I) = {µ[F ] : µ ∈ LM (I)}.

(6.2)

Evidently, K is a submodule of the R-module R M F and by the indexing (6.1) we may consider K as a submodule of R M n . Thus K is a linear n-code over R M. In general not every linear code over R M has the form (6.2), but at the same time we have Proposition 23 [60]. Let R Q be a QF -module. Then for any linear code K < there exist a parameter k ∈ 1, n, a polyhedron F ⊂ Nk0 of cardinality n and a monic ideal I Pk such that

n RQ

K = LF Q (I).

(6.3) n

It is interesting to find the smallest k such that the code K < R Q has the representation (6.3) with the additional condition: F is a Ferrers diagram, i.e. if i ∈ F , j ∈ Nk0 and j ≤ i (in each coordinate) then j ∈ F . Let us call a code K < R M n recursive of dimension k (or k-dim recursive), if it has a presentation (6.2) for some ideal I Pk , Ferrers diagram F ⊆ Nk0 and the ordering (6.1) of their elements. The minimal k with this property will be called the recursive dimension of the code K. First important class of recursive codes gives Theorem 24 (Asturian Theorem [60].) Let K < R M n be a systematic linear code of the rank m with a parity-check matrix  (1)  (1) f0 . . . fm−1 −e 0 . . . 0  (2)  (2) f0 . . . fm−1 0 −e . . . 0    H = . , k = n − m. .. .. . . ..  . .  . . . .  (k)

f0

(k)

. . . fm−1 0

0 . . . −e

k×n

Then K is a k-dim recursive code satisfying the equality (6.2), where I Pk is (1) m−1 − ... − the ideal generated by the polynomials F1 (x1 ) = xm 1 − fm−1 x1 (1) (1) (2) (2) m−1 − . . . − f0 , · · · , Fk (x1 , xk ) = xk − f1 x1 − f0 , F2 (x1 , x2 ) = x2 − fm−1 x1

fm−1 xm−1 − . . . − f0 ; and F ⊂ Nk0 1 {0, e1 , 2e1 , . . . , me1 , e2 , . . . , el }. (k)

(k)

is Ferrers-diagram of the form F =

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379

Recursive and Linear Recursive MDS Codes

The connection between MDS-codes, Latin squares and quasigroups is wellknown. We present here the results of [10,11,12] where these objects are studied under the additional practically useful condition of recursivity for the corresponding code. 7.1. Quality Parameters of Recursive Codes. A code K ⊂ Ω n in an alphabet Ω of q elements is called k-recursive, 1 ≤ k < n, if there exists a function f : Ω k → Ω such that K = K(n, f) consists of all the rows u(0, n − 1) = (u(0), . . . , u(n − 1)) ∈ Ω n with the property u(i + k) = f(u(i, i + k − 1)),

i ∈ 0, n − k − 1.

In other words, K is the set of all output n-sequences of a feedback shift register with a feedback function f. In [10] the MDS-codes of such type, i.e. recursive [n, k, n − k + 1]q -codes are investigated, and the following parameters are considered: n(k, q) – the maximum of lengths of MDS codes K of (combinatorial) dimension k (|K| = q k ) in alphabet Ω of cardinality q; r n (k, q) – the maximum of lengths of k-recursive MDS codes of the same type; l(k, q) – the maximum of lengths of MDS codes K of (combinatorial) dimension k which are linear over an Abelian group (Ω, +) for some operation + (i.e. K is a subgroup of the Abelian group Ω n and |K| = q k , where q = |Ω| and n ∈ N). We also call such codes linear in wide sense. lr (k, q) – the analog of l(k, q) for recursive codes. For the primary (the power of a prime) numbers q one can also define m(k, q) – the analog of l(k, q) for the codes which are linear over the field Fq . mr (k, q) – the analog of m(k, q) for the recursive codes. Moreover, we call the above function f(x) idempotent if it satisfies the identity f(x, . . . , x) = x (this means that all “constants” (a, . . . , a) belong to K(n, f)). Thus, in addition to the above six parameters we can introduce nir (k, q), lir (k, q) and mir (k, q) (only for primary q); “ir” means “idempotent recursive”. So we have the following matrix of parameters  ir  [m (k, q) mr (k, q) m(k, q)] M(k, q) =  lir (k, q) lr (k, q) l(k, q)  nir (k, q) nr (k, q) n(k, q) whose entries are not decreasing from left to right and from up to down. Naturally, the first row of this matrix (put in brackets) is present only when q is primary. It is interesting to estimate and to compare the entries of M(k, q) for various values of k and q. In what follows the equality xy (k, q) = k means that the corresponding code does not exist. A standard argument gives the following source of estimations for the entries of M(k, q):

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Proposition 25 If x ∈ {l, n}, y ∈ {∅, r, ir} and k, q1 , q2 ∈ 2, ∞ then xy (k, q1 q2 ) ≥ min{xy (k, q1 ), xy (k, q2 )}. 7.2. Results for k = 2. It is well-known that n(2, q) = 2 + N (q) where N (q) is the maximal number of mutually orthogonal Latin q × q-squares. The latter were studied extensively (see [15,16,21]). We cite here only the following general conclusion: Theorem 26 Let q ∈ N, q > 1. Then: (a) N (q) ≥ 2 if q ∈ {2, 6}, N (q) ≥ 3 if q ∈ {2, 3, 6, 10}; (b) N (q) ≤ q − 1, if q is primary, then N (q) = q − 1; (c) N (q1 q2 ) ≥ min{N (q1 ), N (q2 )}, in particular, if q = q1 · . . .· qt is the canonical factorization of q then N (q) ≥ min{q1 − 1, . . . , qt − 1}; 10 (d) N (q) ≥ q 143 − 2. For small values of q we have (omitting trivial cases q = 2, 3):       788 566 355 M(2, 4) =  3 5 5  , M(2, 5) =  5 6 6  , M(2, 7) =  7 8 8  , 788 566 355 Proposition 27 If q is a primary number then mr (2, q) = nr (2, q) = n(2, q) = q + 1;  if q is a prime; q mir (2, q) = lir (2, q) = q − 1 if q is not a prime  (A.Abashin (private communication)). Corollary 28 For a prime p and t > 1 there are no cyclic codes among ones permutation equivalent to a Reed–Solomon [pt , 2, pt − 1]− or [pt , pt − 2, 3]−codes. Theorem 29 ([10]) For any q > 2, except q = 6 and possibly q ∈ {14, 18, 26, 42}, nr (2, q) ≥ 4. Really, the last inequality may be sharpened for many values of q. Some of the stronger estimations are easily deduced from Propositions 25,27. We call them standard ones. Other estimations are presented in Theorem 30 ([10]) The following nonstandard lower estimations are valid: nr (2, q) ≥ 8 for q = 80. nr (2, q) ≥ 7 for q ∈ {50, 57, 58, 65, 70, 78, 84, 85, 86, 92, 94, 95, 96, 97, 98}. nr (2, q) ≥ 6 for q ∈ {54, 62, 66, 68, 69, 74, 75, 76, 82, 87, 90, 93}. nr (2, q) ≥ 5 for q ∈ {21, 24, 39, 44, 60}. nir (2, q) ≥ 7 for q ∈ {50, 57, 58, 65, 70, 78, 84, 85, 86, 92, 94, 95, 96, 97, 98}. nir (2, q) ≥ 5 for q ∈ {54, 62, 66, 68, 69, 74, 75, 76, 82, 87, 90, 93}.

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7.3. Results for k > 2. The following estimations in non-recursive case are known. Theorem 31 ([43,21]) (a) If q ≤ k then n(k, q) = k + 1. (b) If k ≤ q and q is even then n(k, q) ≤ q + k − 1. (c) If 3 ≤ k ≤ q and q is odd then n(k, q) ≤ q + k − 2. (d) If k ∈ 1, q + 1 and q is primary (i.e. a power of a prime) then mr (k, q) ≥ q+1. (e) n(3, q) = q + 1 for primary odd q. (f) n(3, q) = n(q − 1, q) = q + 2, for primary even q. The “recursive version” of these results is contained in the following Proposition 32 ([12]) If q ≤ k then lr (k, q) = n(k, q) = k +1 and, for primary q, mr (k, q) = k + 1. The well-known conjecture of Bush [5,43] states that m(k, q) = q + 1 for 2 ≤ k ≤ q; except the case m(3, q) = m(q − 1, q) = q + 2 for even q. For k = 3 we have mr (3, q) = q + 1 and PC calculations give Proposition 33 For every primary q ∈ 4, 128, the number of linear recursive [q + 1, 3, q − 1]-codes is equal to (1/2)ϕ(q + 1)(q − 1). Thus, all the codes enumerated in this Proposition may be constructed in some natural way from the linear cyclic codes indicated in [43, ch.11, Theorem 9]. One can conjecture that it is true for all primary q. For the case k = 3, q = 4 we have the following Proposition 34 mr (3, 4) = 5 < lr (3, 4) = nr (3, 4) = m(3, 4) = n(3, 4) = 6. One of the codes whose existence proves the preceding proposition is linear in wide sense with recursive function f(x1 , x2 , x3 ) = αx21 + αx2 + x23 . over F4 = F2 (α) We call it the Asturian code. The existence of the Asturian code gives some important theoretical corollaries: (1) there exist linear in wide sense recursive codes that are better than any linear in the classical sense recursive code, and (2) for some of the best known linear in classical sense but non recursive codes, there exist linear in wide sense recursive codes with the same parameters. However, for k = 3 the Asturian code is an exclusive example because the PC calculations give lr (3, 8) = 9 = 8 + 1, lr (3, 16) = 17 = 16 + 1, and the following statement is valid: Proposition 35 (A.Abashin, private communication) If t > 4 then lr (3, 2t ) = 2t + 1. Corollary 36 There are no cyclic codes among ones linearly (in the wide sense) equivalent to twice extended Reed–Solomon [2t + 2, 3, 2t ]− or [2t + 2, 2t − 1, 4]−codes.

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Periodic Properties of Polylinear Recurrences over Finite Modules

In the theory of LRS over Galois fields the technique of period calculation for linear recurrences and the constructions of maximal period LRS occupy important place. In [33,36] these results are generalized to the k-LRS over finite rings and modules. Here we present only two series of results that illustrate the advance in this direction. 8.1. Periodic k-Sequences and k-LRS [33,36]. Let µ ∈ M k be a ksequence. For a fixed l ∈ Nk0 , d ∈ Nk0 \0 we call a 1-sequence µ[l,d] (z) = µ(l + dz) a regular (l, d)-extract (extract in the direction d) of the sequence µ. We say that the sequence µ is (l, d)-periodic (periodic in the direction d) if µ[l,d] is a periodic sequence (for any l ∈ Nk0 ). A sequence µ ∈ M k is called a periodic (reversible) sequence if any regular (l, d)-extract of this sequence is a periodic (reversible) 1-sequence. Proposition 37 For µ ∈ M k the following conditions are equivalent: (a) µ is a periodic (respectively, reversible) sequence; (b) the ideal Ann(µ) contains a system of elementary polynomials of the form xl11 (xt11 − e), . . . , xlkk (xtkk − e) (respectively, of the form xt11 − e, . . . , xtkk − e) for some ls ∈ N0 , ts ∈ N, s ∈ 1, k; (c) µ is periodic (respectively, reversible) in each of the directions e1 ,. . .,ek , where es is the s-th row of the identity matrix. Proposition 38 A k-sequence (over a finite module) is periodic if and only if it is a k-LRS. A nonzero vector t ∈ Nk0 is called a vector-period of the sequence µ ∈ M k if x (xt − e)µ = 0 for some l ∈ Nk0 . A subgroup P(µ) ≤ (Zk , +), generated by all vector-periods of µ, will be called its group of periods. If µ has no vector-periods, then P(µ) = 0. l

Proposition 39 The reversibility of the sequence µ ∈ M k is equivalent to the condition ∀i ∈ Nk0 ∃j ∈ Nk0 : xj (xi µ) = µ. If the sequence µ is reversible, then for any i ∈ Nk0 the sequence ν = xi µ is also reversible and P(ν) = P(µ); for any t ∈ P(µ), we have xt µ = µ. The set O(µ) of all k-sequences ν ∈ M k of the form ν = xi µ, i ∈ Nk0 , is called a trajectory of µ. Proposition 40 A sequence µ ∈ M k is periodic if and only if its trajectory O(µ) is finite, and it is reversible if and only if O(µ) = O(xi µ) for any i ∈ Nk0 . For a periodic sequence µ the set T (µ) of all reversible elements of its trajectory O(µ) is called the cycle of the sequence µ, and its cardinality T (µ) = |T (µ)| is called the period of the sequence µ. The set D(µ) = O(µ)\T (µ) is called the set

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of all defect elements of the trajectory O(µ), and its cardinality D(µ) = |D(µ)| is called the defect of the sequence µ. The sequence µ is said to be degenerating if it is periodic and its cycle contains only the zero sequence, i.e., T (µ) = {0}. Thus, D(µ) + T (µ) = |O(µ)|, and a periodic sequence µ is reversible iff D(µ) = 0, i.e., T (µ) = O(µ). A periodic sequence is degenerating iff xi ∈ Ann(µ) for some i ∈ Nk0 . Proposition 41 If µ ∈ M k is a periodic sequence, then T (µ) is the index [Zk : P(µ)] of the subgroup P(µ) of the group (Zk , +), and |O(µ)| ≤ |Pk / Ann(µ)|. Let us denote the Pk -modules of all reversible and degenerating k-sequences over by RM k , and DM k respectively.

RM

·

Proposition 42 LM k = RM k + DM k . The following result gives us an interesting relation between the properties of reversible sequences and the properties of associated rings. Theorem 43 Let µ ∈ M k , and let S = Pk / Ann(µ) be the operator ring of µ (see sec. 5), θs = xs + Ann(µ) for s ∈ 1, k. Then the sequence µ is reversible if and only if θ1 , . . . , θk ∈ S ∗ . If µ is a reversible sequence, then T (µ) = | θ1 , . . . , θk | ≤ |S ∗ | ≤ |S| − 1, where θ1 , . . . , θk is a subgroup of the group S ∗ generated by θ1 , . . . , θk . The equality T (µ) = |S ∗ | holds if and only if S ∗ = θ1 , . . . , θk .

(8.1)

If µ is a reversible sequence and Ann(µ) ∩ R = 0, then T (µ) = |S| − 1 if and only if the following three conditions hold: (a) R = Fq ; (b) Ann(µ) is a maximal ideal of the ring Pk (i.e., S = Fqr for some r ∈ N); (c) the equality (8.1) is true. Let R Q be a QF-module corresponding to the ring R. A reversible sequence µ ∈ Q k is called full-cycle if its annihilator I = Ann(µ) satisfies the conditions I ∩ R = 0; S ∗ = θ1 , . . . , θk ; LQ (I) = Sµ.

Theorem 44 Let µ be a full-cycle k-recurrence over QF-ring and for any ν ∈ LQ (I) = Sµ P(µ) ⊆ P(ν); T (ν)|T (µ);

R Q,

then S = Pk /I is a

(T (ν) = T (µ)) ⇒ (v ∈ T (u)).

Theorem 45 Let R < S where S is a QF-ring. Then there exists a full-cycle LRS over the QF-module R Q with the ring of operators isomorphic to S.

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8.2. Linear Recurrences of Maximal Period over Galois Ring. This is the the most deeply studied and very interesting in various applications class of linear recurrences over rings. Let R = GR(q n , pn ) be a Galois ring of order q n and of the characteristic pn (p is a prime, q = pr ), and let F (x) ∈ P = R[x] be a monic polynomial of degree m. Then for any LRS u ∈ LR (F (x)) the inequality T (u) ≤ (q m − 1)pn−1 holds. If in this situation T (u) = (q m − 1)pn−1 . then we say that the sequence u is a linear recurring sequence of maximal period (MPrecurrence) of rank m over a Galois ring R. For any monic polynomial F (x) ∈ P there exist λ ∈ N0 , t ∈ N such that F (x)|xλ (xt − e). The minimal t with this property is called a period of F (x) and denoted by T (F ). Denote by u and F the images correspondingly of a sequence u and a polynomial F under the natural homomorphism R → R = R/pR. Note, that if degF (x) = m, then T (F ) ≤ T (F (x))pn−1 ≤ (q m − 1)pn−1 . If T (F ) = (q m − 1)pn−1 , then F is called a polynomial of maximal period (MP-polynomial) over R. Proposition 46 Under the above assumptions an LRS u ∈ LR (F ) is an MPrecurrence of rank m if and only if F (x) is an MP-polynomial over R and u = 0. In [33] a simple algorithm for the building an MP-polynomial was presented. It has the following simplest form for p = 2: let R = GR(q n , 2n ), q = 2r , and let G(x) ∈ R[x] be a monic polynomial of degree m such that T (G) = q m − 1. Let the polynomials G[0] (x), G[1] (x), . . . ∈ P be defined recursively: G[0] (x) = G(x), [k] [k] and if G[k] (x) = G(0) (x2 ) + xG(1) (x2 ) then G[k+1] (x) = (−1)m (G(0) (x)2 − xG(1) (x)2 ). [k]

[k]

Theorem 47 Any polynomial of the form F (x) = G[r] (x) + 2∆(x),

deg∆(x) < m,

∆(x) = 0, e

is an MP-polynomial There are some simple conditions for the polynomial over residue ring Zpn to have a maximal period. Theorem 48 ([33]) Let F (x) = xm + am−1 xm−1 + . . . + a0 be a polynomial over Zpn , such that T (F ) = pm − 1. Then F (x) is an MP-polynomial when 1. p > 2 and: ap0 ≡ a0 ( mod p2 ), or F (x) = xm + ak xk + a0 , m ≥ p − 2. 2. p = 2 and: (a) m is even, a0 ≡ e ( mod 4); e + 2a0 a2 if a1 = e; or (b) m is odd and a1 ≡ 2(e + a0 a2 ) if a1 = 0; 4 or (c) F (x) = xm + ak xk + a0 , ak , a0 ∈ {−e, e}, (m, a0 ) = (2k, e).

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385

Pseudorandom Sequences Generated by Polylinear Recurrences

Here we restrict ourselves only with some illustrations of the statement that polylinear sequences over a Galois ring are good source of pseudorandom sequences. More detailed information on this subject is contained in [33,36]. 9.1. Coordinate Sequences of MP-Recurrence. Let u be an MP-recurrence of the period (pm − 1)pn−1 with the minimal polynomial G(x) of the degree m over the ring R = Zpn . Any item u(i) of the sequence u has standard p-ary decomposition: u(i) = uo (i) + u1 (i)p + ... + un−1(i)pn−1 ; us (i) ∈ 0, p − 1. The latter gives us n sequences uo , ..., un−1 over the field Zp . For sufficiently large but acceptable values of m and s the sequence us is a very good source of pseudorandom numbers. Of course us is an LRS over Zp . Let rk us be rank or linear complexity of us: the degree of its minimal polynomial. Apparently we can consider us as an “approximation” of a random sequence only if rk us is large enough. There are some estimations of this parameter for the simplest case p = 2. For k, l ∈ N denote b(k, 0) = k, b(k, l) = 0, if k < 2l, and in other cases  1, if l is even or l = 1, b(k, l) = k − 2l + 2, if l is odd l ≥ 3. Theorem 49 If p = 2 then rk us ≤

ps−1 l=0

(l + 1) ·

b(l)

t=b(l+1)+1

m t

.

Due to the limits on the length of the text, we point out here only one of the earliest and simplest lower estimations of rank. Theorem 50 (A.Nechaev, 1982 [33]) If p = 2 then the polynomial G(x) can be chosen such that for s ∈ 3, n − 1         s−2 m m s−1 m + + s , (9.1) rk us ≥ (2s−1 +1)m+L 2 (2s−1 −2k +1) k+1 +1 2 4 2 k=2

where L ∈ {0, 1} and L = 1 for m ≤ 14 and m = 20. We conjecture that L in (9.1) is always equal to 1. The lower estimations of rk us for p = 2 are also given in [13]; these estimations do not contain the first and the second summands of (9.1). In [33] one can find more precise rank estimations for us for p = 2 as well as for p > 2. The PC-calculations, using these estimations give, for example, that the polynomial G(x) ∈ Zpn [x] can be chosen so that for p = 2 if m = 11, then 3383 ≤ rk u3 ≤ 5340, 59, 703 ≤ rk u7 ≤ 128, 430; if m = 31, then 1.37 · 107 ≤ rk u3 ≤ 1.53 · 107 , 6 · 1010 ≤ rk u7 ≤ 1011 ; for p = 5 if m = 11, then 2 · 108 ≤ rk u3 ≤ 109 , 1011 ≤ rk u7 ≤ 8 · 1011 ; for p = 11 2 · 1010 ≤ rk u7 ≤ 3 · 1011 . if m = 5, then 106 ≤ rk u3 ≤ 2 · 107 ,

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9.2. The Distribution of Elements on a Cycle of an MP-Recurrence. Another important requirement to the pseudorandom sequence is the “uniformity” condition on the distribution of elements and k-tuples on its long enough segments. The results on this topic are summarized in [33,36]. We present here only one of new results. Let F (x) be an MP-polynomial of degree m over a Galois ring R = GR(q n , pn ), and let u ∈ LR (F ) be a MP-recurrence of period T = (q m − 1)pn−1 . Let 0 ≤ i1 ≤ . . . ≤ ik < T be fixed integer numbers, and let α1 , . . . , αk be fixed elements of R. Denote by ν the number of solutions i ∈ 0, T − 1 of the system of equations u(i + i1 ) = α1 , . . . , u(i + ik ) = αk . Theorem 51 ([26]) If the system of residues of polynomials x ¯i1 , . . . , x ¯ik ∈ ¯ ¯ then R[x] modulo F¯ (x) is linearly independent over the field R,   √    ν − T  ≤ p2(n−1)q m/2 ≈ p3(n−1)/2 T .   k |R| More precise results on distribution of elements on cycles of linear recurring sequences over Zpn can be found in [35] and [33, § 27]. 9.3. Weight Characteristics of MP-Recurrence over R = GR(q 2 , 4). In [34,40] the full description was given of the possible variants of distributions of elements on cycles of MP-recurrences not only over Z4 [33] but over any Galois ring R = GR(q 2 , 4) of the characteristic 4. Let F (x) ∈ R[x] be a monic polynomial of the degree m such that its period T = T (F ) is equal to τ = q m − 1 (distinguished polynomial) or to 2τ (MPpolynomial). Let u ∈ LR (F ), u = 0 and Nu (c) be the number of solutions i ∈ 0, T − 1 of the equation u(i) = c for a given c ∈ R. The description of possible types [Nu (c) : c ∈ R] in LR (F ) and their multiplicities is obtained, i.e. the 0,T −1 (F ) is described. These results complete weight enumerator of the code LR were based on the presentation of LRS u using the trace-function in Galois rings [47,33,39] and on the theory of quadrics over Galois fields of characteristic 2 (see [59]). For brevity we give only the description of possible values of Nu (c). Let λ = [m/2] be the integer part of m/2 and δc,0 be the Kronecker delta. Theorem 52 If F (x) is a distinguished polynomial then for any c ∈ R Nu (c) = q m−2 ± wq λ−1 − δc,0 , where w ∈ {1, q − 1} if m = 2λ + 1, and w ∈ {0, 1, q − 1} if m = 2λ. There exists not more than 2q + 1 different types [Nu (c) : c ∈ R] in LR (F ).   [m/2] In particular for a nonzero c we have Nu (c) − q m−2  ≤ q−1 . Note that q q in [31] the trigonometric sums approach gives more rough estimation, with q m/2 in the right part.

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Theorem 53 If F (x) is an MP-polynomial then for any c ∈ R Nu (c) = 2q m−2 ± wq λ−1 − 2δc,0 , where w ∈ {0, 2, q − 2, q, 2(q − 1)} if m = 2λ + 1, and w ∈ {0, 1, 2, q − 1, 2(q − 1)} if m = 2λ. There exist not more than 2q + 1 different types [Nu (c) : c ∈ R] in LR (F ). Acknowledgment. The authors thank the referee for many interesting and helpful comments and suggestions that had improved the explication of the results.

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Calculating Generators for Invariant Fields of Linear Algebraic Groups J¨ orn M¨ uller–Quade and Thomas Beth Institut f¨ ur Algorithmen und Kognitive Systeme Professor Dr. Th. Beth, Arbeitsgruppe Computer Algebra Universit¨ at Karlsruhe, Am Fasanengarten 5, 76 128 Karlsruhe, Germany

Abstract. We present an algorithm to calculate generators for the invariant field k(x)G of a linear algebraic group G from the defining equations of G. This work was motivated by an algorithm of Derksen which allows the computation of the invariant ring of a reductive group using ideal theoretic techniques and the Reynolds operator. The method presented here does not use the Reynolds operator and hence applies to all linear algebraic groups. Like Derksen’s algorithm we start with computing the ideal vanishing on all vectors (ξ, ζ) for which ξ and ζ are on the same orbit. But then we establish a connection of this ideal to the ideal of syzygies the generators of the field k(x) have over the invariant field. From this ideal we can calculate the generators of the invariant field exploiting a field-ideal-correspondence which has been applied to the decomposition of rational mappings before.

1

Introduction

Invariant theory has a long tradition as well as new applications. The branch of constructive invariant theory is mostly interested in questions like finding generators for invariant rings of a given group. Much research has been done for the case of finite groups (see e. g. [10,16,14]). Invariant fields for finite groups are just the quotient fields of the corresponding invariant ring so problems in that area were rather structural like “is a given invariant field rational?” [9] which is an instance of the famous rationality problem. But in the case of linear algebraic groups generating systems of invariant fields become more interesting. Every invariant field has a finite generating system even if the corresponding invariant ring does not. Recently Derksen showed how to compute generators for invariant rings also of infinite reductive groups using ideal theory. This motivated our study of an ideal theoretic approach to invariant fields. This paper presents an algorithm to compute generators for the invariant field of a linear algebraic group which is given by its equations. The paper is structured as follows: Section 2 will explain a correspondence between fields and certain syzygy ideals. This correspondence will later on allow us to pass from the ideal theoretic part of our algorithm back to function fields. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 392–403, 1999. c Springer-Verlag Berlin Heidelberg 1999


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Section 3 introduces the equivalence relation relating those points which are on the same group orbit—the so called graph of the action. The ideal describing the Zariski closure of this equivalence relation can be computed from the equations defining the group. In Section 4 we will present two results relating the ideal describing the graph of the action and the syzygy ideal corresponding to the invariant field we are looking for. Because of the field-ideal-correspondence being constructive we will be able to compute the invariant field following this path. The last section gives four examples which were computed using MapleV.

2

A Field Ideal Correspondence

To be able to use ideal theoretic methods to deal with function fields we shortly restate the field-ideal-correspondence which was used in [13] to find decompositions of rational mappings. Given fields k(f ) = k(f1 , . . . , fr ) and k(x) = k(x1 , . . . , xn ) finitely generated over a field of constants k and both being contained in some field k(V ) = Quot(k[X1 , . . . , Xs ]/I(V )) we formally define the syzygy ideal mentioned in the introduction. Definition 1. Let k(f ) and k(x) be fields lying over a field k of constants and let {x}, {y} denote
the sets of generators of k(f ) and k(x) over k respectively. Furthermore let x∈{x} {Zx } be a set of variables and k(f )[Z] be the ring of polynomials in these variables over the field k(f ). Then the ideal Jk(x)/k(f ) ⊆ k(f )[Z] of all algebraic relations of the set {x} over k(f ) is defined as {x − Zx |x ∈ {x}} ∩ k(f )[Z]. The ideal {x − Zx |x ∈ {x}} used in the definition can also be viewed as a syzygy ideal, namely Jk(x)/k(x) representing the trivial extension k(x)/k(x). There is an alternative characterization of this ideal which was given in [12]. It is the key to the properties of the syzygies ideal which are used in the following. Lemma 1. The ideal Jk(x)/k(f ) equals the kernel of the specialization homomorphism Ψx : k(f )[Zx1 , . . . , Zxn ] → k(f )(x), Zx → x. In this paper we just need a special case of the ideal Jk(x)/k(f ) namely we restrict ourselves to k(f ) being a subfield of k(x). As we can think of the field k(x) and its generating system {x} as being fixed we will just write Jk(f ) instead of Jk(x)/k(f ). In this special case we can express the generators of k(f ) in terms of the x1 , . . . , xr . An alternative characterization of the ideal Jk(f ) can then be given (following [13]). This characterization has computational advantages and for k(x) being fixed we get a correspondence between intermediate fields of k(x)/k and the ideals Jk(f ) . For this correspondence we need a result from [13]:

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Proposition 1. Let k(f ) ≤ k(x) be fields finitely generated over k and let the generators f1 , . . . , fm of k(f ) over k be expressed in x = x1 , . . . , xr as f1 = n1 (x) nm (x) d1 (x) , . . . , fm = dm (x) . Let Z denote Zx1 , . . . , Zxr then we define an ideal 

I=

 n1 (x) nm (x) n1 (Z) − d1 (Z), . . . , nm (Z) − dm (Z) d1 (x) dm (x)

and for d = d1 (Z) · · · · · dm (Z) we get: Jk(f ) = (Jk + I) : d∞ . Furthermore the coefficients of a reduced Gr¨ obner basis of Jk(f ) · k(x)[Z] generate k(f ). In the remaining paper we will not have algebraic relations among the generators x1 , . . . , xr of k(x). Therefore Jk will be the zero ideal and we need not worry about computing Jk . For an ideal I the saturation I : d∞ is effective ([1] Algorithm idealdiv2) and so is the problem of representing a field element in some generators [17,8,13]. Thus the ideals of Proposition 1 can be achieved through Gr¨obner basis computations if the generators for the field are known. We can conclude for the fields k(x), k(f ) being finitely generated over a computable field k and contained in a field k(V ) = Quot(k[X1 , . . . , Xs ]/I(V )) that the ideal Jk(f ) can be computed effectively. Thus the following field ideal correspondence is constructive: Corollary 1. For subfields k(f ) of k(x) being finitely generated over a computable field k and contained in a field k(V ) = Quot(k[X1 , . . . , Xs ]/I(V )) the mapping C:

{k(f )|k ≤ k(f ) ≤ k(x)} → {I|I ✂ k(x)[Z]} k(f ) → Jk(f ) · k(x)[Z]

is inclusion preserving, injective and C as well as C −1 can be computed effectively But the problem in this paper is to calculate field generators. Hence in the next two sections we will develop a method to compute the Ideal Jk(x)/k(x)G · k(x)[Z] from the ideal HG of the defining equations of the graph of the action of a group G. With the above corollary we will then see that generators for k(x)G can be computed effectively from HG .

3

The Graph of the Action of a Linear Algebraic Group

In this subsection we give a formal definition of the equivalence relation which relates two points iff they are on the same G-orbit and we restate the fact that the ideal HG of the defining equations of the graph of the action can be computed effectively. Then we will state a theorem of Derksen which motivated this paper.

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Definition 2. For a group G acting on a finite dimensional vector space V we define the following relation from V to V : {(ξ, ζ)|∃g ∈ G : g · ξ = ζ}. This relation will be called the graph of the action of the group G. The Zariski closure of this relation will be called the closed graph of the action of the group G. The ideal of all polynomials over vanishing on the graph of the action of G will be denoted by HG ✂ k[X, Z] . For the first step in our algorithm we have to compute generators for the ideal HG from the equations defining the group G. Lemma 2. Let p1 , . . . , ps ✂k[m11 , m12 , . . . , mnn ] denote the ideal of those polyn×n nomials over k vanishing on G ⊆ k . Let I ✂ k[m11, m12 , . . . , mnn , X, Z] denote the ideal generated by p1 , . . . , ps and the entries of       m11 . . . mn1 Zx1 X1  .. .   .   .   . · · · ..  ·  ..  −  ..  . mn1 . . . mnn

Zxn

Xn

Then HG = I ∩ k[X, Z]. Hence HG can be computed using elimination. Proof. The proof is obvious from the Extension Theorem [2]: consider the points of the graph of the action of G as all those points which extend to a point on the variety of I. As HG describes the Zariski closure of all partial solutions which can be extended to points on the variety of I the ideal HG also describes the Zariski closure of the graph of the action of G.   But first we will give an additional motivation for the study of the Zariski closure of the graph of the action of a group namely its connection to the invariant ring of that group. Derksen was able to turn the proof of Hilbert’s finiteness theorem into an algorithm (see [4]). The algorithm of Derksen makes use of the ideal defining the graph of the action and thereby establishes the connection between the two structures. Next we will shortly state the result of Derksen following the presentation of Decker and De Jong [3]. Before that we restate the finiteness result of Hilbert: Theorem 1 (Hilbert). Let G be a reductive group, let ∗ : k[X] → k[X]G , f → f ∗ denote the Reynolds operator, and IN be the ideal generated by the invariant ring k[X]G in k[X]. Then for homogeneous polynomials f1 , . . . , fr we have f1 , . . . , fr  = IN iff k[f1∗ , . . . , fr∗ ] = k[X]G . Because of k[X] being noetherian the invariant ring is finitely generated.

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For reductive groups it suffices to know generators of the ideal IN and one can derive a generating system of k[X]G using the Reynolds operator. The algorithm of Derksen computes generators of IN from the ideal defining the graph of the action. Theorem 2 (Derksen). Let HG ✂k[X, Z] denote the defining ideal of the closed graph of the action of G then Z + HG  ∩ k[X] = IN . The proof can be found in [3]. In the next section we will present a variation of Derksen’s algorithm which works for invariant fields of algebraic groups. This algorithm will be independent of the Reynolds operator and will do for non-reductive groups as well.

4

Calculating Generators for Invariant Fields

In this section we want to establish a connection between the Ideal HG describing the graph of the action of a group G and the ideal Jk(x)G corresponding to the invariant field. This connection will then be exploited to calculate invariant fields. But first we look at a result of Rosenlicht connecting generating systems of Invariant fields and the graph of the action. Theorem 3 (Rosenlicht). Let G denote a linear algebraic group and let f = (f1 , . . . , fr ) be a rational mapping for which k(f ) = k(x)G . Then there exists a n G-invariant open subset U of k such that f is regular on U and the relation n n −1 f ◦f ⊂ k ×k restricted to U ×U equals the graph of the action of G restricted to U × U. For a proof see [15, Satz 2.2.] As a corollary we get: Corollary 2. Using the same notation as above we can conclude: 1. We denote the extension of the ideal Jk(x)G to k(x)[Z] by Jk(x)G ·k(x)[Z]. Let (Jk(x)G ·k(x)[Z]∩k[x, Z])|x=X be the ideal of all polynomials in Jk(x)G ·k(x)[Z] where the algebraically independent parameters x1 , . . . , xn are replaced by n variables X1 , . . . , Xn . Then there exists an open nonempty subset U of k such that we get the following equality for the varieties restricted to U × U: V(HG )|U ×U = V((Jk(x)G · k(x)[Z] ∩ k[x, Z])|x=X)|U ×U 2. The ideal HG and (Jk(x)G ·k(x)[Z]∩k[x, Z])|x=X have associated prime ideals in common. 3. Let HG |X=x ✂ k(x)[Z] denote the ideal generated by the image of HG under the specialization homomorphism k[X, Z] → k(x)[Z] defined by Xi → xi . Then the ideal Jk(x) · k(x)[Z] and the ideal HG |X=x have associated prime ideals in common.

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Proof. For an algebraic group G the graph of the action is a projection of a variety. Hence from the Extension Theorem [2, Theorem 4 § 6 Chapter 3] we n can conclude that there exists an open nonempty subset U1 ⊆ k such that the closed graph of the action V(HG ) is on U1 ×U1 equal to the graph of the action of n G. Furthermore it is easy to see that there is an open nonempty subset U2 ⊆ k such that (Jk(x)G · k(x)[Z] ∩ k[x, Z])|x=X describes on the open set U2 × U2 the fibers of a rational mapping f = (f1 , . . . , fr ) for which k(f ) = k(x)G . From the theorem of Rosenlicht we can deduce the existence of an open set U3 such that on U3 × U3 the fibers of (f1 , . . . , fr ) equal the graph of the action. Hence for U = U1 ∩ U2 ∩ U3 we get the equality stated in point 1. of our corollary. To show point 2. let V denote the complement of the open set U × U. To shorten notation we will write J for (Jk(x)G ·k(x)[Z]∩k[x, Z])|x=X and H for HG in the remaining part of the proof. We get V(H) = (V(H) ∩ U × U)∪(V(H)∩V ) and V(J ) = (V(J ) ∩ U × U ) ∪ (V(J ) ∩ V ). As V(J ) ∩ U × U = V(H) ∩ U × U all their associated prime ideals must be equal. Since V(H) ∩ U × U and V(H) ∩ V are disjoint and the latter is closed we can conclude that no associated prime of I(V(H) ∩ V )) is contained in I(V(H) ∩ U × U ). Thus all primes associated to I(V(H) ∩ U × U ) are also associated to H. The same argument applies to J and thus the ideals H and J share all prime ideals which are associated to I(V(H) ∩ U × U ). For the proof of point 3. we first observe that specialization of our variables X to field elements x is equivalent to localizing our ring k[X, Z] with the multiplicatively closed set k[X] \ {0} (and then substituting the symbols X1 , . . . , Xn by the symbols x1 , . . . , xn ). Since V(H) ∩ U × U describes exactly the graph of the action it does not contain any points (0, . . . , 0, ζ) with ζ not equal to the zero vector. Hence no associated prime P of I(V(H)∩U ×U) does contain a polynomial from k[X]\{0}. Thus all prime ideals associated to I(V(H) ∩ U × U) remain associated after localization (see [18, Theorem 16 (b),Theorem 17 § 10 Chapter IV]). It remains to show that these associated prime ideals are also associated to the localization of H and J . All those associated prime ideals which contain an element of k[X]\{0} become the whole ring when localized. Hence they become redundant and all other associated prime ideals remain associated prime ideals according to [18, Theorem 17 § 10 Chapter IV].   Next we shortly look at the action which the group G induces on the associated prime ideals of Jk(x)G · k(x)[Z]. Lemma 3. Let the group G operate on k(x)[Z] by operating on the coefficients from k(x). Then the group G operates transitively on the associated prime ideals of Jk(x)G · k(x)[Z]. G Proof. Let k(x)G alg denote the algebraic closure of k(x) in k(x). Then the field G k(x)alg is the unique maximal field lying algebraically over k(x)G and being contained in k(x). As the fields k(x)G and k(x) are closed under the group

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action of G the field k(x)G alg must by definition be closed under the group action of G, too. The group G operates on k(x)G alg as a group of automorphisms. Hence G operates like a finite group on k(x)G alg because the group of automorphisms of G G k(x)alg leaving k(x) invariant must be finite. From the book of Eisenbud [5, Proposition 13.10’] we know that the group G then operates transitively on the associated primes of Jk(x)G · k(x)G alg [Z]. The claim of our lemma can now be concluded by looking at the algorithm for the computation of primary decompositions given by Gianni, Trager, and Zacharias [6]. This algorithm never introduces transcendental field extensions of the ground field (here k(xG )) and the primary decomposition of the ideal Jk(x)G · k(x)[Z] is already found in k(x)G   alg [Z]. The relation between the ideals HG and Jk(x)G · k(x)[Z] can now be stated: Theorem 4. Let HG |X=x denote the ideal generated by the image of HG under the specialization homomorphism k[X, Z] → k(x)[Z], Xi → xi and let Jk(x)G · k(x)[Z] be the ideal generated by Jk(x)G ✂ k(x)G [Z] in k(x)[Z]. Then we get the following equality: HG |X=x = Jk(x)G · k(x)[Z] Proof. The ideals Jk(x)G · k(x)[Z] and HG |X=x are invariant under the group action of the group G on the coefficients (∈ k(x)). Furthermore they have (at least) one associated prime ideal in common. As the group G operates transitively on the associated primes of Jk(x)G · k(x)[Z] we can conclude that every prime associated to Jk(x)G · k(x)[Z] is also associated to HG |X=x. Hence we have HG |X=x ⊆ Jk(x)G · k(x)[Z] if the two ideals are radical. The ideal HG |X=x is radical as it contains all polynomials vanishing on G · x × G · x. The radicality of Jk(x)G · k(x)[Z] can be deduced from Jk(x)G ✂ k(x)G [Z] being prime and the field extension k(x)/k(x)G being separable [15, Section VI Lemma 1.5], [19, Corollary to the Lemma in § 11 Chapter VII]. The inclusion Jk(x)G · k(x)[Z] ⊆ HG |X=x can be deduced from HG |X=x being by construction the maximal ideal vanishing on G · x × G · x and Jk(x)G · k(x)[Z] also vanishing on G · x × G · x.   This relation between the ideals HG and Jk(x)G · k(x)[Z] can be used to calculate generators for an invariant field k(x)G . Corollary 3. Let C denote the set of coefficients of a reduced Gr¨ obner basis of the ideal HG |X=x. Then k(C) = k(x)G . The proof is obvious from the above theorem and Proposition 1. To make the treatment of the ideals complete we give a method to compute HG from Jk(x)G · k(x)[Z]. For this we need the following result:

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 Lemma 4. Let Jk(x)G · k(x)[Z] ∩ k[x, Z] |x=X be the ideal of all polynomials from Jk(x)G · k(x)[Z] with the algebraically independent parameters x1 , . . . , xn replaced by the variables X1 , . . . , Xn . Furthermore let HG ✂ k[X, Z] be as above. Then we get the equality:

 Jk(x)G · k(x)[Z] ∩ k[x, Z] |x=X = HG

 Proof. Let q(X, Z) be an element of Jk(x)G · k(x)[Z] ∩ k[x, Z] |x=X. Then it vanishes on every point of the graph of the action hence q is in the ideal HG of all polynomials vanishing there. For the other direction let q(X, Z) be contained in HG . Then q(x, Z) is contained in Jk(x)G · k(x)[Z] (according to Theorem 4) and as no denominators occur it is also contained in Jk(x)G · k(x)[Z] ∩ k[x, Z]. Hence q(X, Z) is element 

  of Jk(x)G · k(x)[Z] ∩ k[x, Z] |x=X. Using this lemma we can show the main ingredient for an algorithm to compute the ideal HG from the ideal Jk(x)G · k(x)[Z]. Corollary 4. Let f1 , . . . , fr ∈ k(x)[Z] be a generating system for Jk(x)G ·k(x)[Z] and p1 , . . . , pr ∈ k[x, Z] be this generating set after clearing denominators (multiplying by a suitable polynomial from k[x]). Then HG equals p1 , . . . , pr  : d∞ . d∈k[x]\{0}

Proof. Following Lemma 4 we have to show that d∈k[x]\{0} p1 , . . . , pr  : d∞



equals Jk(x)G · k(x)[Z] ∩ k[x, Z] . Let q be an element of d∈k[x]\{0} p1 , . . . , pr  : d∞ then there exists a polynomial d ∈ k[x] such that dq ∈ p1 , . . . , pr  hence dq ∈ f1 , . . . , fr  and as f1 , . . . , fr  is defined over k(x) we have q ∈ f1 , . . . , fr  and as q is a polynomial we get q ∈ f1 , . . . , fr  ∩ k[x, Z]. Conversely let q be

ran element of Jk(x)G · k(x)[Z] ∩ k[x, Z] . Then there is a representation q = i=1 qi fi and the denominators with a d ∈ k[x]) yields a representation dq =

r (clearing

q p . Hence dq is an element of p1 , . . . , pr  and as d∈k[x]\{0} p1 , . . . , pr  : i i i=1 d∞ is saturated with respect to every polynomial in k[x] we have proven the above claim.   We need two more lemmata (both taken from [11]) to make the above corollary effective. Lemma 5. 1. For a primary ideal Q of a polynomial ring and a polynomial d the ideal Q : d∞ equals Q or 1. 2. For an intersection I1 ∩ · · · ∩ It of ideals I1 , . . . , It of a polynomial ring and a polynomial d we have: (I1 ∩ · · · ∩ It ) : d∞ = I1 : d∞ ∩ · · · ∩ It : d∞ .

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Proof. 1. If the ideal Q contains a power of d then Q : d∞ equals 1. Now let Q be a primary ideal with dn ∈ Q for all n then Q : d∞ = 1. Suppose Q : d∞ = Q then there exists a dl · p ∈ Q with p ∈ Q but because of Q being primary there must be a power of dl which is an element of Q. This contradicts dn ∈ Q for all n. 2. This follows from Proposition 10 in Chapter 4 § 4 of [2] and the fact that there always exists an l large enough such that I : d∞ = I : dl (see Exercise 8 in Chapter 4 § 4 of [2]).   The algorithm can now be derived from Corollary 4 and the following result: Lemma 6. Let I ✂ k[x][Z], I ✂ k[x] be ideals and I prime. Then the ideal

I : d∞

d∈k[x]\I


is effectively computable. Proof. By choosing a term order fulfilling x  Z it is decidable if a given ideal of k[x][Z] contains an element of k[x] \ I . Let I = Q1 ∩ · · · ∩ Qt be a primary decomposition and Q be the set of all those primary components which do not contain an element of k[x] \ I . We claim that  I : d∞ = Q. Q∈Q

d∈k[x]\I




The ideal d∈k[x]\I
I : d∞ equals d∈k[x]\I
(Q1 ∩ · · · ∩ Qt ) : d∞ which is

equal to d∈k[x]\I
Q1 : d∞ ∩ · · · ∩ Qt : d∞ . For every primary component Qi which is not an element of Q there exists a summand where Qi is replaced by the ideal 1. Let dQi denote  a polynomial

from k[x] \ I for which Qi : d∞ Qi = 1. Since I is prime we have Qi ∈Q dQi ∈ I hence there exists a summand   Q1 : ( dQi )∞ ∩ · · · ∩ Qt : ( dQi )∞ Qi ∈Q

Qi ∈Q

in d∈k[x]\I
I : d∞ which contains all other summands. This summand therefore equals the whole sum.  

5

Four Examples

We will give four detailed examples in this section. Our first example will be the invariant field of a representation of the group (C, +).

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 1a |a ∈ C} which is isomorphic to (C, +) 01 the defining equations are the linear equations m11 = 1, m21 = 0, m22 = 1. From the ideal m11 Z1 + m12 Z2 − X1 , m21 Z1 + m22 Z2 − X2 , m11 − 1, m21 , m22 − 1 describing the points (m11 , m12 , m21 , m22 , Z1 , Z2 , X1 , X2 ) for which

Example 1. For the group G1 = {



m11 m12 m21 m22

     Z1 X1 · = Z2 X2

we get by elimination of the variables m11 , m12 , m21 , m22 the defining equations Z2 − X2 for the closed graph of the action. Specializing the variable X2 to the field element x2 and computing a reduced Gr¨obner basis yields C(x1 , x2 )G1 = C(x2 ). For our next example we consider the non-compact group Id3 ⊗GL2 (C) where the group GL2 (C) operates synchronously on three vectors. Example 2. The defining equations of the group GL2 (C) ⊕ GL2 (C) ⊕ GL2 (C) which consist of block matrices are simply mij = 0 for all those entries mij which are not within one of the three 2 × 2-blocks. For the group G2 = Id3 ⊗ GL2 (C) we have the additional equations that the following polynomials must equal zero: m11 − m33 , m12 − m34, m21 − m43 , m22 − m44 , m11 − m55, m12 − m56 , m21 − m65 , m22 −m66 , i. e., the blocks must be equal. Proceeding as in the last example we get the following ideal describing the closed graph of the action: Z2 Z5 X3 − Z2 Z3 X5 − Z4 Z5 X1 + Z4 Z1 X5 + Z6 X1 Z3 + Z6 X3 Z1 , −Z2 X6 Z3 + Z2 X4 Z5 − Z4 X2 Z5 + Z4 Z1 X6 + Z6 Z3 X2 − Z6 Z1 X4 , −X2 Z5 X3 + X2 Z3 X5 + X4 Z5 X1 − X4 Z1 X5 − X6 Z3 X1 + X6 Z1 X3 , Z4 X5 X2 − Z6 X2 X3 − Z2 X4 X5 + Z2 X4 X5 + Z2 X3 X6 + X4 Z6 X1 − Z4 X1 X6 . Again specializing the variables Xi to field elements xi and computing the reduced Gr¨ obner basis {Z1 + Z2 +

x1 x4 − x3 x2 −x1 x6 + x5 x2 Z5 + Z3 , x3 x6 − x4 x5 x3 x6 − x4 x5

−x1 x6 + x5 x2 x1 x4 − x3 x2 Z4 + Z6 } x3 x6 − x4 x5 x3 x6 − x4 x5

G2 3 x2 −x1 x6 +x5 x2 we get the invariant field C( xx13 xx46 −x −x4 x5 , x3 x6 −x4 x5 ) = C(x1 , x2 , x3 , x4 , x5 , x6 )

Our third example has a motivation from physics. In the paper [7] invariants were applied to show that two quantum states are not locally equivalent, i. e., they are not on one orbit of the group SU2 (C) ⊗ SU2 (C) of local unitary transformations. Local in this context refers to the entanglement not being increased by local transformations.

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This leads to the general question how to compute invariants of tensor product groups. We will here consider the group SL2 (C)⊗SL2 (C). The group elements are of tensor product form   ae af be bf  ag ah bg bh     ce cf de df  cg ch dg dh where the variables a, b, . . . , h represent elements of C. This structure is directly reflected in the defining equations. Example 3. The defining equations for the group G3 = SL2 (C) ⊗ SL2 (C) are m11 − ae, m12 − af, m13 − be, m14 − bf, m21 − ag, m22 − ah, m23 − bg, m24 − bh, m31 − ce, m32 − cf, m33 − de, m34 − df, m41 − cg, m42 − ch, m43 − dg, m44 − dh for the group consisting of tensor product matrices plus ad−bc−1, eh−fg −1 for the determinants of the tensor factors being one. Calculating the ideal describing the closed graph of the action of G3 we get Z1 Z4 − Z2 Z3 − X1 X4 + X3 X2 . Specializing the variables Xi to field elements xi , computing a reduced Gr¨ obner basis, and extracting coefficients yields: C(x1 x4 − x3 x2 ) = C(x1 , x2 , x3 , x4 )G3 . An interesting example for calculating the ideal HG from generators of the ideal Jk(x)G comes from the cyclic group G generated by ω · Idn where ω denotes an lth primitive root of unity. The invariant field k(x)G can be generated by n rational functions alone: k(x)G = k(xn1 , xx12 , xx13 , . . . , xxn1 ). But to generate the invariant ring k[x]G one needs all (exponentially many) monomials of degree l. The ring k[x]G proves that the Noether bound on the number of ring generators one needs is tight [16, Proposition 2.1.5]. This change from few field generators to many ring generators can be observed when passing from Jk(x)G to HG . Example 4. For the group G = i · Id3  with the invariant field k(x41 , xx12 , xx13 ) the ideal Jk(x)G ✂C(x1 , x2 , x3 )[Z1 , Z2 , Z3 ] is generated by Z14 −x41 , x1 Z2 −x2 Z1 , x1 Z3 − x3 Z1 (it is already saturated with respect to Z2 Z3 ). Substituting the field elements x1 , x2 , x3 by variables X1 , X2 , X3 and computing a saturation with respect to all d ∈ k[X1 , X2 , X3 ] yields the generating system −X14 +Z14 , −X13 X2 +Z13 Z2 , −X3 X13 +Z13 Z3 , −X22 X12 +Z22 Z12 , −X3 X2 X12 + Z3 Z2 Z12, −X32 X12 + Z32 Z12 , −X23 X1 + Z23 Z1 , −X3 X22 X1 + Z22 Z3 Z1 , −X1 X2 X32 + Z2 Z32 Z1 , −X33 X1 + Z33 Z1 , −X1 Z2 + X2 Z1 , X3 Z1 − X1 Z3 , −X24 + Z24 , −X3 X23 + Z23 Z3 , −X32 X22 + Z22 Z32 , −X33 X2 + Z33 Z2 , X3 Z2 − X2 Z3 , −X34 + Z34 . Following Derksen’s algorithm we set the variables Z1 , Z2 , Z3 to zero and obtain all monomials of degree four. This already is a generating system for the invariant ring. Acknowledgements: We thank Rainer Steinwandt for [18,19].

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References 1. Thomas Becker and Volker Weispfenning. Gr¨ obner Bases: A Computational Approach to Commutative Algebra. In cooperation with Heinz Kredel. Graduate Texts in Mathematics. Springer, New York, 1993. 2. David Cox, John Little, and Donal O’Shea. Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer, New York, 1992. 3. W. Decker and T. De Jong. Gr¨ obner Bases and Invariant Theory. In Bruno Buchberger and Franz Winkler, editors, Gr¨ obner Bases and Applications (Proc. of the Conference 33 Years of Gr¨ obner Bases), volume 251 of London Mathematical Society Lecture Notes Series. Cambridge University Press, 1998. 4. Harm Derksen. Computation of reductive group invariants. Preprint1 , 1997. 5. David Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Springer, New York, 1995. 6. P. Gianni, B. Trager, and G. Zacharias. Gr¨ obner bases and primary decomposition of polynomial ideals. Journal of Symbolic Computation, 6:149–167, 1988. 7. M. Grassl, M. R¨ otteler, and Th. Beth. Computing Local Invariants of Quantum Bit Systems. Physical Review A, 58(3):1833–1839, September 1998. 8. Gregor Kemper. An Algorithm to Determine Properties of Field Extensions Lying over a Ground Field. IWR Preprint 93-58, Heidelberg, Oktober 1993. 9. Gregor Kemper. Das Noethersche Problem und generische Polynome. Dissertation. Universit¨ at Heidelberg, August 1994. 10. Gregor Kemper. Calculating Invariant Rings of Finite Groups over Arbitrary Fields. Journal of Symbolic Computation, 21(3):351–366, M¨ arz 1996. 11. J¨ orn M¨ uller-Quade and Thomas Beth. Computing the Intersection of Finitely Generated Fields. Presented on the ISSAC poster session, 1998. An extended version is submitted for publication. 12. J¨ orn M¨ uller-Quade and Martin R¨ otteler. Deciding Linear Disjointness of Finitely Generated Fields. In Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation. ACM-Press, 1998. 13. J¨ orn M¨ uller-Quade, Rainer Steinwandt, and Thomas Beth. An application of Gr¨ obner bases to the decomposition of rational mappings. In Bruno Buchberger and Franz Winkler, editors, Gr¨ obner Bases and Applications (Proc. of the Conference 33 Years of Gr¨ obner Bases), volume 251 of London Mathematical Society Lecture Notes Series. Cambridge University Press, 1998. 14. L. Smith. Polynomial Invariants of Finite Groups. A. K. Peters, Wellesley, Massachusetts, 1995. 15. T. A Springer. Aktionen Reduktiver Gruppen auf Variet¨ aten. In H.-P. Kraft, P. Slodowy, and T. A. Springer, editors, Algebraic Transformation Groups and Invariant Theory, pages 3–39. Birkh¨ auser, 1989. 16. Bernd Sturmfels. Algorithms in Invariant Theory. Texts and Monographs in Symbolic Computation. Springer, Wien, 1993. 17. Moss Sweedler. Using Groebner Bases to Determine the Algebraic and Transcendental Nature of Field Extensions: return of the killer tag variables. In G´erard Cohen, Teo Mora, and Oscar Moreno, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes 10th International Symposium, AAECC-10, volume 673 of LNCS, pages 66–75, Berlin; Heidelberg, 1993. Springer. 18. O. Zariski and P. Samuel. Commutative Algebra, volume 1. Springer, 1958. 19. O. Zariski and P. Samuel. Commutative Algebra, volume 2. Springer, 1960. 1

see http//www.math.unibas.ch/pub/derksen/preprints.html

Constructing Elements of Large Order in Finite Fields Joachim von zur Gathen1 and Igor Shparlinski2 1

FB Mathematik-Informatik, Universit¨ at Paderborn 33095 Paderborn, Germany [email protected] 2 Department of Computing, Macquarie University Sydney, NSW 2109, Australia [email protected]

Abstract. An efficient algorithm is presented which for any finite field Fq of small characteristic finds an extension Fqs of polynomially bounded degree and an element α ∈ Fqs of exponentially large multiplicative order. The construction makes use of certain analogues of Gauss periods of a special type. This can be considered as another step towards solving the celebrated problem of finding primitive roots in finite fields efficiently.

1

Introduction

One of the most important unsolved and notoriously hard problems in the computational theory of finite fields is to design a fast algorithm to construct primitive roots in a finite field Fq of q elements. It is demonstrated in [12] that one can find a primitive root of Fq in exponential time O(q 1/4+ε ) for any ε > 0 and so far this is the best known result. Even the Extended Riemann Hypothesis (ERH) does not imply any essentially better results, see Chapters 2 and 3 of [13]. On the other hand, for many applications instead of a primitive root just an element of high multiplicative order is sufficient. Such applications include but are not limited to cryptography, coding theory, pseudo random number generation and combinatorial designs. Following this idea, the authors [9] have proposed some algorithms to construct elements of exponentially large order for some sufficiently ‘dense’ sequence of extensions of a prime field Fp of small characteristic. The algorithms are based on studying multiplicative orders of Gauss periods in finite fields. As an additional bonus in many cases the corresponding large order elements are generators of normal bases as well. On the other hand, for several applications it is desirable to provide a large order element either in some given field Fq or at least in a not too large extension Fqs of it. This work as well as its predecessor [9] provides yet another contribution to the theory of Gauss periods over finite fields and their generalizations and analogues, which have already proved to be useful for a number of various applications [1,2,3,4,5,6,7,8,9]. Apparently Gauss periods are worth a further study Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 404–409, 1999. c Springer-Verlag Berlin Heidelberg 1999


Constructing Elements of Large Order in Finite Fields

405

of their properties and areas of possible applications. In particular, it is interesting to remark that numerous experimental results (which can be found in the above papers) indicate that Gauss periods often produce primitive roots and thus generate primitive normal bases. Here we concentrate on the special case of certain analogues of Gauss periods in Fqn of type (n, 2). We let r be an integer relatively prime to q, and n = ϕ(r), the Euler function. Let β be a primitive rth root of unity; this is an element of Fqn . Then we consider the element α = β + β −1 ∈ Fqn . (1) We recall that if r is prime, then α is called a Gauss period of type (n, 2) over Fq . However, for composite r the Gauss period of type (n, 2) are defined in a different way, see [2]. It is well-known that the minimal polynomial of β over Fq is of degree t, where t is the multiplicative order of q modulo r. Thus t|ϕ(r) = n. If we know the minimal polynomial of β over Fq (that is, we have factored xr − 1 over Fq ), then that of α can be determined by linear algebra over Fq in polynomial time. While a deterministic polynomial time algorithm to factor polynomials over finite fields is not known, there are many efficient (probabilistic and deterministic, unconditional and ERH–dependent) methods. We just mention that xr − 1 can be completely factored over the filed Fq of characteristic p with the following number of arithmetic operations in Fq : ◦ (r log q)O(1) probabilistically, ◦ p1/2 (r log q)O(1) deterministically, ◦ (r log q)O(1) deterministically under the ERH. More precise forms of these assertions and further details can be found in [13], Section 1.1. In this paper, as in [9], we do not estimate the cost of constructing α by using (1) (which, as we mentioned above, may depend on the selected computational model or some unproved number theoretic conjecture, but is fairly small anyway). Our result is an explicit formula for parameters of the construction (1) such that the corresponding α is of large period. As in in the previous paper [9], our technique is based on some results about the distribution of exponential functions in residue classes modulo an integer. Namely, we make use some results of Korobov [10] and [11] which provide very precise information about the period and the distribution of exponential functions modulo an integer divisible by a high power of a small prime.

2

Main Result

The following theorem describes how, for a prime power q = pk , one can select the degree s of an appropriate extension of Fq which is polynomially bounded in log q and such that the construction (1) for certain values of parameters produces an element α ∈ Fqs of exponentially large order.

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Theorem 1. Let q = pk be the kth power of a prime number p with gcd(p, k) = 1 and let ∆ > 4 be a constant. We define
5, if p = 3, l= 3, if p = 3,   ∆ log k m= n = ϕ(r), s = n/k. , r = k 2 lm , log l Then s ≤ 4k ∆+1 is an integer and the element α = β + β −1 ∈ Fpn = Fqs , where β is a primitive rth root of unity over Fp , has multiplicative order at least 2ck where

∆/2−1

−10k−1

 3/2  2 /5, c = 2−1/2 (p − 1)−1 ,  1/2 2 2 (p − 1)−1 ,

,

if p = 3, if p ≡ 1 mod 3, if p ≡ 2 mod 3.

Proof. The integrality of s follows from the identity n = ϕ(r) = ϕ(k 2 lm ) = kϕ(klm ). We also have s = ϕ(klm ) ≤ (l − 1)klm−1 < 4klm−1 ≤ 4k ∆+1. To derive the claimed lower bound on the order of α, we denote by r0 the product of all prime divisors of r. Obviously r0 ≤ lk ≤ 5k. Let t and t0 be the multiplicative orders of p modulo r and r0 , respectively, so that t < r and t0 < 5k. Also, let τ and τ0 be the multiplicative orders of p modulo lm and l, respectively. We define γ as the largest power of l which divides pτ0 − 1. Thus lγ ≤ pτ0 − 1. Korobov [10], Remark after Lemma 1, shows that τ ≥ τ0 lm−γ ≥

τ0 lm ; −1

pτ0

see also [11], beginning of Section 1. If p ≡ 1 mod 3, then τ0 = 1, and τ0 1 = . pτ0 − 1 p−1 If p ≡ 2 mod 3, then τ0 = 2, and τ0 2 = 2 . pτ0 − 1 p −1

(2)

Constructing Elements of Large Order in Finite Fields

407

If p = 3, then τ0 = 4, γ = 1, and from (2) we conclude that τ ≥

4 m l . 5

In all cases, we have t ≥ τ ≥ 21/2 clm = 21/2 crk −2 .

(3)

We also have t0 ≤ r0 ≤ 5k.  We define h = (r/2)1/2 − t0 r/t , and consider the set S = {i: 0 ≤ i < t and 1 ≤ (pi rem r) ≤ h}, where (pi rem r) ∈ N is the positive remainder of pi on division by r. We note that gcd(p, r) = 1. Korobov [10], Lemma 2, implies that #S − th/r ≤ t0 ; (4) see also [11], Fact 2. Thus we have #S ≤ th/r + t0 ≤ t(2r)−1/2 , and

 2h · #S ≤ 2 (r/2)1/2 − t0 r/t ·t(2r)−1/2 < t. Now we recall that the degree of the minimal polynomial of a primitive rth root of unity β over Fp equals t. Let U, V ⊆ S be two different subsets, and u=



pi ,

v=

i∈U



pj .

j∈V

Assume that αu = αv . Then

i j (β + β −1 )p − (β + β −1 )p 0 = αu − αv = =β

−u

i∈U

(β

2pi

+ 1) − β

i∈U

−v

j∈V j

(β 2p + 1).

j∈V

We define the sets E = {pi rem r: i ∈ U}, and the numbers A=

e∈E

e,

F = {pj rem r: j ∈ V }, B=



f.

f ∈F

Since i < t = ordr p for each i ∈ S, we have pi ≡ pj mod r for i = j in S. In particular, E = F . If, say, A ≤ B, then we find that β is a root of the polynomial

 2f  Q(x) = xB−A x2e + 1 − x + 1 ∈ Fq [x]. e∈E

f ∈F

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We show that Q(x) is a nonzero polynomial. Indeed, if A < B, this is obvious. Otherwise, we have A = B and the monomial x2g occurs in Q with nonzero coefficient, where g is smallest element which belongs to one and only one of the sets E and F . Furthermore, deg Q ≤ 2h#S < t; thus our assumption αu = αv is false. Therefore, we have at least 2#S distinct powers of α. From (3) and (4) we see that #S ≥ th/r − t0 ≥ t(2r)−1/2 − 2t0 − t/r ≥ cr1/2 k −2 − 2t0 − t/r = clm/2 k −1 − 2t0 − t/r ≥ clm/2 k −1 − 10k − 1 ≥ ck ∆/2−1 − 10k − 1, 

and we obtain the desired statement. In particular, the element α described in Theorem 1 is of order at least 2(c+o(1))k

∆/2−1

,

where “o(1)” tends to zero as k goes to infinity.

3

Remarks

We give several examples of the parameter selection. Since c ≥ 21/2 p−2 , the choice 7 + 4 log2 p ∆=4+ log2 k yields an element of order at least 26k−1 in an extension of degree at most 128p4 k 5 . Selecting ∆ = 6 we obtain an element of order at least 22

1/2

(k/p)2 −10k−1

in an extension of degree at most 4k 7. These estimates are to be compared to the order pks − 1 of a primitive element in Fqs . A natural way to phrase our question is as follows. Given q = pk and t, compute an element of order at least t in an extension of Fq whose degree we want to choose as small as possible. For this task, determining ∆ from the equation 2k ∆ = p4 k 2(10k + 1 + log2 t)2 , (so that ∆ > 4) and applying our construction, we obtain the upper bound s ≤ 2p4 k 3 (10k + 1 + log2 t)2 on the degree s of the corresponding extension; this is polynomial in k and log t. We finish by posing several open problems. Question 1. Obtain lower bounds on multiplicative orders of Gauss periods of type (n, 2) with composite r = 2n+1, see [2] (rather than for their analogues (1).

Constructing Elements of Large Order in Finite Fields

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Question 2. Obtain lower bounds on multiplicative orders of Gauss periods and their analogues of type (n, k) for k > 2, which can be defined similarly, see [1,2,3], [4,5,6,7,8]. Question 3. Find a fast explicit construction of elements of exponentially large multiplicative order which is applicable to all finite fields. The last question is probably quite difficult in its full generality, but the results of [9] and of this work indicate that it may be feasible for large fields of small characteristic.

References 1. I. F. Blake, X.H. Gao, A. J. Menezes, R. C. Mullin, S. A. Vanstone and T. Yaghoobian, Applications of finite fields, Kluwer Acad. Publ., 1993. 2. S. Feisel, J. von zur Gathen and A. Shokrollahi, ‘Normal bases via general Gauss periods’, Math. Comp., 68 (1999), 271–290. 3. S. Gao, J. von zur Gathen and D. Panario, ‘Gauss periods and fast exponentiation in finite fields’, Proceedings LATIN ’95, Springer Lecture Notes in Comp. Sci., 911 (1995), 311–322. 4. S. Gao, J. von zur Gathen and D. Panario, ‘Gauss periods: Orders and cryptographical applications’, Math. Comp., 67 (1998), 343–352. 5. S. Gao and H. W. Lenstra, ‘Optimal normal bases’, Designs, Codes and Cryptography, 2 (1992), 315–323. 6. S. Gao and G. L. Mullen, ‘Dickson polynomials and irreducible polynomials over finite fields’, J. Number Theory, 49 (1994), 118–132. 7. S. Gao and S. Vanstone, ‘On orders of optimal normal basis generators’, Math. Comp., 64 (1995), 1227–1233. 8. J. von zur Gathen and M. J. N¨ ocker, ‘Exponentiation in finite fields: Theory and practice’, Proceedings AAECC’97, Springer Lecture Notes in Comp. Sci., 1255 (1997), 88–113. 9. J. von zur Gathen and I. E. Shparlinski, ‘Orders of Gauss periods in finite fields’, Appl. Algebra in Engin., Commun. and Comp., 9 (1998), 15–24. 10. N. M. Korobov, ‘Exponential sums with exponential functions and the distribution of digits in periodic fractions’, Matem. Zametki, 8 (1970), 641–652. Matem. Notes, 8 (1970), 831–837. 11. N. M. Korobov, ‘On the distribution of digits in periodic fractions’, Matem. Sbornik , 89 (1972), 654–670. Mat. USSR-Sb., 18 (1972), 659–676. 12. I. E. Shparlinski, ‘On finding primitive roots in finite fields’, Theor. Comp. Sci. 157 (1996), 273–275. 13. I. E. Shparlinski Finite fields: Theory and computation, Kluwer Acad. Publ., Dordrecht, 1999.

New Lower Bounds on the Periodic Crosscorrelation of QAM Codes with Arbitrary Energy Serdar Bozta¸s
Department of Mathematics, RMIT University GPO Box 2476V, Melbourne 3001, Australia [email protected]

Abstract. In this paper, we consider a recent technique of Levenshtein [9] which was introduced to prove improved lower bounds on aperiodic correlation of sequence families over the complex roots of unity. We first give a new proof of the Welch [12] lower bound on the periodic correlation of complex roots of unity sequences using a modification of Levenshtein’s technique. This improves the Welch lower bound in the case that the family size is “large enough” compared to sequence length. Our main result is an improved lower bound on the periodic complex valued sequences with QAM-type alphabets. This result improves an earlier result by Bozta¸s [2]. To achieve this result, we extend the Levenshtein technique in a new direction. Here, we assume that sequences are drawn from energy “shells” which are no more than PSK signal sets with the scaling determining the signal energy. It turns out that, if the weights are associated with the energy of the sequence in question, the Levenshtein technique can be modified to obtain lower bounds. The new bound is a non-convex function of a set of suitably chosen “weights.” We demonstrate that our results improve those of [2] with a concrete example.

1

Introduction

n Let Em be the set of vectors of length n over the alphabet

Em = {1, ξ, ξ 2 , . . . , ξ m−1 | ξ = e2πi/m }. Throughout this paper m is an arbitrary positive integer satisfying m ≥ 2. For n n and y = (y0 , y1 , ..., yn−1 ) ∈ Em the periodic any x = (x0 , x1 , ..., xn−1 ) ∈ Em crosscorrelation function θ(x, y; l) is defined as follows: θ(x, y; l) =

n−1


xj yj+l ,

l = 0, 1, ..., n − 1

j=0


Supported by an RMIT Faculty of Applied Science Grant.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 410–419, 1999. c Springer-Verlag Berlin Heidelberg 1999 

(1)

New Lower Bounds on the Periodic Crosscorrelation of QAM Codes

411

where z denotes the complex conjugate of z, and the addition in the subscript of n yj+l is modulo n. Every code C ⊆ Em is characterized by the maximum periodic crosscorrelation, maximum periodic nontrivial autocorrelation and maximum periodic correlation magnitudes respectively: θc (C) = max{|θ(x, y; l)| : x, y ∈ C, x = y, l = 0, 1, ..., n − 1}, θa (C) = max{|θ(x, x; l)| : x ∈ C, l = 1, ..., n − 1}, θ(C) = max{θa (C), θc (C)}. The first lower bound on θ(C) is due to Welch [12] and applies to arbitrary n . In [12] a lower complex sequences with constant energy, and hence to C ⊆ Em bound on θa (C) for such sequences was also derived. Another important lower n bound on θ(C) for C ⊆ Em is that of Sidelnikov [11]. n The class of sequence families which are subsets of Em , for some m are by far the best understood from the point of view of sequence design. For asynchronous CDMA applications, the case of interest is usually M ≥ n2 , where M is the total number of sequences, which allows ≥ n simultaneous users. For m = 2 and n = 2k − 1, k ≥ 3, and k odd, the sequence family due to Gold [5] with  M = n(n + 2), is optimal with θ(C) = 1 + 2(n + 1). For m = 4 and n = 2k − 2, k ≥ 3, k odd, the so-called Family A with M = n(n + 2) (see [1] and the references therein, √ as well as [4] for its connections to Z4 -linearity) is optimal with θ(C) = 1 + n + 1. For m = p, an odd prime, the family due to Kumar √ and Moreno with M = n(n+1) is optimal [8] with n = pk −1, and θ(C) = 1+ n + 1. n This concludes our brief survey of important sequence families in Em from the asynchronous CDMA point of view. See [6], [7] and [10] for more details.

Recently, Levenshtein [9] introduced a new method which improves the aperiodic Welch bound from [12] in the case of sequences in E2n . We have subsequently n shown [3] that Levenshtein’s method works for sequences in Em , for any m ≥ 2. To address the aperiodic case, Levensthein introduced “weights” for shifts of “zero-padded” code words. A suitable choice of weights allowed him to substantially improve Welch’s bound for most cases of interest. In the next section, we demonstrate that the original Welch lower bound on periodic correlation can be obtained by Levenshtein’s method when the weights are uniform. We then modify Levenshtein’s method so that it can be applied to the periodic correlation sequences over more general alphabets which can be n thought of as the disjoint union of scalings of Em . This gives us the freedom n to obtain QAM type of alphabets by modifying the scalings of Em . This will become clearer in the sequel. We make extensive use of the machinery introduced in [9] and believe that this generalization would not be possible without this machinery.

2

n Deriving the Welch Lower Bound on Em

n We will derive the Welch lower bound by considering a code C ⊆ Em to be made up of all the codewords and all their distinct cyclic shifts. Sometimes, it will be

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Serdar Bozta¸s

convenient to associate an ‘exponent’ vector x ∈ Znm with the corresponding n vector Φ(x) ∈ Em given by Φ(x) = (Φ(x1 ), . . . , Φ(xn )) Φ : Zm → E, Φ(u) → ξ u . n For any two subsets A and B of Em define the value F (A, B) :=

1

2 | x, y | , |A||B|

(2)

x∈A y∈B

where u, v denotes the complex inner product of vectors u and v. The desired n results for a code C ⊆ Em will be obtained with the help of finding lower and upper bounds on F (C, C). n Lemma 1. For any code C ⊆ Em of size M,

θ2 (C) ≥

MF (C, C) − n2 . M −1

(3)

Proof. Considering separately in (2) the cases x = y, (in this case | x, y | = n), and x = y we get M 2 F (C, C) ≤ Mn2 + M(M − 1) θ2 (C), which gives the result required. We now prove a Lemma which is going to be crucial in the rest of this paper. Lemma 2. Let U be the unit circle in the set of complex numbers. For any x ∈ U n, n F ({x}, Em )=n Proof. Using (2) we have n F ({x}, Em )=

1
1 2 | x, y | = n Γn,m = n n m m n y∈Em

provided we can show that ΓL,m :=




| x, y | = mL L 2

n y∈Em

L . Note that if we define Φ(v) = (Φ(v1 ), . . . , Φ(vn−l )) = (x1 , . . . , for any x ∈ Em xn−l ), and Φ(z) = (Φ(z1 ), . . . , Φ(zn−l )) = (yl+1 , . . . , yn ), then we can write
ΓL,m = | Φ(v), Φ(z) |2 , Φ(v) ∈ U L arbitrary. L Φ(z)∈Em

Here, zk ∈ Zm for k = 1, . . . , L, while vk ∈ [0, m) for k = 1, . . . , L. We also define sk = vk /m, which allows us to write ξ vk = e2πisk . The sum ΓL,m turns out to be independent of Φ(v) ∈ U L , and is evaluated below.

ΓL,m = Φ(v), Φ(z) Φ(v), Φ(z) = Φ(v), Φ(z) Φ(v), Φ(z) L Φ(z)∈Em

z∈ZL m

New Lower Bounds on the Periodic Crosscorrelation of QAM Codes






=

0≤z1 ,···,zL ≤m−1






=

L


 ξ

k=1

L


 ξ

zl −vl

l=1




L +

0≤z1 ,···,zL ≤m−1

vk −zk

413



e2πi(sk −sl ) ξ zl −zk 

1≤k=l≤L

separating the two sums and interchanging the order of summation in the second sum now gives

e2πi(sk −sl ) ξ zl −zk = L mL + 1≤k=l≤L



and the inner sum proof.

0≤z1 ,···,zL ≤m−1

ξ zl −zk can easily be seen to be zero, which completes the

An immediate result of this Lemma is that n n n , Em ) = F ({x}, Em ). F (Em

n Lemma 3. For any code C ⊆ Em , n n , Em ) = n. F (C, C) ≥ F (Em n Proof. Define functions fi,j (X), i, j = 0, 1, ..., 2n − 2, on Em as follows. If X = (x0 , x1 , ..., xn−1 ), then fi,j (X) = xi xj . Using this notation one can rewrite (2) in the following form:

F (A, B) =



1 fi,j (X)fi,j (Y ). |A||B| X∈A Y ∈B

n , Let us verify (as in [9]) that for any two subsets A and B of Em 2

(F (A, B)) ≤ F (A, A) · F (B, B).

(4)

First change the order of summation and use the Cauchy inequality for codes:

2

2



2n−2







2n−2









fi,j (X)fi,j (Y ) =

fi,j (X) fi,j (Y )

=

X∈A Y ∈B i,j=0

i,j=0 X∈A

Y ∈B ≤







2n−2

i,j=0 X∈A

=

2n−2





i,j=0 X∈A

fi,j (X)



2

2

2n−2





fi,j (X) · fi,j (Y ) =






Y ∈A

i,j=0 Y ∈B

fi,j (Y ) ·

2n−2





i,j=0 X∈B

fi,j (X)


Y ∈B

fi,j (Y ).

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One more change in the order of summation implies (4). Then we note that for n n n n any A ⊆ Em we have F (A, Em ) = F (Em , Em ) by Lemma 2. Therefore the use n of (4) with A = C and B = Em completes the proof.  4 /|A||B| gives a bound | x, y | We remark that using F (A, B) := x∈A y∈B on θ4 (C). This bound is included in the corollary below: Corollary 1. (New ‘Welch’ Lower Bounds) We have θ2 (C) ≥

Mn − n2 , M −1

θ4 (C) ≥ 3n2 − 2n − (n4 /M).

Note that M = nt if there are t cyclically distinct sequences in the code. The new ‘Welch’ bound on θ4 (C) will be tighter than the original Welch bound for θ4 (C) if Mn2 − 2Mn ≥ 0. where the ratio of the two bounds will be (3/2)1/4 ≈ 1.107 as M and n tend to infinity. The corresponding two cases of Welch’s lower bounds are reproduced here for ease of comparison. Theorem 1. (Welch) We have θ2 (C) ≥

3

Mn − n2 , M −1

θ4 (C) ≥

2Mn2 − n4 . M −1

The Main Results

We first restrict ourselves to the case of QAM signals with two “energy shells” for ease of exposition. 3.1

The Lower Bound for Two Shells

In this section, we consider a signal alphabet of the form √ n n X = Em ∪ aEm 0 1

(5)

where the union is assumed to be disjoint, i.e., a codeword belongs either to n n , or to Em but not to, say, Em1 × Em2 × . . . × Em1 . As an example, if we Em 1 2 √ √ 3 take X = E4 ∪ 2E43 , the codewords√(−1 ,√ + i , − i), 2(+1 , − i , − 1) are allowed while √ nthe codeword (−1 , 2 , 2i) is not allowed. To any vector n x ∈ Em ∪ aEm1 , we assign a weight wc(x), where 0 wk ≥ 0, k = 0, 1, and w0 + w1 = 1, where c(x), is the “class” of the vector x, i.e.,  n ,  w0 if x ∈ Em 1 wc(x) = √ n  w1 if x ∈ aEm . 2

(6)

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New Lower Bounds on the Periodic Crosscorrelation of QAM Codes

415

For any two subsets A and B of X , we define the value 1

2 wc(x)wc(y) | x, y | , F (A, B) := |A||B|

(8)

x∈A y∈B

where u, v denotes the complex inner product of vectors u and v. It is also convenient to define an “unnormalized F (A, B)” via G(A, B) := |A||B|F (A, B). We now proceed to prove the generalized versions of the results in the previous section. Lemma 4. For any code C ⊆ X of size M, θ2 (C) ≥

M 2 F (C, C) − (w02 M0 + w12 M1 a2 )n2 (w0 M0 + w1 M1 )2 − (w02 M0 + w12 M1 )

where C = C0 ∪

(9)

√ aC1 , and

n |Ci | = Mi , Ci ⊆ Em , i

i = 0, 1

with M = M1 + M2 . Proof. Considering separately the cases x = y (in this case | x, y | = n for x ∈ C0 , and | x, y | = an for x ∈ C1 , ), and x = y (if x and y are in the same “class” we get the upper bounds Mi (Mi − 1)wi2 θ2 (C), i = 0, 1, while if they are in different classes we get the upper bound 2M0 M1 w0 w1 θ2 (C)) we obtain G(C, C) = M 2 F (C, C) ≤ (w02 M0 + w12 M1 a2 )n2 + (M02 − M0 )w02 θ2 (C) + (M12 − M1 )w12 θ2 (C) + 2M0 M1 w0 w1 θ2 (C) which gives the result required. Lemma 5. We have F ({x}, X ) = where γ =

  2   n γw0 + (1 − γ)w0 w1 a

n if x ∈ Em , 0

   n γw0 w1 a + (1 − γ)w12 a2

n if x ∈ Em , 1

mn 0 n. mn 0 +m1

Proof. Note that √ n n ∪ aEm )= G({x}, X ) = G({x}, Em 0 1

2 | x, y | + wc(x)w1 a = wc(x)w0 n y∈Em 0

which gives G({x}, X ) =

 2  w0 Γn,m0 + w0 w1 aΓn,m1 

w0 w1 aΓn,m0 + w12 a2 Γn,m1

| x, y |

n y∈Em 1

n if x ∈ Em , 0 n if x ∈ Em , 1

2

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Serdar Bozta¸s

and yields the result required after using Lemma 2 (which states that Γn,mi = nmni , i = 0, 1) and noting that F ({x}, X ) = G({x}, X )/(mn0 + mn1 ) . Now, we can investigate the conditions under which √ n √ n n n F (C, C) ≥ F (X , X ) = F (Em ∪ aEm , Em ∪ aEm ) 0 1 0 1 can be made to hold. Lemma 6.      F (C, X ) = n δ γw02 + (1 − γ)w0 w1 a + (1 − δ) γw0w1 a + (1 − γ)w12 a2 √ √ n n ∪ aEm Proof. Note that for any code C = C0 ∪ aC1 which is a subset of Em 0 1 we have √ √ √ n n G(C0 ∪ aC1 , X ) = G(C0 ∪ aC1 , Em ∪ aEm )= 0 1   = n M0 mn0 w02 + M1 mn1 w0 w1 a + M0 mn0 w0 w1 a + M1 mn1 w12 a2 which gives F (C0 ∪

√

aC1 , X ) =

√ G(C0 ∪ aC1 , X ) = (M0 + M1 )(mn0 + mn1 )

  = n γ(δw02 + (1 − δ)w0 w1 a) + (1 − γ)(δw0 w1 a + (1 − δ)w12 a2 ) where δ = M0 /(M0 + M1 ). We now apply Lemma 6 to C = X to obtain F (X , X ) = n [γw0 + (1 − γ)aw1 ]

2

while      F (C, X ) = n δ γw02 + (1 − γ)w0 w1 a + (1 − δ) γw0w1 a + (1 − γ)w12 a2 hence the choice δ = γ, is sufficient to ensure F (C, X ) ≥ F (X , X ) holds (in fact, F (C, X ) = F (X , X ) holds here). We now appeal to an obvious weighted generalization of Lemma 3 to obtain that for the case δ = γ, we still have F (C, X )2 ≤ F (X , X )F (C, C) which now becomes F (X , X )2 ≤ F (X , X )F (C, C) which implies F (X , X ) ≤ F (C, C). Hence, for such codes whose codeword distributions across energy “shells” (i.e., M0 /(M0 + M1 ) ) are equal to the word distributions for the total codeword space, (i.e., mn0 /(mn0 + mn1 )) we have the following:

New Lower Bounds on the Periodic Crosscorrelation of QAM Codes

417

Theorem 2. When δ = γ, we have θ2 (C) ≥ =

max

0≤w≤1

max

0≤w≤1

M 2 F (X , X ) − (w02 M0 + w12 M1 a2 )n2 (w0 M0 + w1 M1 )2 − (w02 M0 + w12 M1 ) Mn(γw + (1 − γ)a(1 − w))2 − (w2 γ + (1 − w)2 a2 (1 − γ))n2 M(γw + (1 − γ)(1 − w))2 − (w2 γ + (1 − w)2 (1 − γ))

Proof. Let w0 = w, w1 = 1 − w, and use Lemma 4. 3.2

The General Lower Bound

Here we consider a signal set with an arbitrary number, say v, of energy “shells.” Let the signal alphabet be of the form √ √ n n X = a0 Em ∪ . . . ∪ av−1 Em (10) 0 v where the union is assumed to be disjoint, i.e., a codeword belongs √ n to a unique √ n n ai Em , for some i = 0, . . . , v − 1. To any vector x ∈ E ∪ aEm1 , we assign m0 i a weight wc(x), where wk ≥ 0, k = 0, 1, . . . , v − 1 and w0 + · · · + wv−1 = 1,

(11)

where c(x), is the “class” of the vector x (compare Section 3.1). The definitions of F (A, B) and G(A, B) are as in Section 1. The quantities Mi , γi , δi , for i = 0, . . . , v − 1, all have the same meanings as in Section 3.1. The arguments to prove Theorem 4 parallel those in Section 3.1 and are omitted. Theorem 3. By choosing δi = γi , i = 0, . . . , v − 1, we have v−1 v−1 Mn( i=0 wi γi ai )2 − n2 ( i=0 wi2 γi a2i ) 2 θ (C) ≥ max v−1 v−1 w0 +...+wv−1 =1 M( i=0 wi γi )2 − ( i=0 wi2 γi ) wi ≥0,i=0,...,v−1

4

Discussion and Conclusions

Note that all the results we have derived apply to constellations which can be obtained arbitrary rotations of the shells relative to each other due to the generality of Lemma 3. The family of bounds we have obtained are in general non-convex as a function of (w0 , . . . , wv−1 ). We now state a special case of a previously proved theorem [2] on CDMA sequences with arbitrary energy distribution. Theorem 4. (Bozta¸s) For any complex sequence family C with arbitrary energy distribution, length n and M cyclically distinct sequences x1 , . . . , xM , we have the lower bound  2  v−1 v−1   2  

1 γi a2i − γi a4i  θ2 ≥ M E − E 2 /(Mn − 1) ≈ n  M i=0

i=0

where E k is the average of the k th energy moment of the sequence family.

418

Serdar Bozta¸s Im(z)

✻ t

t

(0, 1)

(−1, 0)

t

t

t

t

t

(0, −1)

(1, 0)

✲ Re(z)

t

Fig. 1. A two shell constellation corresponding to E4n ∪

√

2E4n .

We now consider an example which uses the symbol constellation of Figure 1. Example 1. Let γi = 1/2, mi = 4, i = 0, 1, a0 = 1, a1 = 2, and M = nt, i.e., t cyclically distinct sequences each of length n. Then the bound in Theorem 3 5 becomes θ2 ≥ n( 94 − 2nt ). However, the new bound in Theorem 2 becomes θ ≥ 2

 2  2 Mn (w + 2(1 − w)) − n2 w2 + 22 (1 − w)2 4 2 M 1 2 2 4 (w + (1 − w)) − 2 (w + (1 − w) )

whichis a ratio oftwo quadratics. w → 0 yields  Simply letting  θ2 ≥ 4Mn − 8n2 /(M − 2) = 4n2 (t − 2) /(nt − 2) ≈ 4n [(t − 2)/t] which is clearly better for typical values t ≥ n of practical interest and tends to 4n as t → ∞.

5

Conclusions

In this paper, we have developed a new lower bound for the periodic correlation of sequence families. It must be emphasized that the new bound, while tighter than the previously derived lower [2] bound on QAM sequences, applies to a more restricted type of sequence, i.e., sequences which are restricted to have symbols in one of the possible “shells” of the constellation. Table 1 compares various performance parameters of different constellations, for the case that M ≈ n2 , which is of most interest for asynchronous √ CDMA applications. Note that it is possible to use the constellation E4n ∪ 2E4n and obtain improved minimum Euclidean distance at the cost of degraded Peak-Off-Peak correlation ratio over 8-PSK. The challenge is to apply algebraic design techniques to design families that approach the bounds derived here.

New Lower Bounds on the Periodic Crosscorrelation of QAM Codes

419

Table 1. Performance parameters of various constellations for M ≈ n2 . The symbol ∗ means that a known sequence family achieves the given value. Sequence Type

Data Rate Minimum Peak √ to (bits/symbol) Off-Peak Ratio E/θ(C)

4-PSK

2

√n n

=

√ n

∗

8-PSK

3

√n n

=

√ n

∗

E4n ∪

√

2E4n

3

≤ or

√ √n = 12 n 4n √ 2n √ = n 4n

Minimum Euclidean Distance d 2 sin(π/4) =

√ 2

2 sin(π/8) ≈ 0.765 1

Acknowledgments Helpful discussions with A. Baliga, G. Hughes (who improved the proof of Lemma 2) and P. Udaya are gratefully acknowledged.

References 1. S. Bozta¸s, R. Hammons, and P. V. Kumar, “4-phase sequences with near-optimum correlation properties,” IEEE Trans. Inform. Theory IT-38 (1992), 1101–1113. 2. S. Bozta¸s, “Lower bounds on the maximum periodic and aperiodic correlation of signal sets with arbitrary energy distribution”, in Proc. of the International Symposium on Information Theory and Its Applications, Sydney, Australia, 20-25 November 1994, pp. 1007–1012. 3. S. Bozta¸s, “New Lower Bounds on the Aperiodic Crosscorrelation of Codes over mth Roots of Unity”, RMIT Mathematics Research Report 13, September, 1998. 4. A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sol´e, “The Z4 -linearity of Kerdock, Preparata, Goethals and related codes,” IEEE Trans. Inform. Theory IT-40 (1994), 301–319. 5. R. Gold, “Optimal binary sequences for spread spectrum multiplexing,” IEEE Trans. Inform. Theory IT-13 (1967), 619–621. 6. S. W. Golomb, Shift-Register Sequences. Aegean Park Press, 1982. 7. T. Helleseth and V. Kumar, “Sequences with low correlation”, in Handbook of Coding Theory, V. S. Pless and W. C. Huffman, Eds., Elsevier, Amsterdam, 1998. 8. P. Kumar and O. Moreno, “Prime-phase sequences with correlation better than binary sequences,” IEEE Trans. Inform. Theory IT-37 (1991), 603–616. 9. V. I. Levenshtein, “New lower bounds on aperiodic crosscorrelation of binary codes”, IEEE Trans. Inform. Theory, IT-45 (1999), 284–288. 10. D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences,” Proc. IEEE 68 (1980), 593–620. 11. V. M. Sidelnikov, “On mutual correlation of codes”, Dokl. Akad. Nauk SSSR, vol. 196, no. 3, pp. 531–534, 1971. English translation in Soviet Math. Dokl., vol. 12, no. 1, pp. 197–201, 1971. 12. L. R. Welch, “Lower bounds on the maximum cross-correlation of signals”, IEEE Trans. on Inform. Theory, IT-20 (1974), 397–399.

Conjectures on the Size of Constellations Constructed from Direct Sums of PSK Kernels Matthew G. Parker
Department of Informatics, University of Bergen, N-5020 Bergen, Norway [email protected]

Abstract. A general equation is given for the size of complex constellations constructed from the direct sum of PSK-like constellation primitives. The equation uses a generating function whose numerator is a power of a ’coordination polynomial’. Conjectures are also given as to the form and value of these coordination polynomials for various PSK. The study has relevance to error-coding, polynomial residue number theory, and the analysis of random walks.

1

Introduction

Communications systems often transmit data by modulating using Binary or Quaternary Phase Shift Keyed (BPSK or QPSK) or Quadrature Amplitude Modulated (QAM) constellations in the complex plane. But larger constellations can be more bandwidth-efficient and lead to efficient hardware implementation of complex arithmetic and algorithms [1,2]. This paper considers the problem of finding the size of constellations constructed from direct sums of {PSK plus the origin}, referred to here as ’PSK⊕’ constellations. These constellations form lattices for 1,2,3, or 6 PSK primitives, but for any other PSK⊕ there will be residue ’folding’ making the determination of constellation size more complicated. This problem can be recast, for mPSK⊕, as finding an expression for the number of non-identical polynomial residues resulting from the reduction, mod Φm (x), of polynomials in x of Coefficient Weight ≤ n, (for some positive integer, n), and degree < m, where Φm (x) is the mth cyclotomic polynomial in x. Although residue folding is, for many applications, undesirable, it is hoped that an algebraic understanding of PSK⊕ will help in the construction of constellations more suited to communications systems which use PSK⊕ as building blocks. Also, from an algebraic point of view, it is useful to be able to enumerate the residues of polynomials, mod Φm (x). The theorem and conjectures to be presented here are based on computational results. During the course of the work integer sequences, relating to the 8PSK⊕ and 16PSK⊕ constellations, were entered into Sloane’s On-Line Encyclopedia of Integer Sequences [3] and were found to refer, in particular, to the paper by Conway and Sloane on Low Dimensional Lattices [4] which, in turn, references work by O’Keefe [5] and


This work was funded by NFR Project Number 119390/431.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 420–429, 1999. c Springer-Verlag Berlin Heidelberg 1999


Conjectures on Constellations

421

others [6]. Their results have applications to crystallography, and use generating functions which require the specification of a ’Coordination Sequence’. This paper conjectures a general solution to a related problem, although a general form for the Coordination Sequence (Polynomial) has yet to be found. The results could be used to help extend the scope of error coding strategies such as [7,8], and may also be useful for the development of ’Random Walk’ statistics.

2

Statement of the Problem

Define mPSK+ as the set of m + 1 points in the complex plane given by, mPSK+ = {0, 1, w, w2 , . . . , wm−1 } 2πi

where w = e m , and i2 = −1. Define mPSK⊕n as the direct sum of n copies of mPSK+, given by, n−1


{0, 1, w, w2 , . . . , wm−1 }

mPSK ⊕ n =

k=0

We wish to find a formula for dn as n varies over the positive integers, where dn is the number of non-identical points in mPSK⊕n, given by, dn = |

n−1


{0, 1, w, w2 , . . . , wm−1 }|

k=0

For instance, let m = 4. The kernel constellation is {0, 1, w, w2 , w3 }, where 2πi w = e 4 , and, d2 = |

1


{0, 1, w, w2 , w3 }| = |{0, ±1, ±w, ±1 ± w, ±1 ∓ w, ±2, ±2w}| = 13

k=0

As another example, for m = 6 and n = 2, d2 = |{0, ±1, ±w, ±w2, ±2, ±2w, ±2w2 , ±1 ± w, ±1 ∓ w2 , ±w ∓ w2 }| = 19 An algebraic description of the same problem is as follows. Definition 1 The ’Coefficient Weight’, (cw), of a polynomial, f(x), is the sum of it’s coefficient values. In other words cw(f(x)) = f(1).  Let g(x) = i gi xi . Let, Gm,n = {g(x) | 0 ≤ deg(g(x)) < m, gi ≥ 0 ∀i, 0 ≤ cw(g(x)) ≤ n} where deg(a(x)) is the degree of a(x). Let x = e

2πi m

, where i2 = −1. Then,

mPSK ⊕ n = {h(x) | h(x) = g(x) Φm (x) , ∀g(x) ∈ Gm,n } where a b is the residue of a mod b, and Φm (x) is the mth cyclotomic polynomial. Therefore, dn = |mPSK ⊕ n| as before.

422

3

Matthew G. Parker

Computational Results

Tables 1 and 2 show some computed values of dn for various n and m. The number of Euclidean distances, D, refers to the size of the set of values for the absolute (straight-line) distance from each point in mPSK⊕n to the origin. The figures for D are not discussed further in this paper, but are included here for the reader’s interest. Table 1. Constellation and Euclidean Distance Enumerations for Various mPSK⊕n. dn -No of points in constellation. D-No of Euclidean distances. n 1 m dn 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 12 13 14 15 15 16 16 17 18 19 20 21 21 22 22 23 24 25 25 26 27 28 30 31 33 34 35 36 36 37 40 41 45 46 48 49 49 50 50 51 54 55 60 61 75 76 90 91

D 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 dn 10 13 21 19 36 41 55 61 73 113 136 145 163 221 253 265 289 351 406 451 595 666 649 841 1081 1153 1275 1301 1459 1801 2926 4051

D 3 4 5 4 6 6 6 7 7 9 9 10 10 12 12 13 13 15 15 16 18 20 19 22 24 25 27 27 28 31 39 46

3 dn 19 25 56 37 120 129 217 211 253 575 811 833 865 1521 2017 2047 2089 3276 4051 3901 7129 8436 7237 11441 17281

D 5 6 8 6 14 13 17 77 16 29 33 35 33 46 59 59 54 78 89 81 125 138 118 161 213

4 dn 31 41 126 61 330 321 685 551 661 2171 3751

5 6 D dn D dn 7 46 9 64 9 61 12 85 17 9 91 12 127 30 29 681 53 1289 46 1837 99 38 1201 72 38 1441 72 96 132 14176 440

3313

114

7 8 9 10 D dn D dn D dn D dn D 12 85 15 109 18 136 22 166 26 16 113 19 145 24 16 169 20 96 2241

3649

5641

8361

12496 322 63946 1396 11969 310 10825 258 31213 4296 22831 425 65671 1072

22051 246 24949 258 33901 310

And here are a few more partial results for the case m = 8. Table 2. Constellation Enumerations for More 8PSK⊕n. n 11 12 13 14 15 m dn dn dn dn dn 8 11969 16641 22569 29961 39041

4

Some Conjectures

We shall form a generating function for the sequences, dn , where dn is different ∞ for every m. Thus define dm (x) = n=0 dn xn . The following conjecture satisfies all numerical results quoted above,

Conjectures on Constellations

Conjecture 1

423

m

ch (x) h dm (x) = (1 − x)φ(m)+1 where φ is Euler’s Totient Function, h is the square free part of m, and ch (x) is referred to as the hth coordination polynomial. ch (x) is palindromic and deg(ch (x)) = φ(h). The above conjecture omits to specify exactly the form of ch (x). This is an area of further research. However the following theorem determines ch (x) where h is a prime, and two following conjectures satisfy the computational results for h = 2p, p an odd prime, and h = 15, respectively, Theorem 1 cp (x) = Φp (x),

p prime

Theorem 1 was conjectured by the author based on numerical computation. A proof was found by T.Kløve and it is given in Appendix A. Conjecture 2 p−3

c2p (x) =

2


k=0

k

p−1−k

x +x

k
p i

p−1

+x

p−1 2

i=0

2
p i

,

p an odd prime

i=0

Conjecture 3 c15(x) = (1 + x8 ) + 7(x + x7 ) + 28(x2 + x6 ) + 79(x3 + x5 ) + 130x4 The following observation was also made, Conjecture 4

 m|

mn+1 − 1 − dn m−1



From the computational results values of ch (x) have also been partially ascertained for various h as shown in Table 3. All preceding coordination polynomials were computed from the dn sequences using the following strategy. For instance, for m = 6 the dn sequence is computed to be 1,7,19,37,61,91,127,169,..... Thus d6 (x) = 1+7x+19x2 +37x3 +61x4 +91x5 + 127x6 + 169x7 + . . . . Note that φ(6) + 1 = 3 so, from Conjecture 1, we multiply d6 (x) (truncated to degree 7) by (1 − x)3 to get c6 (x) = e(x) + x2 + 4x + 1, where e(x) is some error term due to having truncated d6 (x) to degree 7. In this case e(x) = −217x8 + 380x9 − 169x10 , which is evidently an error term so c6 (x) = x2 + 4x + 1. The same strategy can be used to compute ch (x) for all dn sequences in the table, and hence arrive at the preceding Conjectures 2 - 3 on the form of ch (x).

424

Matthew G. Parker

Table 3. Incomplete Coordination Polynomials for Various Composite h. h 21 30 33 35

5

ch (x) 1 + 9x + 45x2 + 158x3 + 432x4 + 909x5 + . . . ? 1 + 22x + 208x2 + 874x3 + 1480x4 + . . . ? 1 + 13x + 91x2 + 444x3 + 1677x4 + . . . ? 1 + 22x + 208x2 + 874x3 + 1480x4 + . . . ?

Triangle Patterns

An examination of number triangles may give a clue as to how to extend the previous conjectures on coordination polynomials to the more general case. On page 14 of [4] it was observed that the coordination polynomials for the dual lattice, A∗d , satisfy the following ’coordinator’ triangle.

1

1

10

9

8

7

6

46

37

29

1 5 22

1 4 16

93

64

1

1

5 22

6

93

29

1 7

1

37

8

1

1 1 130 256 130 46 10 1 11 56 176 386 386 176 56 11 1 12 67 232 562 1024 562 232 67 12 1

1

1

1

1

1

1

9

1

The pth line of the above triangle, p prime, also provides the coordination polynomials, c2p(x), for Conjectures 1 and 2 of this paper. In the same way we can construct a partial triangle for the c3p (x) case, using our previous computational results. Thus,

1

1 13

1 12 91

1 11 78 444

1 1 8 1 9 45 10 55 212 66 277 902 354 1245 3525 1677 5124 ?

1 7 36 158 635 2346 ? ?

1 6 28 114 432 1499 ? ? ?

1 5 21 79 282 909 ? ? ? ?

1 4 ? ? 130 ? ? ? ? ? ?

1 5 21 79 282 909 ? ? ? ?

1 6 28 114 432 1499 ? ? ?

1 7 36 158 635 2346 ? ?

1 8 45 212 902 3525 ?

1 9 55 277 1245 5124

1 10 66 354 1677

1 11 78 444

1 12 91

1 13

1

where each entry apart from those of the middle three columns seems to be the sum of the three entries immediately above, e.g. 158 = 8 + 36 + 114. Note that the only triangle entries directly computed from computational results are the sequences, 1,4,1, and 1,7,28,79,130,79,28,7,1, and 1,9,45,158,432,909, and 1,13,91,444,1677. All other numbers in the above triangle are nominally filled in to fit the ’sum of three’ conjecture. The c3p (x) coordination polynomial can be read from the pth line of the previous triangle for p prime. For instance, c15 (x) = (1 + x8 ) + 7(x + x7 ) + 28(x2 +x6 )+79(x3 +x5 )+130x4 . Although we do not currently have an equation for c3p(x) it is worth noting that the following triangle is similar to the previous triangle,

Conjectures on Constellations

1

1 13

1 12 91

1 11 78 444

1 1 6 1 7 28 1 8 36 114 1 9 45 158 432 10 55 212 635 1562 66 277 902 2409 5470 354 1245 3588 8781 18714 1677 5187 13614 31083 62907

1 5 21 79 282 972 3273 10385 35412 114586

1 4 15 52 175 576 1869 6000 19107 60460 190333

1 5 21 79 282 972 3273 10385 35412 114586

1 6 28 114 432 1562 5470 18714 62907

1 7 36 158 635 2346 8781 31083

1 8 45 212 902 3588 13614

1 9 55 277 1245 5187

1 10 66 354 1677

1 11 78 444

1 12 91

1 13

425

1

and satisfies the equation, Pr (x) =

r−1


(3x)k (1 + x + x2 )r−k−1

k=0

for each line of the triangle, r, thus providing a clue as to the true form of c3p (x). Appendix B sketches out an alternative strategy for the rapid computation of the coefficients, dn , of dm (x) for general m, and is a good starting place for further research. It is hoped that the strategy of Appendix B may lead to a proof of the remaining conjectures, in particular Conjecture 1, and may also lead to a theorem for the construction of ch (x) in the general case.

6

Conclusion

This paper has presented computational results relating to the size of constellations formed from direct sums of PSK-type constellations. A theorem and a number of conjectures have been offered, comprising formulae for the rapid computation of sizes of such ’direct-sum’ constellations. These formulae have application to error-control coding, random-walk statistics, algebraic number representations, and (polynomial) residue number theory. It remains to verify the conjectures.

7

Appendix A- Proof of Theorem 1 by T. Kløve

Let pr (n) denote the number of ordered partitions of n into r parts, that is pr (n) = |{(a1 , a2 , . . . , ar ) ∈ N r | a1 + a2 + . . . + ar = n}| where N = {0, 1, 2, . . . }. If, n = a1 + a2 + . . . + ar−1 + ar and ai ≥ 1 for i = 1, 2, . . . , r − 1, then n − (r − 1) = (a1 − 1) + (a2 − 1) + . . . + (ar−1 − 1) + ar and vice versa. Hence the number of ordered partitions of n into r parts such that the first r − 1 parts are positive is pr (n − (r − 1)).

426

Matthew G. Parker

Lemma 1 We have ∞


pr (n)xn =

n=0

1 (1 − x)r

∞


and

pr (n − (r − 1))xn =

n=r−1

xr−1 (1 − x)r

Proof of Lemma 1: These are standard results from the theory of partitions: ∞


pr (n)xn = (1 + x + x2 + x3 + . . . )r =

n=0

1 (1 − x)r

and ∞

n=r−1

pr (n − (r − 1))xn = xr−1

∞

n=r−1

pr (n − (r − 1))xn−(r−1) =

xr−1 (1−x)r .

Lemma 2 Let m be an odd prime. Then dn = pm+1 (n) − pm+1 (n − m). Proof of Lemma 2: dn counts the number of distinct sums a1 w + a2 w2 + . . . + am wm + am+1 · 0

(1)

where ai ≥ 0 for i = 1, 2, . . . , m + 1 and a1 + a2 + . . . + am+1 = n. Noting that w + w2 + . . . wm = 0 we get dn by counting all sums (1), this number is pm+1 (n), and subtracting the number of sums where ai ≥ 1 for i = 1, 2, . . . , m, this number is pm+1 (n − m) (as explained above). Theorem 1 now follows from the two lemmas: ∞

n n=0 dn x

=

∞

n n=0 pm+1 (n)x

−

∞

n=0 pm+1 (n

− m)xn =

1−xm (1−x)m+1

=

Φm (x) (1−x)m

since Φm (x) = xm + xm−1 + . . . + 1.

8

Appendix B - A General Strategy for Computing the Size of PSK⊕ Constellations

Here a technique is proposed for the fast computation of the coefficients of dm (x) in the general case. Hopefully this may lead to a general proof of the conjectures of this paper, and a fast way to construct ch (x) in the general case, at least for m up to some large value. The technique will be illustrated by looking at the case where m = 6. Note that Φ6 (x) = x2 − x + 1. The steps of the technique are the following subsection headings.

Conjectures on Constellations

8.1

427

Find all Forbidden Binary Patterns

Φ6 (x) implies the following polynomial equivalences: x2 + 1 = x

pattern is 101000

x +1=0

pattern is 100100

3

These are the two binary patterns (polynomials) which are ’forbidden’ for m = 6. The forbidden polynomials are the set of polynomials which are equivalent, mod Φm (x), to polynomials of lower hamming weight. Note that, for example, x2 −x+1 is not included as a ’forbidden’ polynomial as it includes the polynomial x2 + 1 as a sub-polynomial. In general, for m = 2p, p prime, there are only two forbidden polynomials, namely, xp−1 + xp−3 + xp−5 + . . . + x2 + 1, and, xp +1. More generally, for large, composite m, there may be non-binary forbidden polynomials. 8.2

Enumerate all Length m Binary Words which Avoid the Forbidden Patterns

For m = 6, and for Hamming Weights (hw) 0-6 we have the following cyclically distinct binary strings which avoid the forbidden patterns or any cyclic shift of the forbidden patterns. hw = 0 000000 hw = 1 100000 hw = 2 110000 hw = 3 none hw = 4 none hw = 5 none hw = 6 none Each string of non-zero Hamming Weight has cyclic shift order 6. We will refer to the set of length m strings which avoid the forbidden patterns as the ’foundation’ polynomials. These ’foundation’ polynomials form the set E. For m = 6 |E| = 3. We will define there to be ehw,m cyclically distinct length m binary words in E, 0 ≤ hw ≤ m. For m = 6, e0,6 = 1, e1,6 = 1, e2,6 = 1, e3,6 = 0, e4,6 = 0, e5,6 = 0, e6,6 = 0. Note that e0,m = 1 ∀m. 8.3

Use Each Member of E as a ’Foundation’ for Building All Length m Inequivalent Polynomials of Coefficient Weight n, mod Φm (x)

The ’1’ positions of the ’foundation’ polynomials of E mark the positions where we are allowed to add ’coefficient weight’ to construct our inequivalent polynomials. It therefore follows that the number of inequivalent polynomials, dn , satisfies,  n m 

k−1 e dn = 1 + m (2) k − hw hw,m k=1 hw=1

428

Matthew G. Parker

For m = 6, d0 = 1 d1 = 1 + 6 = 7 d2 = 1 + 6 + 6(1 + 1) = 19 d3 = 1 + 6 + 6(1 + 1) + 6(1 + 2 + 0) = 37 d4 = 1 + 6 + 6(1 + 1) + 6(1 + 2 + 0) + 6(1 + 3 + 0 + 0) = 61 d5 = 1 + 6 + 6(1 + 1) + 6(1 + 2 + 0) + 6(1 + 3 + 0 + 0) + 6(1 + 4 + 0 + 0 + 0) = 91 . . . etc

These numbers agree with those of Table 1. The number of r-way ordered partitions adding to n is pr (n), and   n+r−1 pr (n) = n Therefore we can rewrite (2) in terms of partitions as, dn = 1 + m

m n

k=1 hw=1

8.4

phw (k − hw)ehw,m

(3)

Comments on the Technique

The technique assumes that all polynomials in E have cyclic order m. It seems likely that this is true in general as dn appears to satisfy m|(dn − 1) for all cases computed in Tables 1 and 2. A proof of Conjecture 1, and a proof of the general form of ch (x) may well follow if one can do the following for a given m, 1. Derive an efficient method to compute the ’forbidden’ polynomials. 2. Derive an efficient method to compute the elements ehw,m of E from the forbidden polynomials. For large m (e.g. perhaps m = 105?) there may be non-binary forbidden polynomials for which the above technique must be modified as follows: Consider, as an example, a ’hypothetical’ forbidden polynomial, F (x), of the following form: F (x) = x5 + 3x2 + x + 2 Then it has an associated binary forbidden polynomial, f(x), where, f(x) = x5 + x2 + x + 1 We wish to disallow all polynomials built from the foundation F (x) not f(x). Let the cyclic order (over m) of F (x) and f(x) be v. Then we should include γn polynomials in our count for dn , where n n−3 n


p4 (k − 4) − p4 (k − 4)) = v p4 (k − 4) γn = v( k=1

k=1

k=n−2

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where the ’3’ in the summation limit of the previous equation is the coefficient weight (cw) of F (x) minus the hamming weight of F (x). In general, for a given forbidden polynomial F (x) we include γn in our count for dn where γn satisfies, n


γn = v

phw(F (x))(k − hw(F (x)))

k=n+hw(F (x))−cw(F (x))+1

In the case where the forbidden polynomial is a binary polynomial hw(F (x)) = cw(F (x)) and γn for F (x) is 0, as expected. Things will be further complicated if the cyclic order of F (x) is lower than that of f(x).

Acknowledgments The author thanks S.J.Shepherd and D.A.Gillies for helpful discussions, and D.A.Gillies for writing software which independently confirmed results for the m = 8 case, and provided extra data for this case.

References 1. Parker, M.G.: VLSI Algorithms and Architectures for the Implementation of Number-Theoretic Transforms, Residue and Polynomial Residue Number Systems. PhD thesis, School of Eng, University of Huddersfield, March 1995 2. Safer, T.: Polygonal Radix Representations of Complex Numbers. Theoretical Computer Science. 210, (1999) 159–171 3. Sloane, N.J.A.: An On-Line Version of the Encyclopedia of Integer Sequences. http://www.research.att.com/~njas/sequences/index.html, The Electronic Journal of Combinatorics. 1, (1994) 1–5 4. Conway, J.H., Sloane, N.J.A.: Low Dimensional Lattices VII: Coordination Sequences. Proc. Royal Soc. A453 (1997) 2369–2389 5. O’Keeffe, M.: Coordination Sequences for Lattices. Zeit. f. Krist. 210, (1995) 905– 908 6. Grosse-Kunstleve, R.W., Brunner, G.O.: Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites. Acta Crystallographica. Section A. A52, (1996) 879–889 7. Huber, K.: Codes Over Gaussian Integers. IEEE Trans. on Inf. Theory. 40, No 1, Jan. (1994) 207–216 8. Huber, K.: Codes Over Eisenstein-Jacobi Integers. Contemporary Mathematics. 168, (1994) 165–179

A New Criterion for Normal Form Algorithms B. Mourrain INRIA, SAGA 2004 route des Lucioles B.P. 93, 06902 Sophia Antipolis [email protected] http://www.inria.fr/saga/mourrain/

Abstract. In this paper, we present a new approach for computing normal forms in the quotient algebra A of a polynomial ring R by an ideal I. It is based on a criterion, which gives a necessary and sufficient condition for a projection onto a set of polynomials, to be a normal form modulo the ideal I. This criterion does not require any monomial ordering and generalizes the Buchberger criterion of S-polynomials. It leads to a new algorithm for constructing the multiplicative structure of a zerodimensional algebra. Described in terms of intrinsic operations on vector spaces in the ring of polynomials, this algorithm extends naturally to Laurent polynomials.

1

Introduction

Solving polynomial equations is ubiquitous in many domains like robotics, computer vision, signal processing, . . . One of the algebraic approaches for treating safely this problem stands with constructing a normal form function in the quotient algebra A of the ring of polynomial R by the ideal I generated by the polynomials that one wants to solve. Once this normal form is known, one can solve the polynomial system, either by using symbolic methods like elimination of variables [7], [10], [9], rational representation [14][p. 88], [20], [1], [12], [21] . . . and solving a univariate polynomial equation or by matrix manipulations and eigenvalue or eigenvector computations [13], [2], [15], [16], . . . In practice, many of the polynomial systems are given with approximate coefficients, which means that we have to consider not just a single polynomial system but its neighborhood. Despite this, the aforementioned methods proceed pointwise and do not take into account the continuity that exists in this neighborhood. Our motivation here is to develop a method which can exploit as much as possible, the continuity in the coefficients of the input system. It is well-known that the geometry of the solutions is not a global continuous function of the parameters of a polynomial system. In some specific configurations, by a small perturbation of the coefficients, we can for instance blow-up points into curves. However, in many practical situations such as in robotics, in computer vision, . . . , the geometry of the solutions is not changing in a neighborhood of the input system. We are interested in algorithms, which can take into account this continuity in a given neighborhood. From a practical point of view, such a Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 430–442, 1999. c Springer-Verlag Berlin Heidelberg 1999


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requirement is necessary for developing safe and numerically stable methods for solving polynomial equations with approximate coefficients. As is illustrated below, one of the customary tools used nowadays to compute normal forms (ie. Gr¨obner bases) is not very adapted to this situation. Consider for instance a system of equations in two variables of the form
p1 := a x21 + b x22 + l1 (x1 , x2 ) = 0 p2 := c x21 + d x22 + l2 (x1 , x2 ) = 0 where a, b, c, d ∈ C are complex numbers, a d − b c = 0 and l1 , l2 are linear forms. Form a Gr¨ obner basis of these polynomials for a monomial order refining the degree order. The initial ideal is generated by (x21 , x22 ) and the corresponding basis of A = C[x1 , x2 ]/(p1 , p2 ) is {1, x1 , x2 , x1 x2 }.

3

2 y

1

-1

0

-0.5

0.5

x

1

-1

Fig. 1. The two conics with horizontal and vertical axis and the basis (1, x1 , x2 , x1 x2 ) of A = C[x1 , x2 ]/(p1 , p2 ), deduced from a Gr¨ obner basisxs computation for a monomial ordering refining the degree ordering.

Consider now a small perturbation of this system
p˜1 = p1 + 1 x1 x2 p˜2 = p2 + 2 x1 x2 , where 1 , 2 ∈ C are “small” parameters. The zero-set is also the points of intersection of two conics, which are slightly deformed.

3

2 y

1

-1

-0.5

0

0.5

x

1

-1

Fig. 2. The perturbed conics and the basis {1, x1 , x2 , x22 } of A = C[x1 , x2 ]/(p1 , p2 ), deduced from a Gr¨ obner basis computation for the same monomial ordering.

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In the result of a small perturbation, basis may “jump” from one set of monomials to another, though the two situations are very closed to each other from a geometric point of view. Moreover, some of the polynomials of the Gr¨ obner basis have large coefficients, for we have to divide by coefficients of the order 1 , 2 . Thus, this computation has to be carried out with exact and multi-precision arithmetic, inducing an unnecessary additional cost in the resolution process. As we can see from this example, Gr¨obner basis computations may introduce artificial discontinuities, due to the choice of a monomial order. Indeed, computing a Gr¨ obner basis for a given monomial order only allows “flat” deformations of the initial ideal of the input system (see [8]). In this work, we want to consider more geometric deformations, by removing the constraints induced by monomial orders. We present a new method, for computing the normal form with respect to a set of polynomials, provided that this set of polynomials satisfies a property of connectivity1 . This construction is based on a new criterion, which gives a necessary and sufficient condition for a projection onto this set of polynomials, to be a normal form modulo the ideal I generated by the equations that we want to solve. This criterion generalizes the Buchberger criterion. Indeed, if the projection is compatible with a monomial order and if the basis is a set of monomials not in the initial of the ideal, we recover the S-polynomial criterion. Moreover, it extends naturally to Laurent polynomials, as we will see. This leads us to new algorithms for solving polynomial equations, which are described in intrinsic terms, involving operations on vector-spaces of polynomials instead of pairs of polynomials. Consequently, the structure of the objects involved in the computation can be handled more efficiently [18]. The paper is organized as follows. In section 2, we define the reduction onto a vector space B and the notion of connectivity. In section 3, we describe and prove the criterion, relating the canonicity of the B-reduction and the commutativity of the operators of multiplication. In section 4, we present a new algorithm, based on this criterion, for constructing the multiplicative structure of a zerodimensional algebra. Its natural and direct generalization to Laurent polynomials is given in section 5.

2

Definitions

In this section, we introduce the notations that will be used hereafter. Let K be a field. We denote by R = K[x1 , . . . , xn ] = K[x] the ring of polynomials in the variables x1 , . . . , xn with coefficients in K. For any α = (α1 , . . . , αn ) ∈ Nn , αn α 1 we denote by xα the monomial xα 1 · · · xn , by |α| its degree deg(x ) = |α| =  α1 + · · · + αn and by (x) the set of all monomials in the variables x1 , . . . , xn . In order to have “homogeneous” notations, we will write x0 = 1. For any subset G ⊂ R, we denote by G the vector space generated by the elements of G and, by (G) the ideal generated by its element in R. 1

This is the case, for instance, for the set of monomials which are not in initial of an ideal.

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Definition 2.1. For any linear subspace V of R = K[x1 , . . . , xn ], we write V + = V + x1 · V + · · · + xn · V the vector space generated by the elements v, xi v for v ∈ V, i = 1, . . . , n. If m ⊂ (x) is a monomial subset of R, we also denote by m+ the union of m, x1 · m, . . ., xn · m, so that m+ = m+ . We denote by V [d] = (V [d−1] )+ , the dth power of the operator + on V . By convention, V [0] = V . The set V [] = ∪d≥0 V [d] is also the ideal generated by V and denoted by (V ). Definition 2.2. For any polynomial p ∈ R and any vector space B ⊂ R, the Bindex of p is the minimum of d such that p ∈ B [d] if p ∈ (B) and −∞ otherwise. The B-index of p corresponds to the ”distance” between p and B:

1

B

2

3

Next, we define the reduction of a polynomial on B, taking into account the “geometry” of B, i.e. the B-index of the polynomial: Lemma 2.3. Let B and K be vector spaces of R such that B + K = B + . Then any element p of B [] of B-index d can be reduced by an element k ∈ K [d−1] to an element r ∈ B : p = r + k. The polynomial r is called a B-remainder of p along K. Proof. We prove it by induction on the B-index of a polynomial. The property is true for the elements in B [1] = B + , which are of B-index ≤ 1 and which can be reduced by K = K [0] to elements of B. Assume that the property is true for polynomials of B-index ≤ d − 1 and let p be a polynomial of B-index d. Then p ∈ B [d] is of the form p=

n 

xi pi ,

i=0

with pi ∈ B [d−1] . By induction hypothesis, as pi is of index ≤ d − 1, there exist ri ∈ B, ki ∈ K [d−2] such that pi = ri + ki , so that p= n

n  i=0

xi ri +

n 

xi ki = r + k 

i=0

n

with r = i=0 xi ri ∈ B + and k  = i=0 xi ki ∈ K [d−1] . By hypothesis r = r+k  with r ∈ B, k  ∈ K. Therefore, p = r+k with r ∈ B, k = k  +k  ∈ K [d−1] .

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Remark 2.4. If 1 ∈ B then B [] = R and any polynomial p ∈ R can be B-reduced along K. If B = m, where m is a set of monomials containing 1, then B [d] is generated



by monomials of the form xα = xα xα with xα ∈ m and |α | ≤ d. The Bindex of xα is the minimum of the degree of |α | in such a decomposition. We describe an algorithm of reduction, in this case. Algorithm 2.5 Reduction onto B = m with 1 ∈ m ⊂ (x) . For any monomial xα of B-index d,





1. Decompose xα as xα = xi xα with xα
∈ B [d−1] , for an appropriate i. 2. Compute the B-reduction r ∈ B of xα . 3. Compute the B-reduction r ∈ B of xi r ∈ B + by projection of B + on B along K. The polynomial r is a B-remainder of xα along K. Note that the remainder r is not necessarily unique. This algorithm gives one of the possible B-remainder. Definition 2.6. The B-reduction along K is canonical modulo (K) iff for any n ∈ N and any p ∈ B [n] , there exists a unique pair of r ∈ B and k ∈ K [n−1] such that p = r + k : B [] = B ⊕ K [] . We introduce now a notion which will be used extensively hereafter. It is a property of connectivity, which allows us to reach any element of a vector space from a single element, by multiplication by the variables xi , i = 1, . . . , n. Definition 2.7. Let B be a vector space of R. We say that B is connected to e0 ∈ B, if for any b ∈ B, there exists b1 , . . . , bn ∈ B such that b=

n 

xi bi ,

i=1

with bi a multiple of e0 and deg(bi ) < deg(b). A typical situation is when B is generated by a set of monomials m such that for any m ∈ m, there exists i1 , . . . , ik with m = xi1 · · · xik and xi1 · · · xil ∈ B, for l = 1, . . . , k. In other words, we can go from m0 ∈ m to any element m ∈ m, by multiplication by a variable and staying in m. This situation occurs, for instance, with Gr¨obner basis when m is the set of monomials not in the initial of an ideal I for a fixed monomial order. It is a basis of the quotient ring R/I (see Macaulay’s theorem [7]). In our approach we can imaging more complicated basis.

3

Criterion

In this section, we consider a vector space B connected to 1 and a projection N from B + onto B. As we have seen in the previous section any polynomial p ∈ R can be B-reduced along the kernel of N . The problem that we address next is the following:

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How to decide if the reduction is canonical, using the only “local” information that we have, ie. the projection N from B + onto B? Theorem 3.1. Let B be a vector space of R connected to 1. Let N : B + → B be a K-linear map such that N|B = I. Let I = (ker(N )) be the ideal generated by the kernel of N . We define Mi : B → B b → N (xi b). The two properties are equivalent: 1. For all 1 ≤ i, j ≤ n, Mi ◦ Mj = Mj ◦ Mi . 2. R = B ⊕ I. If this holds, the map B-reduction along ker(N ) is canonical. Theorem 3.1 will be proved in two steps: First, we describe the normalization N in terms of the operators Mi . For any α = (α1 , . . . , αn ) ∈ Nn , we denote by Mα the operator M1α1 ◦ · · · ◦ Mnαn . Note that if the operators Mi commute, Mα is independent of the order in which we compose the operators. By linearity, for any p ∈ R, we also define p(M), which is obtained by substitution of the variables xi by the operators Mi in p. By convention, we will take M0 = I. Proposition 3.2. Assume that the vector space B is connected to 1 and that the operators Mi commute. Then for any b ∈ B + , N (b) = b(M)(1). Proof. We first prove by induction on the degree that for b ∈ B, we have b(M)(1) = b. If b = 1, then we have b(M)(1) = M0 (1) = 1, which proves the property for polynomials in B of degree 0. Assume that the proposition is true for all b ∈ B of degree < n. By hypothesis, for any polynomial b ∈ B of den gree ≤ n, there exists b1 , . . . , bn ∈ B such that b = i=1 xi bi with deg(bi ) < n. Then by induction hypothesis, we have  n  n   b(M)(1) = xi bi (M)(1) = Mi ◦ bi (M)(1) =

n  i=1

i=1

Mi (bi ) =

n  i=1

i=1 n 

N (xi bi ) = N (

xi bi ) = N (b) = b,

i=1

for N = I on B. We deduce now the property for the elements of the form b = xi b with b ∈ B. By definition, N (xi b ) = Mi (b ) = Mi (b (M))(1) = b(M)(1). By linearity, it extends to B + . Secondly, we describe the ideal I generated by ker(N ) in terms of the operators Mi .

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Proposition 3.3. Assume that the vector space B is connected to 1 and that the operators Mi commute. Then the ideal I = (ker(N )) is – p(x) − p(M)(1)p∈R , – {p ∈ R st. p(M)(1) = 0}. Proof. For any p ∈ R, let σ(p) = p(M)(1) ∈ B and let J = p(x) − σ(p)p∈R . We first prove that J coincide with the ideal J  = {p ∈ R st. p(M)(1) = 0}. According to proposition 3.2, we have (p(x) − σ(p))(M)(1) = σ(p) − σ(p)(M)(1) = σ(p) − σ(p) = 0, which proves that J ⊂ J  . Conversely, for any p ∈ J  , such that σ(p) = p(M)(1) = 0, we have p(x) = p(x) − σ(p) ∈ J, which proves that J = J  is an ideal of R. Let us show now that J = I. As N ◦ N = N , the kernel ker(N ) is generated by the polynomials b − N (b), b ∈ B + . Therefore according to proposition 3.2, I = (ker(N )) is generated by the polynomials b − b(M)(1), for b ∈ B + , which implies that I ⊂ J . Let us prove conversely by induction on the degree, that J ⊂ I. For any
α = 0 ∈ Nn , there exists i ∈ N and α ∈ Nn such that xα = xi xα . We have

















xα − Mα (1) = xi (xα − Mα (1)) + xi Mα (1) − Mi ◦ Mα (1) By induction, the element (xα − Mα (1)) ∈ J is also in I. Moreover denoting
bα
= Mα (1), we have xi bα
− Mi (bα
) = xi bα
− N (xi bα
) ∈ ker(N ) ⊂ I, which proves that xα − Mα (1) ∈ I. By linearity, this implies that J ⊂ I, and I = J . Proof of Theorem 3.1. 2 ⇒ 1. If R = B ⊕ I then B ∩ I = {0}. As for any b ∈ B, we have Mi (b) ≡ xi b modulo I, we deduce that for any 1 ≤ i, j ≤ n, we have (Mi ◦ Mj − Mj ◦ Mi )(b) ≡ (xi xj − xj xi )b ≡ 0 modulo I. But we also have (Mi ◦ Mj − Mj ◦ Mi )(b) ∈ B. Therefore, (Mi ◦ Mj − Mj ◦ Mi )(b) = 0. 1 ⇒ 2. Consider the map σ:R→B p → p(M)(1) which is surjective, according to proposition 3.2: ∀b ∈ B, σ(b) = b. By proposition 3.3, its kernel is the ideal I = (ker(N )), which proves that B is isomorphic to R/I and R = B ⊕ I.  

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Remark 3.4. The hypothesis that B is connected to 1 is necessary in the proof of the theorem. Consider, for instance, the case where B = 1, x, x3  and N defined on B + = 1, x, . . . , x4  by N (1) = 1, N (x) = x, N (x2 ) = 0, N (x3 ) = x3 , N (x4 ) = x3 . Then, the ideal generated by ker(N ) is I = (x2 ). There is only one operator M1 , so that the hypothesis of commutation is obviously satisfied, but {1, x, x3 } is not a basis of R/I. B ∩ I = x3 . Remark 3.5. Criterion 3.1 generalizes the Buchberger criterion [5], [6], [3], [7], which says that a set of polynomials {g1 , . . . , gs } is a Gr¨ obner basis for a given monomial order if for any pair 1 ≤ i1 , i2 ≤ s, the S-polynomial of gi1 , gi2 reduces to 0 by G. Let mi1 , (resp. mi2 ) be the leading term of gi1 (resp. gi2 ), m = lcm(mi1 , mi2 ), xα1 (resp. xα2 ) a variable dividing mi1 (resp. mi2 ), with α1 = α2 . Thus we have m = xα1 xα2 m . Let us consider a pair (i1 , i2 ) such that m is not in the initial (Buchberger’s criterion 3) of G. For any polynomials p ∈ R, let N (p) be the remainder of p in the reduction by G. Then we can check that (Mα1 ◦ Mα2 − Mα2 ◦ Mα1 )(m ) is the remainder of the S-polynomial S(gi1 , gi2 ) = m (

gi1 gi − 2 ), mi1 mi2

by reduction by G.

4

Computing the Multiplicative Structure of Zero-Dimensional Algebra

In this section, we describe a new algorithm for computing the multiplication table of a quotient algebra of dimension 0, based on the criterion of theorem 3.1. More precisely, we compute the normalization from B + to B, which yields directly the multiplication tables by the variables xi , and consequently the roots of the system by eigenvector computations [2], [15], [16]. Compared with Gr¨obner basis computations, instead of considering pairwise reductions, we manipulate coefficient matrices of polynomials, offering a global view of their structure. This enables us to exploit this structure involved in the computation, like sparsity [4], quasi-Toeplitz structure [18], [17], . . . and it leads to algorithms for polynomials with approximate coefficients whose numerical stability can be analyzed more clearly. Similar ideas are also used in [11], where an extension of Buchberger algorithm using linear algebra tools is proposed. Let f1 , . . . , fm ∈ R be m polynomials and L a finite vector space connected to 1 and containing f1 , . . . , fm . Let I be the ideal of R generated by f1 , . . . , fn . We assume here that A = R/I is zero-dimensional. Let us define by induction the following vector spaces: – K0 = f1 , . . . , fm , – Kn+1 = Kn+ ∩ L, n ≥ 0.

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Since L is a finite vector space, the increasing sequence Kn is stationary. We denote by K∗ the union of all the vector spaces Kn . By construction, we have K∗ ⊂ I. We assume that 1 ∈ K∗ , otherwise the ideal I = (f1 , . . . , fm ) is trivial. Then we have the following property: Proposition 4.1. Let B be a supplementary space of K∗ in L, connected to 1 and assume that B + ⊂ L. Let N be the projection of L onto B along K∗ and Mi : B → B b → Mi (b) = N (xi b) Then we have Mi ◦ Mj = Mj ◦ Mi , for 1 ≤ i, j ≤ n. Proof. Let ρ = I − N be the projection onto K∗ along B. Since for any b ∈ B and for all 1 ≤ i < j ≤ n, we have Mj ◦ Mi (b) = Mj (xi b − ρ(xi b)) = xj (xi b − ρ(xi b)) − ρ(xi xj b − xj ρ(xi b)) = xi xj b + xj k1 + k2 with k1 , k2 ∈ K∗ . Similarly, we have Mi ◦ Mj (b) = xi xj b + xi k1 + k2 , with k1 , k2 ∈ K∗ , so that (Mj ◦ Mi − Mi ◦ Mj )(b) = xj k1 + k2 − xi k1 − k2 ∈ K∗+ . By definition of the operators Mi , we also have (Mj ◦ Mi − Mi ◦ Mj )(b) ∈ B but B ∩ K∗+ = B ∩ K∗+ ∩ L = B ∩ K∗ = {0}. Consequently, the operators Mi commute. Theorem 4.2. Let B be a supplementary space of K∗ in L, connected to 1. Assume that B + ⊂ L. Then R = B ⊕ I and the B-reduction along K = K∗ ∩ B + is canonical modulo I = (f1 , . . . , fm ). Proof. Assume that B ⊕ K∗ = L and B + ⊂ L. Let N be the projection of L onto B along K∗ and Mi : B → B such that ∀b ∈ B, Mi (b) = N (xi b). Denoting by K = K ∗ ∩ B + , we have B + = B ⊕ K. According to proposition 4.1, the operators Mi commute. Thus by theorem 3.1, we have R = B ⊕ J , where J is the ideal generated by K = K∗ ∩ B + = ker(N ) ∩ B + . By construction, as K ⊂ K∗ ⊂ I, we have J ⊂ I. Let us prove by induction on the degree, that for any p ∈ L, we have N (p) = p(M)(1). The property is obviously true for polynomials of degree 0. Assume that it is true for polynomials of degree < d and let p ∈ L be a polynomial of degree d. As L is connected to 1, there exist p1 , . . . , pn ∈ L such that p=

n  i=1

xi pi ,

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with deg(pi ) < d. We deduce that p − p(M)(1) =

n 

xi (pi − pi (M)(1)) + (xi pi (M)(1) − Mi (pi (M)(1))) .

i=1

By induction, pi −pi (M)(1) ∈ K∗ . Moreover as pi (M)(1) ∈ B, ki = xi pi (M)(1)− Mi (pi (M)(1) is in K so that p − p(M)(1) ∈ (K∗ )+ ∩ L = K∗ . This implies that, for any p ∈ L, the projection N (p) of p onto B along K∗ is p(M)(1): N (p) = p(M)(1). As fj ∈ K∗ ⊂ L and N (fj ) = 0 = fj (M)(1), we deduce from proposition 3.3, that fj ∈ J , so that I = J . This proves that R = B ⊕ I where I = (K). Theorem 4.2 leads to the following algorithm, for constructing a normal form function in A = R/I. Algorithm 4.3 Normal form fo A. Input: Let f1 , . . . , fm ∈ R and I = (f1 , . . . , fm ) and assume that A = R/I is zero-dimensional. Let L be a finite vector space of R containing f1 , . . . , fm and connected to 1. (1) K0 = f1 , . . . , fm ; (2) While Kn = Kn−1 , compute Kn+1 = Kn+ ∩ L, replace n by n + 1; (3) Compute a supplementary vector space B of K∗ (= Kn0 ), which is connected to 1; (4) If B + ⊂ L, replace L by L+ and go to step (2). Otherwise stop. Output: The B-reduction along K∗ is canonical modulo the ideal I = (f1 , . . . , fm ). This algorithm necessarily stops. Otherwise, L would have contained the multiples of f1 , . . . , fm of degree k, with k as large as we want. Consequently, L would have contained the S-polynomials and the remainders involved in the computation of the Gr¨ obner basis of f1 , . . . , fm for a fixed monomial order. In other words, L would have contained a Gr¨ obner basis of I = (f1 , . . . , fm ). Thus a supplementary space B of K∗ in L, connected to 1 would have had dimension D = dimK (R/I) so that B + ⊂ L if k > D. According to theorem 4.2, when the algorithm stops, the B-reduction along K = K∗ ∩ B is canonical modulo I. Since L is connected to 1, the vector space B connected to 1 and supplementary to K∗ can be computed incrementally, starting from 1 (in the case where 1 ∈ K∗ ).

5

The Case of Laurent Polynomials

Generalization of Gr¨ obner basis to Laurent polynomials has already been considered, but they require specific adjustments (monomial order, special Spolynomials [19], introduction of new variables [22], . . . ). In this section, we

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describe a direct and natural extension of algorithm 4.3, to zero-dimensional ± ± quotient algebra of Laurent polynomials. Let S = K[x± 1 , . . . , xn ] = K[x ] be the ring of Laurent polynomials in the variables x1 , . . . , xn , with coefficients in K. Let f1 , . . . , fm ∈ S and let I be the ideal of S generated by these polynomials. We assume that A = S/I is zero-dimensional. For any vector space V of S, we define −1 V × = V + x1 V + · · · + xn V + x−1 1 V + · · · + xn V.

Let L be a finite vector space of R ⊂ S connected to 1 and containing f1 , . . . , fm . We define by induction: – K0 = f1 , . . . , fm , – Kn+1 = Kn× ∩ L, n ≥ 0. As L is a finite vector space, the increasing sequence Kn is stationary and we denote by K the maximum (or union) of all the vector space Kn . We also have K ⊂ I. Moreover, we have the following property: Proposition 5.1. Let B be a supplementary space of K in L, connected to 1 and assume that B + ⊂ L. Let N be the projection of L× onto B along K× and Mi : B → B b → N (xi b)

M−i : B → B b → N (x−1 i b)

Then we have Mi ◦ Mj = Mj ◦ Mi and M−i ◦ Mi = I, for 1 ≤ i, j ≤ n. Proof. The proof that the operators commute is similar to the proof of proposition 4.1 (V + ⊂ V × ). Let us prove that M−i ◦ Mi = I. For ρ = I − N and b ∈ B, we have M−i ◦ Mi (b) = M−i (xi b − ρ(xi b)) −1 = x−1 i (xi b − ρ(xi b)) − ρ(b − xi ρ(xi (b))) −1 −1 = b − xi ρ(xi b)) − ρ(b − xi ρ(xi (b))) = b + x−1 i k1 + k2 , with k1 , k2 ∈ K , so that M−i ◦ Mi (b) − b ∈ K+ ∩ B = K  ∩ B = {0}, which proves that the inverse of Mi is M−i . We deduce the following theorem: Theorem 5.2. Let B be a supplementary space of K in L, connected to 1. If B + ⊂ L then S = B ⊕ I and (K) = I where K = K∗ ∩ B + Proof. Let K = B + ∩ K and J ⊂ I be the ideal of S generated by K. Similarly as in theorem 4.2, we prove that for j = 1, . . . , m, the polynomial fj is in the ideal generated by K in R, so that J = I. Let us prove now that S = B ⊕ I. Consider the map σ:S→B p → p(M)(1)

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By proposition 3.2, ∀b ∈ B, σ(b) = b(M)(1) = b and σ is surjective. Let p ∈ ker(σ). Then, there exists α ∈ Zn such that p(x) = xα q(x), with q ∈ R. According to proposition 5.1, Mi is invertible so that q(M)(1) = M−α σ(p) = 0. By proposition 3.3, q is in the ideal of R generated by K, which implies that p ∈ (K) = I. This proves that ker(σ) = I so that S = B ⊕ I. This leads to an algorithm, similar to algorithm 4.3: Algorithm 5.3 Normal form for A (case of Laurent polynomials). ± Input:Let f1 , . . . , fm ∈ S = K[x± 1 , . . . , xn ] and I = (f1 , . . . , fm ), and assume that A = S/I is zero-dimensional. Let L be a finite vector space of R ⊂ S connected to 1 and containing gj = xαj fj for some αj ∈ Zn and 1 ≤ j ≤ m. (1) K0 = g1 , . . . , gm . (2) While Kn = Kn−1 compute Kn+1 = Kn× ∩ L. (3) Compute a supplementary vector space B of K (= Kn0 ) which is connected to 1. (4) If B + ⊂ L, replace L by L+ and go to step (2). Otherwise stop. Output: The B-reduction along K is canonical modulo the ideal I ∩ K[x]. Remark 5.4. A similar result can be obtained by replacing V × by −1 V ∗ = V + x1 V + · · · + xn V + x−1 r+1 V + · · · + xn V ± and S by K[x1 , . . . , xr , x± r+1 , . . . , xn ], for any r = 1, . . . , n.

6

Concluding Remarks

We end with some of the improvements or extensions that can be bring to this approach. Our presentation focuses on zero-dimensional quotient algebras. However the criterion that we give is valid for quotient algebras of any dimension and can lead to similar algorithms, by modifying the test in the while loop according to the “shape” of B. In algorithms 4.3 and 5.3, instead of extending L in all directions (step (4)), one can extend it using only a subset of the variables (say L+ = L + x1 · L + · · · + xr · L, 1 ≤ r ≤ n), in order to compute polynomials of I in this subset of variables. The hypothesis of connectivity of B can be removed in the theorem 3.1, if 1 ∈ B and dimK (B) = dimK (A) = D. In this case, B is a generating set of A, of the right dimension and thus isomorphic to A.

References 1. M.E. Alonso, E. Becker, M.F. Roy, and T. W¨ ormann. Zeros, multiplicities and idempotents for zero dimensional systems. In L. Gonz´ alez-Vega and T. Recio, editors, Algorithms in Algebraic Geometry and Applications, volume 143 of Prog. in Math., pages 1–15. Birkh¨ auser, Basel, 1996.

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2. W. Auzinger and H.J. Stetter. An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations. In Proc. Intern. Conf. on Numerical Math., volume 86 of Int. Series of Numerical Math, pages 12–30. Birkh¨ auser Verlag, 1988. 3. T. Becker, V. Weispfenning, and H. Kredel. Gr¨ obner Bases. A Computational Approach to Commutative Algebra, volume 141 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1993. 4. D. Bondyfalat, B. Mourrain, and V.Y. Pan. Controlled iterative methods for solving polynomial systems. In Proc. ISSAC, pages 252–259. ACM Press., 1998. 5. B. Buchberger. An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Polynomial Ideal (German). PhD thesis, Math. Inst, Univ. of Innsbruck, Austria, 1965. (also in Aequationes Math. 4/3, 1970). 6. B. Buchberger. Gr¨ obner bases: An algebraic method in ideal theory. In N.K. Bose, editor, Multidimensional System Theory, pages 184–232. Reidel Publishing Co., 1985. 7. D. Cox, J. Little, and D. O’Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer, New York, 1992. 8. D. Eisenbud. Commutative Algebra with a view toward Algebraic Geometry, volume 150 of Graduate Texts in Math. Springer-Verlag, 1994. 9. I.Z. Emiris and B. Mourrain. Matrices in Elimination Theory. J. of Symbolic Computation, 28(1&2):3–44, 1999. 10. J.C. Faugere, P. Gianni, D. Lazard, and T. Mora. Efficient computation of zerodimensional groebner bases by change of ordering. J.S.C, 16(4):329–344, 1993. 11. Faug`ere J.C. A new efficient algorithm for computing Gr¨ obner Basis (F4). J. of Pure and Applied Algebra, 139:61–88, 1999. 12. M. Giusti, J. Heintz, J.E. Morais, and L.M. Pardo. When Polynomial Equation Systems can be “Solved” Fast ? In AAECC’95, LNCS. Springer-Verlag, 1995. 13. D. Lazard. R´esolution des syst`emes d’´equations alg´ebriques. Theo. Comp. Science, 15:77–110, 1981. 14. F.S. Macaulay. On the Resolution of a given Modular System into Primary Systems Including some Properties of Hilbert Numbers. Proc. London Math. Soc., pages 66–121, 1912. 15. H. M. M¨ oller. Systems of algebraic equations solved by means of endomorphisms. Lect. Notes in Comp. Sci., 673:43–56, 1993. 16. B. Mourrain. Computing isolated polynomial roots by matrix methods. J. of Symbolic Computation, Special Issue on Symbolic-Numeric Algebra for Polynomials, 26(6):715–738, Dec. 1998. 17. B. Mourrain and V.Y. Pan. Asymptotic acceleration of solving multivariate polynomial systems of equations. In Proc. STOC, pages 488–496. ACM Press., 1998. 18. B. Mourrain and V.Y. Pan. Multivariate polynomials, duality and structured matrices. J. of Complexity, 1999. To appear. 19. F. Pauer and S. Zampieri. Groebner Bases with Respect to Generalized Term Orders and their Application to the Modelling Problem. J.S.C., 21:155–168, 1996. 20. J. Renegar. On the computational complexity and geometry of the first order theory of reals (I, II, III). J. Symbolic Computation, 13(3):255–352, 1992. 21. F. Rouillier. Algorithmes efficaces pour l’´etude des z´eros r´eels des syst`emes polynomiaux. PhD thesis, Universit´e de Rennes, 1996. 22. B. Sturmfels. Gr¨ obner bases of toric varieties. Tˆ ohoku Math. J., 43:249–261, 1991.

Discrete Fourier Transform and Gr¨ obner Bases
A. Poli1 , M.C. Gennero1, and D. Xin2 1 2

AAECC/IRIT, University Paul Sabatier, 31062 Toulouse cedex 4, France Department of Computer Sciences, Fudan University, Shanghai, PR of China

Abstract. Using multivariate polynomials, Gr¨ obner bases have a great theoretical interest in decoding cyclic codes beyond their BCH capability [1,2], but unfortunately have a high complexity [7]. From engineers point of view, the complexity comes in particular from the number of needed indeterminates, from the maximal number of needed polynomials during Buchberger’s algorithm (this number is unknown), and from the maximal number of attempts before recovering the error polynomial e(X). In this paper we propose a new algorithm, using Gr¨ obner bases and Discrete Fourier Transform. In most of the cases this algorithm needs fewer indeterminates than Chen et al. algorithm [1], and at most as many as for XP algorithm [9] (sometimes less). In some cases the maximal number of needed polynomials for calculations is reduced to 1. Finally, it is shown that only one attempt is needed for recovering e(X).

Introduction For cyclic codes, many known algorithms are very efficient for decoding up to the designed decoding capability tBCH [3]. For decoding up to the true decoding capability ttrue , the best algebraic method uses Gr¨ obner bases [1,2]. Unfortunately, the corresponding algorithm (i.e. Buchberger’s algorithm) has such a hight complexity [7] that it cannot be considered for implementation. In order to decrease the complexity, we propose a new algorithm, using Gr¨ obner bases we call FG-algorithm, which seems to be possibly implemented. Let us consider GF (q r ), with α some primitive element. Set Ar = GF (q r )[X]/(X n − 1); (n, q) = 1; β a primitive nth root of unity. Set T for the DFT
in this algebra : for every polynomial a(X) in Ar one has A(X) = T (a(X)) = i=0,...,n−1 a(β i )X i . As ordinary, each location of a vector of length n is labeled with a power of β, explicitely : β 0 , β 1 , . . . , β n−1 . If labeled with β j we say that the location is j. We consider all elements S(X) in Ar whose known coefficients are given, in locations i1 , i2 , . . . , is . Using multivariate polynomial equations, we examine what can be known about the respective corresponding s(X). Then we apply to error correcting codes, giving FG-algorithm. We show how to use it for decoding up to ttrue errors, sometimes up to ttrue + 1 ones.


This work was partly done under PRA9605.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 444–453, 1999. c Springer-Verlag Berlin Heidelberg 1999 

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In Part 1, after a few algebraic developments, we introduce multivariate polynomial equations. Part 2 introduces Gr¨ obner bases, and for codes satisfying ttrue = tBCH , we describe the FG-algorithm. We show why it is less complex than Chen et al. algorithm [1], and less than XP algorithm [9]. We prove that the number of needed attempts of the whole algorithm is reduced to 1. In some cases, FG-algorithm runs with polynomials with one indeterminate. Then, dealing with Gr¨ obner bases is equivalent to deal simply with GCD’s. In this case, the maximal number of needed polynomials during buchberger’s algorithm is given. Xin prediction is suggested to decrease the complexity.

1

Basics

Throughout this paper dega(X) means “degree of a(X)”, and WH (a(X)) is for the Hamming weight of a(X). Lemma 1. 1) The set I of all a(X) in Ar , such as T (a(X)) = 0 on locations i1 , i2 , . . . , is is an ideal of Ar generated by the minimum polynomial (over GF (q r )) of {β i1 , β i2 , . . . , β is } (I depends on the set of locations). 2) If the set {i1 , ...is } is some union of cyclotomic classes in ZZ/(q v − 1) for v dividing r, then I contains an ideal of Av . 3) The set of all polynomials a(X) in Ar with determined values for T (a(X)) in locations i1 , i2 , . . . , is is a coset of I. Proof. Points 1) and 3) are direct. 2) The minimum polynomial of {β i1 , . . . , β is } has its coefficients from GF (q v ).   For every pair of polynomials V (X) = V0 + V1 X + . . . + Vn−1 X n−1 and U(X) = U0 + U1 X + . .
. + Un−1 X n−1 , we define their (ordinary) scalar product by < V (X), U(X) >= i=0,...,n−1 Vi Ui . We say that U(X) is V (X)-orthogonal iff < V (X), X i U(X) >= 0 for i = 0, 1, . . . , n − 1. We also say that the polynomial U(X) is (Vj , t)-orthogonal if < V (X), X i U(X) >= 0 for i = j, j + 1, . . . , j + t − 1.
Lemma 2. Given some s(X) = j=1,...,w sij X ij (sij = 0) in Ar we call locator of s(X) the monic polynomial Ls (X) whose roots are β i1 , . . . , β iw . Set S(X) = T (s(X)). a) A polynomial L(X) is S(X)-orthogonal iff L(β ij ) = 0, j = 1, . . . , w. b) The set of the L(X)’s which are S(X)-orthogonal is the ideal (Ls (X)). Proof. a) and b) are direct.  

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Example 1. n = 7, q = 8, 1 + α + α3 = 0. a) s(X) = 1 + X 3 , S(X) = αX + α2 X 2 + α6 X 3 + α4 X 4 + α3 X 5 + α5 X 6 . The locator of s(X) is Ls (X) = α3 + αX + X 2 . Ls (X) is S(X)-orthogonal. For example < (0, α, α2 , α6 , α4 , α3 , α5 , ), (1, 0, 0, 0, 0, α3 , α) >= 0 holds. Also, α3 + αX + αX 2 + αX 3 + X 4 is S(X)-orthogonal. b) s(X) = 1 + X + X 3 + X 4 , S(X) = α4 X + αX 2 + X 3 + α2 X 4 + X 5 + X 6 . L(X) = 1 + α4 X + X 2 is (S1 , 2)-orthogonal, but not (S1 , 3)-orthogonal. In the following, we first suppose that all coefficients of S(X) are known, and then only some of them are known. 1.1

All Coefficients of S(X) Are Known: Linear Algebra

We have a little refinement of Lemma 2. Proposition 1. Let some S(X) be given, as well as some integer j ≤ n − 1. Suppose there exists some polynomial R0 (X) with degree t, which is (Sj , t)orthogonal. a) Suppose that some R1 (X), with degree at most t, is (Sj , t)-orthogonal. Then the GCD (R0 (X), R1 (X))(= RM (X)) is (Sj , 2t − degRM (X))-orthogonal. b) The set E of all polynomials which have degree at most t and which are (Sj , t)-orthogonal is the set of all multiples of some polynomial R(X). Each polynomial B(X) which is (Sj , t)-orthogonal then is (Sj , 2t − degB(X))-orthogonal. Proof. a) The GCD RM (X) can be obtained from successive Euclidian divisions : R0 (X) = Q1 (X)R1 (X) + R2 (X), R1 (X) = Q2 (X)R2 (X) + R3 (X), . . . , RM −1 (X) = QM (X)RM (X). By hypothesis R1 (X) is (Sj , degR0 (X))-orthogonal. Moreover, from R0 (X) = Q1 (X)R1 (X) + R2 (X), and degR0 (X) ≤ t, it is clear that Q1 (X)R1 (X) is (Sj , t−degR0(X)+degR1 (X))-orthogonal. So, R2 (X) is (Sj , degR1 (X))-orthogonal. Now, suppose that Ri−1 (X) is (Sj , degRi−2 (X))-orthogonal, as well as Ri (X) is (Sj , degRi−1 (X))-orthogonal. It is a direct induction to prove that Ri+1 (X) is (Sj , degRi (X))-orthogonal. We conclude that RM (X) is (Sj , degRM −1 (X))orthogonal. Whatever degRM (X) is, RM (X) is (Sj , degRM −1 (X)−degRM (X))orthogonal. As RM (X) divides RM −1 (X) which is (Sj , degRM −2 (X))-orthogonal, it follows that the polynomial RM (X) is (Sj , degRM −2 (X)− degRM (X))-orthogonal. Iterating, RM (X) is (Sj , degR0 (X) − degRM (X))-orthogonal, and from hypothesis on R0 (X), RM (X) is (Sj , 2t − degRM (X))-orthogonal. b) As RM (X) divides Ri (X) for i = 0, . . . , M − 1, then Ri (X) is (Sj , 2t − degRi (X))-orthogonal. Moreover, from the property GCD(a1 , a2 , a3 ) = GCD(GCD(a1 , a2 ), a3 ) the statement comes from point a).   Observe that RM (X) is not necessarily the locator of s(X), as shown in the following example.

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Example 2. Consider the field GF (16), with α a primitive element such as 1 + α + α4 = 0. In A4 , consider S(X) whose coefficients are : 1, 0, α, α2 , α9 , α4 , α6 , α3 , α12 , 0, α8 , 1, α9 , 1, α. Set t = 4, j = 1, R0 (X) = 1+X+α5X 2 +X 3 +X 4 and R1 (X) = α+α8 X+X 3 . RM (X) is equal to 1 + αX + X 2 . As it is irreducible over GF (16) it cannot be Ls (X). Corollary 1. With the same notations as in proposition 1 : a) If R0 (X) is (Sj , t+1)-orthogonal, then RM (X) is (Sj , 2t+1−degRM (X))orthogonal. As a consequence, if R0 (X) is S(X)-orthogonal then RM (X) is S(X)-orthogonal. b) If R0 (X) is the locator of s(X) then RM (X) = R0 (X). In other words, there does not exist any other polynomial which is (Sj , t)-orthogonal, up to scalar multiplication. c) If R0 (X) is not the locator of s(X) then X t−degLs (X)Ls (X) has degree t, and is S(X)-orthogonal, and has its t − degLs (X) first coefficients which are 0.  Sj Sj+1 · · · Sj+λ  Sj+2 · · · Sj+λ+1   Sj+1 , with integers j ≥ 0, µ − d) Set Mj =  ..   . Sj+µ−1−λ · · · · · · Sj+µ−1 1 − λ ≥ 0, µ ≤ 2λ. Set WH (s(X)) = u. If u ≥ µ − λ then Mj has rank equal to µ − λ. Proof. a) Direct, because RM (X) divides R0 (X). b) From proposition 1, RM (X) divides the locator. From a) lemma 2 the degree of the locator is minimal. c) Direct. This remark will be used for decoding with FG-algorithm (see proposition 4).   Sj+λ Sj+λ−u+1 · · ·   .. d) By point b) the matrix   has rank u because . Sj+λ · · · Sj+λ−u+1 Ls (X) is (Sj+λ−u+1, u)-orthogonal, and by hypothesis u ≥ µ − λ.   1.2

Some Coefficients of S(X) Are Known: Polynomial Equations

Now suppose that S(X) has its known coefficients in locations i1 , i2 , . . . , is . We show how to obtain all possible explicit polynomials S(X). Choose some integer t strictly less than n, and consider a polynomial L(X) = Y0 + Y1 X + Y2 X 2 + . . . + Yt−1 X t−1 + X t whose coefficients Y0 , Y1 , . . . , Yt−1 are indeterminates. For every integer u within {0, . . . , n − 1} we define a scalar product Pu by Pu =< (Su , Su+1, . . . , Su+t−1), (Y0 , Y1 , . . . , Yt−1 ) >. Pu is a polynomial expression (over the field GF (q r )) which contains indeterminates Y0 , . . . , Yt−1 , and possibly with some others, say Sj1 , . . . , Sjv for some

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v. It means that in the range {u, u + 1, . . . , u + t − 1} the unknown coefficients of S(X) are Sj1 , . . . , Sjv . Note that v may depend on u and t. Choose some value u0 for u. Set SE(t, u0) for a set of equations, empty at the beginning, into which we will add equations, using the following algorithm : For j = 0, . . . , n − 1 do If Su0 +t+j is known then add Pu0 +j − Su0+t+j = 0 to SE(t, u0). else substitute −Pu0+j for Su0+t+j . end For (u0 + k is performed modulo n, for every k). Now, to SE(t, u0) add the polynomial equations : r r Yiq − Yi = 0 (i = 0, . . . , t − 1), and Sjqm − Sjm = 0 (m = 1, . . . , v). Every solution of this system is a (v + t)-tuple (sj1 , . . . , sjv , y0 , . . . , yt−1 ), whose components are from GF (q r ). Each one gives rise to an explicit polynomial L(X) = y0 +y1 X+y2 X 2 +. . .+yt−1 X t−1 +X t and to an explicit polynomial S(X) such as, by construction, L(X) is S(X)-orthogonal. The number of solutions may be equal to 0. Example 3. Consider a cyclic code of length 7, over GF (8), with roots α1 , α3 , α6 . Suppose s(X) = α + α2 X 3 , with S(X) known at locations 1,3,6 : S1 = α6 , S3 = α2 , S6 = α5 . Choose u0 = 1, t = 2. We have L(X) = Y0 + Y1 X + X 2 . SE(2,1) contains six equations : α6 Y0 + Y1 S2 + α2 = 0, α5 + S2 Y02 + S2 Y0 Y12 + α2 Y13 = 0, α6 + α5 Y0 + α5 Y12 + 2 2 α Y0 Y1 + S2 Y02 Y12 + α2 Y0 Y13 = 0, Y08 − Y0 = 0, Y18 − Y1 = 0, S28 − S2 = 0. Proposition 2. a) The number of solutions of the system SE(t, u0 ) is the number of distinct polynomials L(X). b) Suppose we obtain some explicit polynomial S(X). The number of polynomials L(X) we obtain which are S(X)-orthogonal is q c , with c = t − WH (s(X)). Proof. a) Two distinct polynomials L(X) correspond to two distinct solutions of the system, by construction. b) A polynomial L(X) is a multiple of the locator Ls (X) of s(X) (see point b) of lemma 2). The number of monic multiples of Ls (X) with degree t (it is the degree of each L(X)) is q c , with c = t − degLs (X) = t − WH (s(X)).  

2

Application to Codes

In the theory of Gr¨ obner bases, a system of polynomial equations (as SE(t, u0)) is associated with a basis of some ideal J . Buchberger’s algorithm constructs a sequence of bases (that corresponds to constructing a sequence of equivalent systems of polynomial equations). At the end of the process one has a basis of J , called a Reduced Gr¨ obner Basis (RGB). When there are only finitely many solutions that RGB is a system of polynomial equations with one equation having only one indeterminate. A zero of J is simply a solution of each system.

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RGB are used in Chen et al. [1], in order to correct errors up to the true capability ttrue . The authors use a code over GF (q), with length n, and
known syndromes Sj1 , . . . , Sjw . The error polynomial is represented as e(X) = i=1,u Yi Ziji . They find the RGB of the ideal Jm given below. (Y1q−1 − 1, . . . , Ymq−1 − 1, Z1n − n j1 j1 − 1, Y1 Z1j1 + . . . + Ym Zm − Sj1 , . . . , Y1 Z1jw + . . . + Ym Zm − Sjw ) 1, . . . , Zm Each zero of Jm is a (2m)-tuple which gives the values of the Yi ’s and Zi ’s. From these values one can recover e(X) if WH (e(X)) = m. As WH (e(X)) is not known, Chen et al. have to run their algorithm from m = tBCH +1 to m = ttrue (in the worst case). The maximal number of attempts is then ttrue − tBCH . As we already said, the concrete complexity in implementing Gr¨ obner bases for decoding mainly comes from the number of indeterminates, the maximal number of polynomials appearing into the successive bases during Buchberger’s algorithm, and the number of attempts. In the following we also propose to use Gr¨ obner bases, but with fewer complexity. Let C be a code over GF (q), with length n relatively prime to q. When binary, the considered codes are from [4] (these codes satisfy dBCH = dtrue ). Set GF (q r ) for the smallest extension field of GF (q), containing β a primitive n-th root of unity, and α a primitive element. Suppose that c(X) is the emitted codeword. During transmission, errors occur, and the receiver receives c(X) + e(X)(= R(X)), where e(X) is the error polynomial. As usual, we set Si for the syndrome R(β i ). In the polynomial T (e(X)) = S0 + S1 X + · · · + Sn−1 X n−1 (= S(X))) the coefficient Si is known as soon as β i is a root of the code. Suppose these locations are i1 , i2 , . . . , is and suppose also that the syndromes Su0 , Su0 +1, . . . , Su0 +dBCH −2 are known, for some j. Set L(X) = Y0 + Y1 X + . . .+ Yttrue−1 X ttrue −1 + X ttrue , where Y0 , . . . , Yttrue −1 are indeterminates. Example 4. (continuation of example 3). The RGB we obtain is : {α + α4 Y0 + α4 Y02 , α4 + αY0 + α2 S2, α6 + α4 Y0 + αY1 }. The zeroes (Y0 =, Y1 =, S2 =) are {(α, 1, α6 ), (α3 , α, 0)}. The zero (α3 , α, 0) gives rise to Ls (X) = α3 + αX + X 2 . We now determine the number of needed indeterminates for L(X). Proposition 3. Suppose ttrue + 1 ≤ dBCH − 1. a) The number of needed indeterminates is 2ttrue − dBCH + 1. b) If dBCH is odd the number of needed indeterminates is 2(ttrue − tBCH ). c) If dBCH is even the number of needed indeterminates is 2(ttrue −tBCH )−1. Proof. From the hypothesis ttrue + 1 ≤ dBCH − 1, only the Yi ’s are needed indeterminates. In other words, in the range u0 , u0 + 1, . . . , u0 + ttrue all syndromes are known.

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a) In point d) corollary 1, set λ = ttrue and µ = dBCH − 1. Then, if WH (s(X)) ≥ dBCH − ttrue − 1 then the rank of Mj is dBCH − ttrue − 1. We can eliminate dBCH − ttrue − 1 indeterminates. b) If dBCH is odd we can eliminate 2tBCH − ttrue indeterminates, and the number of remaining ones is 2(ttrue − tBCH ). c) If dBCH is even we can eliminate 2tBCH + 1 − ttrue indeterminates, and the number of remaining ones is 2(ttrue − tBCH ) − 1.   Proposition 3 means that we will run Buchberger’s algorithm with only 2(ttrue − tBCH ) or 2(ttrue − tBCH ) − 1 indeterminates. We note that Xin prediction [8] is less expensive than Gaussian elimination for obtaining the needed indeterminates. Now we consider binary codes, from [4]. One problem arises : when using FG-algorithm, even e(X) has binary coefficients it could be possible to obtain a zero which gives rise to some s(X) with non binary coefficients. In order to avoid this situation we have the following lemma. Lemma 3. Suppose to add to SE(t, u0) the equations P2k − Pk2 = 0 for every k such as Pk is already calculated. Then the zeroes we get from the RGB give rise to polynomials s(X) which have binary coefficients. Proof. It is because every explicit polynomial S(X) must satisfy S2k = Sk2 .

 

Now we prove that only one attempt of the complete algorithm is necessary to find Le (X). Proposition 4. a) If the error is uncorrectable then the RGB reduces to a constant. b) Suppose the error is correctable, with Hamming weight ttrue . Then each polynomial in the RGB has only one indeterminate, and has degree one. c) If the error polynomial has Hamming weight ttrue − λ, then add the coefficients L0 , . . . , Lλ−1 to the RGB. The new RGB we obtain contains only polynomials with only one indeterminate, and degrees equal to 1. Proof. a) Suppose to have a zero of the ideal. Then there exists a polynomial L(X) and a polynomial S(X) such as L(X) is S(X)-orthogonal. From b) proposition 1 the degree of L(X) is greater than the Hamming weight of the error polynomial. From lemma 3, the (binary) error would be correctable, contrarily to the hypothesis. b) As the error is the only (binary) polynomial of weight at most ttrue in its own coset, there exists only one zero. Moreover, the ideal must contain one polynomial of degree 1 for each indeterminate. c) Among the polynomials L(X) we obtain, there is the polynomial X c Le (X) with c = t − WH (e(X)) (see point c) corollary 1).   Let us give FG algorithm : 1) Reduce the number of indeterminates (see proposition 3). Get L(X), with some indeterminates.

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2) Construct the initial basis (see lemma 3). 3) Run buchberger’s algorithm, and get the RGB. – If the RGB is reduced to a constant, then uncorrectable error. – If the RGB contains only polynomials with degrees 1 and 1 variable each, then correct. – In other cases, add to the RGB some coefficients of L(X), get a new RGB, and correct. When ttrue + 1 ≤ dBCH − 1, FG algorithm compares favorably with Chen et al. method which needs ttrue indeterminates. If ttrue + 1 > dBCH − 1 then FG-algorithm cannot run (see Step 1). Fortunately, this case only holds for the 8 following codes with parameters (length,dimension,dtrue, dBCH ), among the 147 ones given in [4] : (33, 13, 10, 5), (43, 15, 13, 7), (47, 24, 11, 5), (47, 23, 12, 6), (51, 25, 10, 5), (51, 17, 12, 6), (55, 30, 10, 5), (57, 21, 14, 6). In these cases, Chen et al. method uses less indeterminates than FG algorithm. Example 5. Consider the code (21, 7, 8, 5). Set β for an element of order 21 in GF (64). The cyclotomic classes containing the roots of the code are the respective classes of β 1 , β 3 , β 7 , β 9 . One chooses u0 = 1. Suppose e(X) = 1 + X + X 4 . We need 2 indeterminates. After step 1 of FG-algorithm, one has L(X) = α9 + α24 Y0 + α30 Y1 + Y1 X + Y2 X 2 + X 3 . The RGB we obtain is {α28 +α39 Y0 , α9 +α48 Y1 }. The unique zero is (α52 , α24 ). The polynomial L(X) we have is then L(X) = α9 + α24 α52 + α30 α24 + α52 X + α24 X 2 + X 3 . Its roots are α0 , α3 , α12 , which correspond to β 0 , β, β 4 . Example 6. Consider the same code as in example 5, namely the (21, 7, 8, 5) code. It is a code with ttrue = 3. The polynomial L(X) has degree 3. Suppose the error polynomial is 1 + X 4 . First we obtain L(X) = (α6 + α2 Y0 + α52 Y1 ) + Y0 X + Y1 X 2 + X 3 . The RBG is {α30 + α41 Y0 + α43 Y1 }. It is not reduced to a constant, so the error is correctable. It is not a pair of polynomials with degree 1 with one variable. It means that the degree of the locator is less than 3. We add α6 + α2 Y0 + α52 Y1 to the RBG, and we run buchberger’s algorithm. We then get {α54 + α52 Y1 , α20 + α8 Y0 }. The unic zero is (α12 , α2 ). The explicit L(X) is L(X) = X(α12 + α2 X + X 2 ), whose roots are 1, α12 . As β = α3 it means that erroneous locations are 1, 4, which is correct. With FG-algorithm it is possible to correct sometimes up to ttrue + 1, as it is shown now. Proposition 5. Suppose the code has an even minimal distance 2ttrue + 2, and the number of its minimum weight codewords is Nmin . If we choose t = ttrue +

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1 and if we use the previous algorithm, then we can correct some errors with Hamming weight ttrue  true +2 + 1. In casen of hard decoding, the probability to not correct is at most 2t ttrue +1 × Nmin / ttrue +1 . Proof. If e(X) is the only element in its own coset, with Hamming weight less that ttrue + 1, then we can correct. If not, it means that e(X) is a part of some codeword of minimum weight 2ttrue + 2.   Example 7. Consider the binary (15, 6, 6, 6) code, with 1, α, α2 , α3 , α4 as roots, and 1 + α + alpha4 = 0. Suppose e(X) = 1 + X + X 3 . We have L(X) = (α12 + α7 Y0 ) + Y0 X + α7 X 2 + X 3 . The RGB we obtain is α5 + α6 X, and the unic zero is α14 . This zero gives rise to Le (X) = α4 + α14 X + α7 X 2 + X 3 . The error can be corrected. Sometimes every word of Hamming weight ttrue + 1 is a part of a minimum weight 2(ttrue + 1) codeword (as for even Hamming codes). Then hard decoding cannot run. Observe that if hard decoding cannot run, then soft decoding may correct most errors with Hamming weight t = ttrue + 1. A Table for FG-Algorithm up to 4 Indeterminates Even using a powerful DSP such as the Texas-Inst. TMS 320C62 along with a SDRAM of 64 MgO dynamic memory, it looks difficult to implement FGalgorithm with more than 4 indeterminates. The following Table shows what could be realistic for implementing FGalgorithm. It also compares the number of needed indeterminates for FGalgorithm (column 1) and for Chen et al. algorithm (column 2). Table 1. Comparison of number of needed indeterminates. FG

Chen

1 1 2 3 4

tBCH tBCH tBCH tBCH tBCH

remarks +1 +1 +1 +2 +2

ttrue ttrue ttrue ttrue ttrue

= tBCH = tBCH = tBCH = tBCH = tBCH

> 1, both even + 1, dBCH even + 1, dBCH odd + 1, both even + 2, dtrue odd

example

correction

(31, 20, 6, 6) (35, 16, 7, 6) (21, 7, 8, 5) (51, 26, 8, 6) (55, 25, 11, 7)

some (tBCH + 1) tBCH + 1 tBCH + 1 some (tBCH + 2) tBCH + 2

Conclusion The proposed FG-algorithm, which uses Discrete Fourier Transform as well as Gr¨ obner bases, is mainly dedicated to decoding cyclic codes up to ttrue for codes having ttrue = ttrue and ttrue ≤ dBCH − 2.

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In the binary case FG algorithm needs 2(ttrue −tBCH ) (resp. 2(ttrue −tBCH )− 1) indeterminates when dBCH is odd (resp. even). It is less than Chen et al. algorithm which needs ttrue indeterminates, and sometimes less than XP algorithm, which needs 2(ttrue − tBCH )) indeterminates whatever is dBCH . For non binary codes the number of needed indeterminates for FG-algorithm remains the same as for binary codes, when for Chen et al. it is twice. FG algorithm is used for each received word. One referee considers that the only realistic way to use Gr¨obner bases is to have some preprocessing step for the code itself. We agree with him, except when one uses 1, 2 or 3 indeterminates. On the other hand FG algorithm reduces the number of needed indeterminates. That looks interesting whatever is the used decoding algorithm with Gr¨ obner bases. Also, another practical advantage of FG-algorithm is to need only one attempt for decoding one word. We are now working on how to implement some particular codes from [4], with FG-algorithm.

References 1. X. Chen et al. : Use of Gr¨ obner bases to decode binary cyclic codes up to the true minimum distance, IEEE IT-40 (5), pp. 1654-1661, Sept. 1994.— General Principles for the Algebraic decoding of cyclic codes, IEEE IT-40 (5), pp. 1661-1663, Sept. 1994. 2. Cooper : Finding BCH codes error locator polynomial in one step, Electronics Letters, vol. 27, (22), pp. 2090-2091, 1991. 3. Imai H. : Essentials of Error-Control coding techniques, Acad. Press Inc. 1990. 4. van Lint J.H., Wilson R.M. : On the minimum distance of cyclic codes, IEEE Trans on IT, vol. IT-32, (1), January 1986, pp. 23-40. 5. Manhun S. : Decoding RS codes beyond the error correction bound, Journal of complexity, (13), pp. 180-193, 1997. 6. Poli A., Huguet Ll. : Error correcting codes : theory and applications, Prentice Hall eds, 1992. 7. Sakata S. : Gr¨ obner bases and coding theory, in “Gr¨ obner bases and applications” B. Buchberger and F. Winkler eds. Cambridge University Press, 1998. 8. Xin D. : Homogeneous minimum interpolation and key equation for decoding RS codes, Science in China, vol. 37, (11), pp. 1387-1398, 1994. 9. Xin D., Poli A., Gennero M.C. : Xin algorithm and Gr¨ obner bases, Proc of ISITA, vol. 1, pp. 204-208, Mexico, August 1998

On the State Complexities of Ternary Codes Sylvia Encheva1 and G´erard Cohen2 1

Stord/Haugesund College, Bjørnsonsgt. 45, 5528 Haugesund, Norway [email protected] 2 ENST, Dept. of Informatics and CNRS 46 rue Barrault, 75634 Paris Cedex 13, France [email protected]

Abstract. The paper studies self-orthogonal codes over GF (3). The state complexities of such codes of lengths ≤ 20 with efficient coordinate ordering are found.

1

Introduction

A good deal of effort has been devoted to finding the minimum “trellis complexity” of linear codes. It is well known that the minimum trellis size depends on the order of the code coordinates [6]. We consider in this paper only linear codes over GF (3), whose generator matrices have no zero columns. An [n, k, d] code C has length n, dimension k and minimum Hamming distance d. A self-orthogonal code is maximal if it is not contained in a larger selforthogonal code of the same length [2]. The support of a vector a = (a(1) , a(2) , ..., a(n)) in GF (3)n is defined by χ(a) = {j|a(j) = 0}. The minimum support weight, dr , of a code C is the size of the smallest support of any r-dimensional subcode of C. In particular d1 = d. The weight hierarchy of C is the set {d1 , d2 , ..., dk }. The concepts of chain condition and two-way chain condition were introduced by Wei and Yang [13] and Forney [5] respectively. Definition 1. An [n, k] code C satisfies the chain condition if it is equivalent to ˜ D1 ⊂ D2 ⊂ · · · ⊂ Dk = a code C˜ such that there exists a chain of subcodes of C, ˜ where, for 1 ≤ r ≤ k, we have dim(Dr ) = r, and χ(Dr ) = {1, 2, . . . , dr }. C, Theorem 1. (Duality) [13] Let C be an [n, k] code and C ⊥ be its dual code. Then {dr (C) : 1 ≤ r ≤ k} = {1, 2, ..., k} − {n + 1 − dr (C ⊥ ) : 1 ≤ r ≤ n − k}. Definition 2. An [n, k] code C satisfies the two-way chain condition (TCC) if it is equivalent to a code C˜ with the following property: there exist two chains ˜ the left chain DL ⊂ DL ⊂ · · · ⊂ DL = C, ˜ and the right of subcodes of C, 1 2 k R R R ˜ where, for 1 ≤ r ≤ k, we have dim(DL ) = chain D1 ⊂ D2 ⊂ · · · ⊂ Dk = C, r R L dim(Dr ) = r, χ(Dr ) = {1, 2, . . . , dr }, and χ(DrR ) = {n−dr +1, n−dr +2, . . . , n}. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 454–461, 1999. c Springer-Verlag Berlin Heidelberg 1999 

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Lemma 1. [3] If C = C1 ⊕ C1 ⊕ ... ⊕ C1 , where C1 satisfies the two-way chain condition, then C satisfies the two-way chain condition. A code has an efficient coordinate ordering (eco) if and only if it satisfies the two-way chain condition [5]. If C satisfies the TCC, then si (C) = k − ki − kn−i , where ki is the maximum dimension of any subcode of C whose effective length (support size) is i, 1 ≤ i ≤ n [5]. The maximum component of s(C) is denoted by smax = max si (C). i

The minimum smax (C) over all coordinate orderings is called the state complexity s(C) of C (see [12] for a nice survey). Lemma 2. [7] A coordinate ordering is efficient for a linear code C if and only if it is efficient for its dual code C ⊥ . We shall use notations of [2], [9], [10] whenever we refer to these papers. The direct sum C1 ⊕ C1 we will be denoted by 2C1, etc. The outline of this paper is as follows. Section 2 presents a necessary and sufficient condition for some particular codes to have an eco. All codes of length ≤ 12 with eco are found. Section 3 considers coordinate orderings of all self-dual codes of length 16 with weights divisible by 3. All self-dual codes of length 20 with eco are described in Section 4. A necessary graphical condition for a code to admit an eco is provided.

2

Maximal Self-Orthogonal Codes of Length ≤ 12

The maximal self-orthogonal codes of length ≤ 12 have been completely classified in [9]. In [2], [9], and [10] codes are efficiently described as being made up of various components held together by glue vectors. The following component codes shall be used in this paper. 1) e3 is the [3, 1, 3] code with generator matrix ( 1 1 1 ) and the coordinate ordering is obviously efficient. 2) e4 is the [4, 2, 3] code with generator matrix 1110 0121. Its weight hierarchy is {3, 4} and its coordinate ordering is efficient. 3) g8 is a [8, 2, 6] code and may be formed by doubling each codeword of e4 . Its weight hierarchy is {6, 8} and its coordinate ordering is efficient.

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4) g9 is the [9, 3, 6] code with generator matrix 111111000 111000111 120120120. Its weight hierarchy is {6, 8, 9} and its coordinate ordering is efficient. 5) The “free” code fn consists of the zero vector of length n. A code C is reversible if (c(0), c(1), ..., c(n−1)) ∈ C implies (c(n−1), ..., c(1), c(0)) ∈ C [8]. Lemma 3. [4] If the rows of a generator matrix of a reversible code generate the subcodes in the left chain from Definition 2, then the code satisfies the TCC. Proposition 1. Let C = ae3 ⊕ be4 where a, b are nonegative integers. C has eco iff a + b > 0 and ab = 0. Proof. If a + b > 0 and ab = 0 then C has eco by Lemma 1. Let C have eco and C = ae3 ⊕ be4 . If ab = 0, then the weight hierarchy of C will be d2p+1 = 3 + 4p, 0 ≤ p ≤ b − 1 d2p = d2p−1 + 1, 1 ≤ p ≤ b d2b+i = d2p + 3i, 1 ≤ i ≤ a. There is only one subspace of dimension 2b with weight hierarchy {d1 , d2 , ..., d2b }, generated by the last 2b rows of C. In order to obtain an eco for C we would need two such subspaces with supports in the first and last 2b positions, respectively.  A code C is indecomposable if it cannot be written as a direct sum of other codes. Corollary 1. All indecomposable self-orthogonal codes generated by vectors of weight 3 have eco. Proof. According to Theorem 11 [9] the only indecomposable self-orthogonal codes generated by vectors of weight 3 are e3 and e4 . Then the proof follows by Proposition 1.  Corollary 2. All self-dual codes generated by vectors of weight 3 have eco. Proof. According to Corollary 12 [9] the only self-dual codes generated by vectors of weight 3 are e4 ⊕ e4 ... ⊕ e4 . Then the proof follows by Proposition 1. 

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Lemma 4. There are exactly 3 decomposable maximal self-orthogonal codes of length ≤ 12 that do not have eco and exactly 3 decomposable maximal selforthogonal codes of length ≤ 12 that have eco. Proof. The first three codes are e3 ⊕ e4, 2e3 ⊕ e4 and e3 ⊕ 2e4 . They do not have an eco by Proposition 1. The other three are 2e3 , 2e4 and 3e4 . Their coordinate ordering is efficient by Lemma 1.  A hexad of a self-orthogonal code C of minimum weight 6 is the binary vector of length n which is the support of a pair of codewords c, c ∈ C of weight 6 [10]. Theorem 2. There are exactly six inequivalent codes that do not have eco and exactly nine inequivalent codes that have eco among maximal self-orthogonal codes of length ≤ 12. Proof. The three decomposable maximal self-orthogonal codes of length ≤ 12 which do not have eco have been discussed in Lemma 4. The other three which do not have eco are the indecomposable codes given below. i) The [10, 4, 3] code has generator matrix 1110000000 0001110000 1200001111 0001201212. The first two elements of its weight hierarchy are 3 and 6. If the code had eco then it would contain two codewords c1, c2 of weight 6 with supports in the first and last six positions, respectively, and two codewords of weight 3 with supports in the first and last three positions. Consider any two vectors of weight 6 which intersect in exactly two positions {c(i), c(j) } = A. It turns out by inspection that vectors of weight 3 cover at least one of these two positions for each couple in A. Therefore, the [10, 4, 3] code does not have eco. ii) The [11, 5, 3] code has generator matrix 11100000000 00011100000 00000011100 12012000012 21000012012. If the code had eco then it would contain two codewords c1 , c2 of weight 6 with supports in the first and last six positions, respectively, and wt(c1 + c2 ) would be 10 or 11.

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The weight distribution of C [9] shows that all weights in C are divisible by 3. Therefore, C does not have eco. iii) Next code has parameters [11, 5, 6] and is the set of vectors in the ternary [12, 6, 6] Golay code G12 which are zero in one fixed position. We need two hexads which meet in one position only. By [10] two distinct hexads meet in 0,2,3 or 4 coordinates. This concludes the proof for codes not having an eco. Codes of length 3 and 4, i.e.e3 and e4 are shown to have eco in the beginning of this subsection. The other three decomposable codes which have eco are discussed in Lemma 4. A direct check shows that the coordinate orderings of 3C3(9) and 4C3(12) codes are efficient. An eco for G12 may be seen from the following generator matrix 111111000000 001221110000 010122101000 021012100100 010110200110 000000211111. The last code g10 has parameters [10, 4, 6] and is the set of vectors in G12 which are zero on two fixed coordinates. Elements of its weight hierarchy are {6, 8, 9, 10}. They are found applying Theorem 1. Finally an eco for g10 may be seen from the following generator matrix 1002001212 0101020222 0011021011 0000111221. The right chain can be seen immediately. The left chain of subcodes {(DrL )}, 1 ≤ r ≤ 4 is generated by the first r vectors from {c1 + c2 + c3 + c4 , c1 + c4 , c1 +  c3 , c4 } where the rows are denoted by ci , 1 ≤ i ≤ 4.

3

Type III Codes of Length 16

A theorem of Gleason and Pierce (see [11]) implies that a self-dual code over GF (q) can only have all weights divisible by some integer t > 1 in five cases I: q = 2, II : q = 2, III : q = 3, IV : q = 4, V : q arbitrary,

t = 2, t = 4, t = 3, t = 2, t = 2, weight enumerator (x2 + (q − 1)y 2 )n/2 .

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In the literature they are known as codes of types I-V [2]. In [2] and [10] a code is regarded as consisting of a number of components which are held together by glue. The glue vectors are a = 120, a ¯ = 210, x, y and t0 [10]. Vectors x and y are chosen so that 11x ∈ G12 and 12y ∈ G12 . Vectors t0 , t1 , ..., t12 are the cyclic shifts of vector t0 = (1101000001000) and ⊥ they generate a [13, 7] code P13 [8]. Theorem 3. There are two codes that have eco and five codes that do not have eco among type III codes of length 16. Proof. All type III codes of length 16 are enumerated in [2]. The 4e4 code has eco by Lemma 1. The unique [16, 8, 6] code has generator matrix [I|H8 ], where H8 is the Hadamard matrix 11111111 12221211 12212112 12121122 11211222 12112221 11122212 11222121. If the columns of [I|H8 ] are numbered from 1 to 16 then the following column permutation 1,2,10,11,12,14,6,8,13,15,16,9,7,3,4,5 shows that the code has eco. From this column permutation the left chain of subcodes {(DrL )}, 1 ≤ r ≤ 8 is generated by the first r vectors from {c1 + 2c2, c6 + 2c8 , c1 + 2c8 , c6 , c7 , c3 , c4 , c5 } and the right chain of subcodes is generated by the vectors {c3 + c4 + c5 + c7 , c4 + c5 , c6 + c8 , c5 + c6 , c6 + c7 , c3 , c2 , c1 }. aaa0 does not have eco. Looking The 4e3 ⊕ e4 code with glue words aaa00, 0¯ at the weight distribution of the code [2] we notice that all codewords of weight 3 are generated by the first 6 rows of the code. The first 2 elements of its weight hierarchy are 3 and 4. Applying Proposition 1 we complete the proof. The G12 ⊕ e4 code does not have eco. Note that all codewords of weight 3 are generated by the last 2 rows of its generator matrix. The proof is similar to the one in the previous case. The 4e3 ⊕ f4 code with glue (a000), (2111) does not have eco because there are no 2 different subspaces of dimension 4 with support {3, 6, 9, 12}. The 2e3 ⊕ g10 code with glue words a0x, 0ay, does not have eco since there is only one subspace of dimension 2 and support {3, 6}. The last code is e3 ⊕ P13 with glue at0 . It does not have eco because it contains exactly 2 vectors of weight 3 and one of them is obtained from the other by multiplication by 2. 

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Corollary 3. The state complexities of the two previous codes that have eco are s(4e4 ) = 2 and s([16, 8, 6]) = 5.

4

Self-Dual Codes of Length 20

Lemma 5. Among the seven decomposable self-dual codes of length 20 there is one with eco and six without. Proof. All decomposable self-dual codes of length 20 are enumerated in [10]. The coordinate ordering of 5e4 is efficient by Lemma 1. a a00 and G12 ⊕ 2e4 code do The 4e3 ⊕ 2e4 code with glue words aaa000, 0a¯ not have eco because they do not contain 2 different subsets of dimension 4 and weight hierarchy {3, 4, 7, 11}. The last 4 codes do not have eco since they do not contain 2 different subsets of dimension 2 and weight hierarchy {3, 4}.  Corollary 4. The state complexity of the previous eco code is s(5e4 ) = 2. Let Bi = {b1 , b2 , ..., bd2 } denote the support of a two-dimensional subspace of C containing at least one codeword of weight d1 . Note that there might exist no such Bi ; such codes are studied in [1] under the name of antichain codes. We define an undirected graph Γ with
vertex set {Bi }, two nodes Bi and Bj being joined by an edge if and only if Bi Bj = ∅. Proposition 2. Let C be of length n ≥ 2d2 . If the associated graph Γ is complete then C does not have eco. Proof. In order to find eco for C we should find two subspaces of dimension two with disjoint supports and weight hierarchies {d1 , d2 }. Since Γ is complete such an ordering does not exist.  Example 1. Consider all hexads in the code 10f2 (code 19 in Table III [10]). Any two hexads with support 8 form a set Bi . There are exactly five different sets Bi and any two of these sets meet in exactly two positions. Therefore, the associated graph is complete, regular and its valency is 4. By Proposition 2 we conclude that the code 10f2 does not have eco. Remark 1. 10f2 is the only self-dual code of length 20 and minimum distance 6 that does not have eco. Let C = [g(k, d) + 2l, k, d], l ≤ 2 be a self-dual code which satisfies the chain condition. W.l.o.g. we may assume that its coordinates are arranged in such a way that the rows of its generator matrix generate the left chain of subcodes from Definition 2.

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Let C be a self-dual code of length na + nb . Partition its generator matrix as follows A0 0 B D E, where subcodes A and D have even length na . Let the dimension of D be kd and the last kd rows be the glue words for A and B. Corollary 5. The state complexities of the previous eco codes are: s(6e3 ⊕ f2 ) = s(4e3 ⊕ g8 ) = 3, s(5e3 ⊕ f5 ) = 4, s(4f4 ⊕ 2f2 ) = s(5f4 ) = s(2g9 ⊕ 2f2 ) = s(2g10 ) = 5, s(4f5 ) = 6.

References 1. G. Cohen, S. Encheva and G. Z´emor: Antichain codes. Designs, Codes and Cryptography (to appear) 2. J. H. Conway, V. Pless and N. J. A. Sloane: Self-Dual Codes over GF (3) and GF (4) of Length not Exceeding 16. IEEE Trans. Inform. Theory 25 (1979) 312–316 3. S. Encheva: On binary linear codes which satisfy the two-way chain condition. IEEE Trans. Inform. Theory Theory 42 (1996) 1038–1047 4. S. Encheva: On repeated-root cyclic codes and the two-way chain condition. Springer-Verlag LNCS 1255 (1997) 5. G. D. Forney, Jr.: Density/Length Profiles and Trellis Complexity of Lattices. IEEE Trans. Inform. Theory 40 (1994) 1753–1772 6. P.A. Franaszek: Sequence-state coding for digital transmission. Bell Syst. Tech. J. 47 (1968) 143-157 7. T. Kasami, T. Takata, T. Fujkiwara and S. Lin: On the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes. IEEE Trans. Inform. Theory 39 (1993) 242-245 8. F. J. MacWilliams and N. J. A. Sloane: The Theory of Error-Correcting codes. North-Holland Amsterdam (1977) 9. C. L. Mallows, V. Pless and N. J. A. Sloane: Self-Dual Codes over GF (3) SIAM J. Appl. Math. 31 (1976) 649-666 10. V. Pless, N. J. A. Sloane and H. N. Ward: Ternary codes of Minimum Weight 6 and the Classification of the Self-Dual Codes of Length 20 IEEE Trans. Inform. Theory 26 (1980) 305-316 11. N.J.A. Sloane: Self-dual codes and lattices. In Relations between Combinatorics and other Parts of Mathematics Amer. Math. Soc. Proc. Sympos. Pure Math. vol. XXXIV (1979) 273-308 12. A. Vardy: Trellis Structure of Codes. In Handbook of Coding Theory II. Elsevier (1998) 1989-2117 13. V. K. Wei and K. Yang: On the generalized Hamming weights of product codes. IEEE Trans. Inform. Theory 39 (1993) 1709-1713

Binary Optimal Linear Rate 1/2 Codes Koichi Betsumiya1 , T. Aaron Gulliver2, and Masaaki Harada3 1

Nagoya University, Graduate School of Mathematics Nagoya 464–8602, Japan [email protected] 2 University of Victoria, Dept. of Electrical & Computer Eng. P.O. Box 3055, STN CSC, Victoria, B.C., Canada V8W 3P6 [email protected] 3 Yamagata University, Department of Mathematical Sciences Yamagata 990–8560, Japan [email protected]

Abstract. In this paper, we classify the optimal linear [n, n/2] codes of length up to 12. We show that there is a unique optimal odd formally self-dual [20, 10, 6] code up to equivalence. We also show that at least one optimal linear [n, n/2] code is self-dual or formally self-dual for lengths up to 48 (except 38 and 40).

1

Introduction

A binary linear [n, k] code C is a k-dimensional vector subspace of Fn2 , where F2 is the finite field of two elements. The rate of a linear [n, k] code C is defined as k/n. The elements of C are called codewords. The weight wt(x) of a codeword x is the number of non-zero coordinates. The minimum weight of C is the smallest weight among all non-zero codewords of C. An [n, k, d] code is an [n, k] code with minimum weight d. Two codes C and C
are equivalent if one can be obtained from the other by permuting the coordinates. The automorphism group of C is the set of permutations of
the coordinates which preserve C. The weight n enumerator of C is WC (y) = i=0 Ai y i , where Ai is the number of codewords of weight i in C. The dual code C ⊥ of C is defined as C ⊥ = {x ∈ Fn2 | x · y = 0 for all y ∈ C} where x · y denotes the standard inner-product of x and y. A code C is self-dual if C = C ⊥. A code C is formally self-dual if C and C ⊥ have identical weight enumerators. Self-dual codes are by definition also formally self-dual. A formally self-dual code is called even if the weights of all codewords are even, otherwise the code is odd. A linear [n, k] code C is optimal if C has the highest minimum weight among all linear [n, k] codes (see [4] for current bounds on the highest minimum weight). It is well known that there is a unique optimal linear code up to equivalence for parameters [8, 4, 4] and [24, 12, 8], namely the extended Hamming code and the extended Golay code. Recently it has been shown that there is a unique optimal rate 1/2 linear [n, n/2] code for n = 16, 18, 22 and 28, up to equivalence [3,10,12]. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 462–471, 1999. c Springer-Verlag Berlin Heidelberg 1999 

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A natural question that now arises is how many inequivalent optimal linear rate 1/2 codes are there for other small lengths? In this paper, we present the classification of optimal linear [n, n/2] codes of length up to 12. We show that at least one optimal linear [n, n/2] code is self-dual or formally self-dual for lengths up to 48 (except 38 and 40 for which the highest minimum weight has not yet been determined). We also show that there is a unique optimal odd formally self-dual [20, 10, 6] code up to equivalence. Throughout this paper, we denote the highest minimum weight among all linear [n, n/2] codes, by dmax (n) (see Table 3 for the actual values of dmax (n)).

2

Classification of Length up to 12

In this section, we classify the optimal linear [n, n/2, dmax(n)] codes up to length 12. Let NS (n) NE (n) and NO (n) denote the numbers of inequivalent optimal linear [n, n/2, dmax (n)] codes which are self-dual, even formally self-dual and odd formally self-dual, respectively, and let NT (n) denote the total number of inequivalent optimal linear [n, n/2, dmax (n)] codes. 2.1

Lengths 2, 4, 6 and 8

It is trivial that there is a unique linear code for parameters [2, 1, 2] and [6, 3, 3], and there are exactly three inequivalent [4, 2, 2] codes, namely the unique selfdual code, the unique even formally self-dual code and the unique odd formally self-dual code. The extended Hamming code is the unique [8, 4, 4] code and the code is self-dual. Hence NT (2) = 1, NT (4) = 3, NT (6) = 1 and NT (8) = 1. 2.2

Length 10

First note that dmax (10) = 4. The classifications of even and odd formally selfdual [10, 5, 4] codes are given in [11] and [3], respectively. Thus NS (10) = 0, NE (10) = 1 and NO (10) = 1. We describe how the optimal linear [10, 5, 4] codes C were classified. Every [10, 5] code is equivalent to a code with generator matrix of the form ( I , A ) where A is a 5 × 5 (1, 0)-matrix. Thus we only need to consider the set of 5 × 5 (1, 0)-matrices A, rather than the set of generator matrices. The set of matrices A was constructed, row by row, using a back-tracking algorithm under the condition that the first row is (00111), since the minimum weight of C is 4. Permuting the rows of A gives rise to different generator matrices which generate equivalent codes. We consider only those matrices A which are smallest among all matrices obtained from A by permuting its rows, where the ordering involved is lexicographical on the binary integers corresponding to the rows of the matrix.

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Our computer search shows that an optimal linear [10, 5, 4] code which is not formally self-dual is equivalent to one of the two codes C10,1 and C10,2 , with generator matrices G10,1 and G10,2 where     10000 00111 10000 00111  01000 01011   01000 01011         G10,1 =  00100 01101  and G10,2 =   00100 01101  ,  00010 01110   00010 10011  00001 10011 00001 10101 respectively. Some equivalences were verified by Magma. The weight enumerators of C10,1 and C10,2 are 1 + 18y 4 + 8y 6 + 5y 8 and 1 + 16y 4 + 12y 6 + 3y 8 , respectively. Hence we have the following: Proposition 1 There are exactly four inequivalent optimal linear [10, 5, 4] codes. 2.3

Length 12

There is a unique self-dual [12, 6, 4] code, up to equivalence. The classifications of even and odd formally self-dual [12, 6, 4] codes are given in [1] and [3], respectively. Thus NS (12) = 1, NE (12) = 2 and NO (12) = 5. Similar to length 10, we found all distinct 6 × 6 (1, 0)-matrices A such that the matrices ( I , A ) generate optimal linear [12, 6, 4] codes. The set of matrices A was constructed, row by row, using a back-tracking algorithm under the condition that the first row is (000111) and considering a lexicographical ordering of the rows. The codes are divided into the following 35 distinct weight enumerators: W12,1 = 1 + 6y 4 + 24y 5 + 16y 6 + 9y 8 + 8y 9 W12,2 = 1 + 8y 4 + 20y 5 + 14y 6 + 8y 7 + 7y 8 + 4y 9 + 2y 10 W12,3 = 1 + 9y 4 + 18y 5 + 13y 6 + 12y 7 + 6y 8 + 2y 9 + 3y 10 W12,4 = 1 + 10y 4 + 15y 5 + 16y 6 + 11y 7 + 5y 8 + 5y 9 + y 11 W12,5 = 1 + 10y 4 + 16y 5 + 12y 6 + 16y 7 + 5y 8 + 4y 10 W12,6 = 1 + 10y 4 + 16y 5 + 16y 6 + 8y 7 + 5y 8 + 8y 9 W12,7 = 1 + 10y 4 + 18y 5 + 10y 6 + 12y 7 + 9y 8 + 2y 9 + 2y 10 W12,8 = 1 + 10y 4 + 20y 5 + 8y 6 + 8y 7 + 13y 8 + 4y 9 W12,9 = 1 + 11y 4 + 14y 5 + 15y 6 + 12y 7 + 4y 8 + 6y 9 + y 10 W12,10 = 1 + 11y 4 + 16y 5 + 8y 6 + 16y 7 + 11y 8 W12,11 = 1 + 12y 4 + 13y 5 + 12y 6 + 15y 7 + 7y 8 + 3y 9 + y 11 W12,12 = 1 + 12y 4 + 14y 5 + 12y 6 + 12y 7 + 7y 8 + 6y 9

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W12,13 = 1 + 12y 4 + 16y 5 + 6y 6 + 16y 7 + 11y 8 + 2y 10 W12,14 = 1 + 12y 4 + 20y 5 + 6y 6 + 8y 7 + 11y 8 + 4y 9 + 2y 10 W12,15 = 1 + 13y 4 + 12y 5 + 11y 6 + 16y 7 + 6y 8 + 4y 9 + y 10 W12,16 = 1 + 14y 4 + 36y 6 + 9y 8 + 4y 10 W12,17 = 1 + 14y 4 + 12y 5 + 8y 6 + 16y 7 + 9y 8 + 4y 9 W12,18 = 1 + 15y 4 + 32y 6 + 15y 8 + y 12 W12,19 = 1 + 15y 4 + 33y 6 + 12y 8 + 3y 10 W12,20 = 1 + 15y 4 + 10y 5 + 15y 6 + 12y 7 + 10y 9 + y 10 W12,21 = 1 + 15y 4 + 16y 5 + 16y 7 + 15y 8 + y 12 W12,22 = 1 + 16y 4 + 30y 6 + 15y 8 + 2y 10 W12,23 = 1 + 16y 4 + 9y 5 + 12y 6 + 15y 7 + 3y 8 + 7y 9 + y 10 W12,24 = 1 + 16y 4 + 10y 5 + 12y 6 + 12y 7 + 3y 8 + 10y 9 W12,25 = 1 + 17y 4 + 27y 6 + 18y 8 + y 10 W12,26 = 1 + 18y 4 + 24y 6 + 21y 8 W12,27 = 1 + 18y 4 + 28y 6 + 13y 8 + 4y 10 W12,28 = 1 + 18y 4 + 8y 5 + 8y 6 + 16y 7 + 5y 8 + 8y 9 W12,29 = 1 + 19y 4 + 24y 6 + 19y 8 + y 12 W12,30 = 1 + 19y 4 + 25y 6 + 16y 8 + 3y 10 W12,31 = 1 + 20y 4 + 22y 6 + 19y 8 + 2y 10 W12,32 = 1 + 22y 4 + 20y 6 + 17y 8 + 4y 10 W12,33 = 1 + 23y 4 + 16y 6 + 23y 8 + y 12 W12,34 = 1 + 25y 4 + 27y 6 + 10y 8 + y 10 W12,35 = 1 + 26y 4 + 24y 6 + 13y 8 . We verified that all [12, 6, 4] codes are equivalent for each weight enumerator except W12,12 , W12,18 , W12,22 , W12,26 , W12,27 and W12,31 . For these six weight enumerators, the numbers of inequivalent codes are 2, 3, 3, 2, 2 and 2, respectively. Some equivalences were verified by Magma. Note that the codes with W12,i , (i = 1, 2, . . . , 5) are odd formally self-dual and the codes with W12,18 are even formally self-dual codes including the unique self-dual code. We consider only the remaining codes, and complete the classification by listing generator matrices ( I , G12,i ) for the inequivalent codes where the weight enumerator of G12,i is W12,i . To save space, we list the matrix G12,i using the form g1 , g2 , . . . , g6 where gj is the j-th row of G12,i . G12,6 = 000111, 001011, 010011, 011101, 100101, 111011 G12,7 = 000111, 001011, 010011, 011101, 100101, 111001 G12,8 = 000111, 001011, 001101, 011110, 101110, 110001 G12,9 = 000111, 001011, 010101, 011001, 100110, 111010 G12,10 = 000111, 001011, 010011, 100101, 111001, 111100

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G12,11 = 000111, 001011, 010011, 100101, 101001, 111100 G12,12−1 = 000111, 001011, 001101, 010011, 101110, 110001 G12,12−2 = 000111, 001011, 010011, 011101, 100101, 101001 G12,13 = 000111, 001011, 010011, 011101, 100011, 101101 G12,14 = 000111, 001011, 010011, 011101, 011110, 100101 G12,15 = 000111, 001011, 001101, 010011, 100101, 111001 G12,16 = 000111, 001011, 010011, 100101, 111000, 111110 G12,17 = 000111, 001011, 001101, 010011, 100011, 110101 G12,19 = 000111, 001011, 010101, 011001, 100110, 111000 G12,20 = 000111, 001011, 010101, 011001, 011111, 101101 G12,21 = 000111, 001011, 010011, 011101, 011110, 100011 G12,22−1 = 000111, 001011, 001101, 010011, 100011, 111101 G12,22−2 = 000111, 001011, 001101, 010011, 100101, 111000 G12,22−3 = 000111, 001011, 010011, 100101, 101001, 110100 G12,23 = 000111, 001011, 001101, 010011, 010101, 111001 G12,24 = 000111, 001011, 001101, 010011, 010101, 101110 G12,25 = 000111, 001011, 001101, 010011, 100101, 110001 G12,26−1 = 000111, 001011, 001101, 010011, 100011, 110001 G12,26−2 = 000111, 001011, 010011, 100101, 101001, 110001 G12,27−1 = 000111, 001011, 001101, 010011, 010101, 111011 G12,27−2 = 000111, 001011, 010011, 100101, 101001, 101111 G12,28 = 000111, 001011, 001101, 001110, 010011, 110101 G12,29 = 000111, 001011, 001101, 010011, 010101, 111000 G12,30 = 000111, 001011, 001101, 010011, 010101, 101001 G12,31−1 = 000111, 001011, 001101, 001110, 010011, 110001 G12,31−2 = 000111, 001011, 001101, 010011, 010101, 100011 G12,32 = 000111, 001011, 001101, 001110, 010011, 100101 G12,33 = 000111, 001011, 001101, 001110, 010011, 100011 G12,34 = 000111, 001011, 001101, 010011, 010101, 011001 G12,35 = 000111, 001011, 001101, 001110, 010011, 010101 Therefore we have the following: Proposition 2 There are exactly 43 inequivalent optimal linear [12, 6, 4] codes.

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3 3.1

467

Lengths 14 and 20 On the Classification of Length 14

There is a unique self-dual [14, 7, 4] code, up to equivalence. Recently the classification of even formally self-dual [14, 7, 4] codes has been given in [2] and [7], independently. The classification of odd formally self-dual [14, 7, 4] codes is given in [3]. Thus NS (14) = 1, NE (14) = 9 and NO (14) = 112. As with other lengths, we found all optimal linear [14, 7, 4] codes which should be checked to complete the classification. However, we found there were too many distinct weight enumerators (518 in total) to classify. These weight enumerators are available on the world wide web at http://www.math.nagoya-u.ac.jp/~koichi/data. We were able to determine that there are at least 1360 inequivalent [14, 7, 4] codes, and there appear to be many more. Some examples of these codes are listed below as the right halves of the generator matrices. G14,1 = 0000111, 0001011, 0011101, 0101101, 0110011, 1010001, 1101111 G14,2 = 0000111, 0001011, 0011101, 0101101, 0110001, 1001110, 1110010 G14,3 = 0000111, 0001011, 0011101, 0101101, 0110011, 1001110, 1010001 G14,4 = 0000111, 0001011, 0011101, 0101101, 0110011, 0111110, 1001110 G14,5 = 0000111, 0001011, 0010101, 0101101, 0110011, 1011000, 1101110 3.2

Odd Formally Self-Dual Codes of Length 20

Since optimal linear codes with parameters [16, 8, 5] and [18, 9, 6] are unique from [3] and [12], the next open case is length 20. However it seems infeasible to classify all optimal linear [20, 10, 6] codes. Recently it has been shown in [8] that there are exactly six inequivalent even formally self-dual [20, 10, 6] codes. Thus we consider the odd formally self-dual [20, 10, 6] codes. We constructed all distinct 10 × 10 (1, 0)-matrices A such that the matrices ( I , A ) generate odd formally self-dual [20, 10, 6] codes using the technique described earlier. From these, we obtained 4872 distinct odd formally self-dual [20, 10, 6] codes. Our search determined that these codes are equivalent to the double circulant code with generator matrix ( I , R ) where R is the circulant matrix with first row (1111100100). The weight enumerator of this code is 1 + 40y 6 + 160y 7 + 130y 8 + 176y 10 + 320y 11 + 120y 12 + 40y 14 + 32y 15 + 5y 16 . Combining the results in [8] with this classification, we have the following: Proposition 3 There are exactly seven inequivalent formally self-dual [20, 10, 6] codes, one of which is odd. Combining the results in [3] with this classification, we have the following:

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Corollary 1. The optimal formally self-dual codes (that is, the codes having the highest minimum weight among all formally self-dual codes of that length) are classified for all lengths up to 24. The actual numbers of inequivalent optimal formally self-dual codes are listed in Table 1. 3.3

Shortened Codes

If an optimal linear [12, 6, 4] code has a coordinate such that the codewords of minimum weight are 0 in this coordinate, then the shortened code obtained by deleting this coordinate is an optimal [11, 6, 4] code. Thus our classification of optimal linear [12, 6, 4] codes implies the classification of optimal [11, 6, 4] codes. Therefore we have the following: Corollary 2. There are exactly two inequivalent optimal [11, 6, 4] codes. 3.4

Summary

As a summary, we list in Table 1 the actual values NT (n), NS (n), NE (n) and NO (n) for 2 ≤ n ≤ 24. From Table 1, we have the following observation. Corollary 3. Let n(d) be the smallest length for which there is a linear rate 1/2 code with minimum weight d. For d ≤ 8, there is a unique [n(d), n(d)/2, d] code, up to equivalence. There arises a natural question, namely is this true for larger d? The next minimum weight is 9. It is not known if there is a linear [38, 19, 9] code. But there are inequivalent linear [40, 20, 9] codes, for example, D40,1 and D40,2 in Table 2.

4

Optimal Formally Self-Dual Codes

In this section, we show that at least one optimal linear [n, n/2, dmax (n)] code is self-dual or formally self-dual for lengths up to 48. We use double circulant codes as examples of formally self-dual codes. A pure double circulant code of length 2n has a generator matrix of the form ( I , R ), where I is the identity matrix of order n and R is an n × n circulant matrix. A code with generator matrix of the form   α β ··· β   γ    , .  I .. R
 γ

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Table 1. NT (n), NS (n), NE (n) and NO (n) for 2 ≤ n ≤ 24. Length n NT (n) dmax (n) NS (n) NE (n) NO (n) 2 1 2 1 0 0 4 3 2 1 1 1 1 3 0 0 1 6 8 1 4 1 0 0 10 4 4 0 1 1 12 43 4 1 2 5 14 ≥ 1360 4 1 9 112 1 5 0 0 1 16 18 1 6 0 1 0 20 ≥8 6 0 7 1 22 1 7 0 0 1 24 1 8 1 0 0

Table 2. New optimal double circulant codes.

Code D26 D40,1 D40,2 D46 D50

Parameters [26, 13, 7] [40, 20, 9] [40, 20, 9] [46, 23, 11] [50, 25, 10]

First Row |Aut(Dn)| 1111001000100 78 00000000101111010011 20 00000001010111100011 20 00000110110111110101111 23 0000000000010100111111011 25

where R
is an (n − 1) × (n − 1) circulant matrix, is called a bordered double circulant code of length 2n. These two families of codes are collectively called double circulant codes. Note that all double circulant codes are formally self-dual. In Table 2, we present several new optimal pure double circulant codes which are odd formally self-dual. The weight enumerators WDn of the codes Dn in Table 2 are given below. WD26 = 1 + 117y 7 + 273y 8 + 338y 9 + 598y 10 + 923y 11 + 1105y 12 + 1340y 13 WD40,1

+1300y 14 + 923y 15 + 598y 16 + 338y 17 + 182y 18 + 117y 19 + 39y 20 , = 1 + 320y 9 + 1012y 10 + 2140y 11 + 5400y 12 + 11660y 13 + 21300y 14 +37844y 15 + 60970y 16 + 84900y 17 + 107880y 18 + 126040y 19 +131376y 20 + 123960y 21 + 107400y 22 + 84840y 23 + 61160y 24 +38984y 25 + 1540y 26 + 11020y 27 + 5240y 28 + 2300y 29 + 964y 30 +260y 31 + 45y 32 + 20y 33 ,

WD40,2 = 1 + 280y 9 + 1010y 10 + 2420y 11 + 5425y 12 + 10880y 13 + 21200y 14 +38824y 15 + 61115y 16 + 84740y 17 + 107980y 18 + 124920y 19 +130690y 20 + 125360y 21 + 108520y 22 + 84320y 23 + 60210y 24

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+38704y 25 + 21970y 26 + 11380y 27 + 5165y 28 + 2160y 29 + 952y 30 WD46

+280y 31 + 50y 32 + 20y 33 , = 1 + 3312y 11 + 9660y 12 + 121440y 15 + 235290y 16 + 991760y 17 +1338876y 20 + 1961280y 23 + 1879560y 24 + 991760y 27 + 672980y 28 +121440y 31 + 56925y 32 + 3312y 35 + 1012y 36 ,

WD50 = 1 + 350y 10 + 1175y 11 + 3425y 12 + 11325y 13 + 28650y 14 + 65575y 15 +144725y 16 + 294525y 17 + 542200y 18 + 903875y 19 + 1400185y 20 +2009425y 21 + 2645350y 22 + 3226275y 23 + 3625875y 24 + 3750241y 25 +3619400y 26 + 3231525y 27 + 2645275y 28 + 2008375y 29 + 1407350y 30 +900725y 31 + 533350y 32 + 295575y 33 + 150000y 34 + 66625y 35 +27275y 36 + 10875y 37 + 3450y 38 + 1025y 39 + 305y 40 + 75y 41 + 50y 42 . In Table 3, we list the values for dmax (n). We also indicate in this table whether one of the optimal linear codes is self-dual or formally self-dual. In the third (resp. fourth) column “Y” means that one of the optimal linear codes is a self-dual code (resp. formally self-dual code which is not self-dual) and “N” means that there is no self-dual [n, n/2, dmax (n)] code (resp. formally self-dual [n, n/2, dmax (n)] code which is not self-dual). For information on self-dual codes, see [5, Table I]. We list the references only for formally self-dual codes. Thus we have the following: Proposition 4 Let dmax (n) be the highest minimum weight among all linear [n, n/2] codes. At least one optimal linear [n, n/2, dmax(n)] code is self-dual or formally self-dual for length n ≤ 36 and n = 42, 44, 46, 48. Remark 1. For n = 38, 40 and 50, the exact value of dmax (n) has not yet been determined. Let dkmax (n) be the highest minimum weight among all known linear [n, n/2] codes. Then at least one optimal linear [n, n/2, dkmax (n)] code is self-dual or formally self-dual for length n ≤ 50.

References 1. C. Bachoc, On harmonic weight enumerators of binary codes, Designs, Codes and Cryptogr., (to appear). 2. K. Betsumiya, T.A. Gulliver and M. Harada, On binary extremal formally self-dual even codes, Kyushu J. Math. (to appear). 3. K. Betsumiya and M. Harada, Binary optimal odd formally self-dual codes, (submitted). 4. A.E. Brouwer, Bounds on the size of linear codes, in Handbook of Coding Theory, V.S. Pless and W.C. Huffman (Eds.), Elsevier, Amsterdam (1998) pp. 295–461. 5. J.H. Conway and N.J.A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory Vol. 36 (1990) pp. 1319–1333.

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Table 3. The highest minimum weights dmax (n). Length n dmax (n) SD FSD Reference (FSD) 2 2 Y Y 4 2 Y Y [3], [11] 6 3 N Y [3] 8 4 Y N 10 4 N Y [3], [9], [11] 12 4 Y Y [1], [3], [9] 14 4 Y Y [2], [3], [7], [9] 5 N Y [3] 16 18 6 N Y [12] 20 6 N Y [2], [8], [9] 22 7 N Y [3] 24 8 Y N 7 N Y Section 4 26 28 8 N Y [9], [10], [11] 30 8 N Y [9], [11] 32 8 Y Y [9] 34 8 N Y [9] 8 Y Y [9] 36 38 8–9 ?, Y (d = 8) ?, Y (d = 8) [9] 40 9–10 N ?, Y (d = 9) Section 4 42 10 N Y [9] 44 10 N Y [9] 11 N Y Section 4, [6] 46 48 12 Y ? 50 10–12 ?, Y (d = 10) ?, Y (d = 10) [9]

6. S.T. Dougherty and M. Harada, Shadow optimal self-dual codes, Kyushu J. Math., (to appear). 7. J.E. Fields, P. Gaborit, W.C. Huffman and V. Pless, On the classification of extremal even formally self-dual codes, (submitted). 8. J.E. Fields, P. Gaborit, W.C. Huffman and V. Pless, On the classification of extremal even formally self-dual codes of lengths 20 and 22, (submitted). 9. T.A. Gulliver and M. Harada, Classification of extremal double circulant formally self-dual even codes, Designs, Codes and Cryptogr. Vol. 11 (1997) pp. 25–35. 10. D.B. Jaffe, Optimal binary linear codes of length ≤ 30, Discrete Math., (to appear). 11. G. Kennedy and V. Pless, On designs and formally self-dual codes, Designs, Codes and Cryptography Vol. 4 (1994) pp. 43–55. 12. J. Simonis, The [18, 9, 6] code is unique, Discrete Math. Vol. 106/107 (1992) pp. 439–448.

On Binary/Ternary Error-Correcting Codes with Minimum Distance 4 ¨ Patric R.J. Osterg˚ ard Department of Computer Science and Engineering Helsinki University of Technology P.O. Box 5400, 02015 HUT, Finland [email protected]

Abstract. We consider error-correcting codes over mixed alphabets with n2 binary and n3 ternary coordinates, and denote the maximum cardinality of such a code with minimum distance d by N (n2 , n3 , d). We here study this function for short codes (n2 +n3 ≤ 13) and d = 4. A computeraided method is used to settle 14 values, and bounds for 34 other entries are improved. In the method used, codes are built up from smaller codes using backtracking and isomorphism rejection. Codes of short length are further classified.

1

Introduction

We consider codes in the Hamming space with n2 binary coordinates and n3 ternary coordinates. A code C ⊆ F2n2 F3n3 with minimum distance greater than or equal to d is said to be an (n2 , n3 , d)|C| code. The maximum cardinality of a code in F2n2 F3n3 with minimum distance d is denoted by N (n2 , n3 , d), and corresponding codes are called optimal. This paper is one in a series of papers where a promising approach for classifying error-correcting codes has been considered. In the seminal paper [7], an old open problem in combinatorial coding theory is settled by showing that for binary codes, N (10, 0, 3) = N (11, 0, 4) = 72 and N (11, 0, 3) = N (12, 0, 4) = 144. In [6] the approach is carried over to binary/ternary mixed codes with d = 3, and many new bounds on N (n2 , n3 , 3), n2 + n3 ≤ 13 are obtained. We here continue the research and consider N (n2 , n3 , 4) for n2 + n3 ≤ 13. A code can occur in many equivalent forms, and in studying optimal codes we are only interested in one of these forms. Two codes are said to be equivalent if a permutation of the coordinates and permutations of the coordinate values, one for each coordinate, map the codewords of one code onto those of the other. For mixed alphabets, we are only allowed to permute coordinates over the same alphabet. A mapping of a code onto its own codewords is an automorphism of the code. The set of all automorphisms forms the full automorphism group of a code. A procedure for showing code equivalence is discussed in Section 2. In Section 3 the classification approach, which is based on backtrack search and rejection of Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 472–481, 1999. c Springer-Verlag Berlin Heidelberg 1999 

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equivalent codes, is considered. The results of the search are presented in Section 4, where an updated table of N (n2 , n3 , 4), n2 + n3 ≤ 13, is given. The new table contains 48 improved entries compared with the previous published table [1]. Classified, optimal codes are listed in the Appendix; these can also be obtained electronically.

2

Code Equivalence

Proving equivalence of codes is a nontrivial task. In this work, we transform this problem into the graph isomorphism problem, and we can then use the program nauty [4] written by Brendan McKay. The program nauty can be used to get, for example, the full automorphism group (the size and generators) of a graph. We will expressly need the feature that nauty can produce canonical labellings of graphs. The canonical labellings of two graphs coincide exactly when the graphs are isomorphic. We will thus have to transform codes into graphs so that two codes are equivalent exactly when the corresponding graphs are isomorphic. Then, by comparing the canonical labellings of the graphs, equivalence or inequivalence can be established. In the following way, such a transformation into vertexcolored graphs is carried out (the purpose of the vertex-coloring is that an automorphism is not allowed to map a vertex into another vertex with a different color). For every codeword of a given code, we create a vertex and color these with one color. For each of the coordinates, one vertex is created for every possible coordinate value (here, 2 or 3 vertices). All these are colored with one color that differs from that of the codeword vertices. Edges are finally inserted. There is an edge from a codeword vertex to a coordinate vertex exactly when that codeword has the corresponding value in that coordinate. Moreover, edges are inserted so that the subgraph induced by all coordinate vertices of any coordinate is a complete graph. To illustrate the mapping described above, we show in Fig. 1 the graphs that correspond to all three inequivalent (4, 1, 4)2 codes. The reader is encouraged to ascertain that the graphs are indeed nonisomorphic—and therefore that the codes are inequivalent—and that there are no other inequivalent codes.

3

Searching for Codes

The algorithm outlined here is analogous to that presented in [6,7]. The procedure for proving code equivalence plays a central role in our backtrack search for codes. One difficulty in a computer search for codes is the multitude of equivalent codes. There are in fact exactly 2n2 6n3 n2 !n3 !/|Aut(C)| codes equivalent to a given binary/ternary mixed code C. By fixing subcodes of a code that we are looking for, we can eliminate most of the branches in the search tree that lead to equivalent codes. This can be done in the following way.

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C1(0)

C1(0)

C1(1)

C1(1) C2(0)

C2(0) 00000

00000 C2(1)

C2(1)

C3(0)

C3(0)

C3(1)

C3(1)

C4(0)

C4(0) 11110

11111 C4(1)

C4(1)

C5(0)

C5(0)

C5(1)

C5(1)

C5(2)

C5(2)

C1(0) C1(1) C2(0) 00000 C2(1) C3(0) C3(1)

C4(0) 11101 C4(1) C5(0) C5(1) C5(2)

Fig. 1. Code equivalence graphs of the three optimal (4, 1, 4)2 codes.

On Binary/Ternary Error-Correcting Codes with Minimum Distance 4

475

Delete any one of the binary coordinates of an (n2 , n3 , d)M code with n2 ≥ 1, and let C0 and C1 be the shortened codes which had a 0 and a 1, respectively, in the deleted coordinate. Trivially, the minimum distance of both C0 and C1 is at least d, and one of the codes has at least M/2 codewords. Therefore, if we have all inequivalent (n2 − 1, n3 , d)M
codes with M  ≥ M/2, we can get all inequivalent (n2 , n3 , d)M codes by lengthening these. The approach used is thus to recursively build up longer codes, starting off from very small codes—with one, two, or three words—where all inequivalent codes can be constructed by hand. In each step, one binary coordinate is added to the code (this is not always the case; see later discussion), and all codes with prescribed parameters are constructed by backtrack search (and equivalent codes are removed using the procedure described earlier). This process is continued until the limit of the computers used is reached—as for computing time or memory requirement. The backtrack search is still to be described. To find all (n2 , n3 , d)M codes, we fix C0 to be an (n2 − 1, n3 , d)M
code with M  ≥ M/2, and this step is done for all such subcodes that have been obtained earlier in this recursive process. Having fixed C0 , we search for words in C1 using backtracking. Actually, we then have an instance of the maximum independent set problem, which is an NP-complete problem [2]. (This problem is computationally equivalent to several other important problems such as the maximum clique problem.) An independent set in a graph G = (V, E) is a subset of the vertices V , such that no two vertices in the set are adjacent. For our problem, we create a graph with a vertex for each word with a 1 in the new coordinate that is at distance at least d from C0 (that is, the word is a potential word in C1 ). Moreover, since the words in C1 must be at distance at least d from each other, edges are inserted into our graph whenever the corresponding words are at distance less than d from each other. We are now facing the problem of finding an independent set of size M − M  in this graph. In the search for an independent set (the words of C1 ), we developed the following algorithm. The algorithm works surprisingly well and is yet to be compared with other published algorithms for this graph problem. To apply the algorithm, we need to impose a total order on the vertices of the graph (and the vertices are indexed 1, 2, . . . , |V | starting from the smallest vertex). For codes, the lexicographic order appears to be a good choice. For all independent m (< M − M  = M  ) vertices, we want to know the maximum value possible of the minimum index in such a set. We denote this function—for a given graph—by g(m). Thus, g(1) = |V |, and for m ≥ 2 we obtain g(m) in a backtrack search by trying to find an independent set where the smallest index is g(m − 1) − 1, g(m − 1) − 2, . . .. The values of g(m ) for m < m are used to effectively prune the search. A value of g(m) is obtained by finding a corresponding independent set. In particular, when we obtain g(M  ) we have a solution to the original problem, and we proceed to find all solutions (and remove equivalent codes using the procedure described earlier).

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Another function h(m) denoting the minimum value possible of the maximum index in an independent set with m vertices (for m ≤ M  /2) turns out to be useful if there is no independent set with M  vertices. Namely, if we first calculate the first few values of h(m), then in calculating the values of g(m), we can stop the whole search if for some value of m, g(m) ≤ h(M  − m). For codes with only ternary coordinates, we add one ternary coordinate at a time. Analogously to the case of adding one binary coordinate at a time, (n2 , n3 − 1, d)M
codes with M/3 ≤ M  ≤ N (n2 , n3 − 1, d) are lengthened to get (n2 , n3 , d)M codes. Other possibilities for lengthening codes are discussed in [6].

4

The Results

In the binary case, codes with even minimum distance 2d can be obtained by extending codes with minimum distance 2d − 1; cf. [7]. With mixed binary/ternary codes with at least one ternary coordinate such an extension is not always possible, but we have to consider the odd-distance and even-distance cases separately. The following (obvious) inequality, however, can in many cases be applied to improve upper bounds: N (n2 , n3 + 1, d) ≤ N (n2 , n3 , d − 1) .

(1)

The parameters of the classified codes that were needed to get the new results in this paper are given in Table 1. The number of codes with given parameters is shown, with a 0 indicating that no such codes exist (such an entry is included only if it improves on an earlier upper bound). The codes were built up by adding one binary coordinate at a time as described in the previous section. Codes with only ternary coordinates, however, were built up by adding one ternary coordinate at a time. With a few exceptions, the classified optimal (n2 , n3 , 4)M codes are listed in the Appendix. All codes with parameters given in Table 1 can be obtained electronically from URL: http://www.tcs.hut.fi/~pat/234.html. The results were obtained using approximately 3 months of CPU time, and the search was sped up by distributing it over several computers using the program autoson [5]. We present exact values and best known bounds on N (n2 , n3 , 4), n2 +n3 ≤ 13, n3 ≥ 1 in Table 2. For the case n3 = 0, see [7]. A previous table for the same function was published in [1, Table II-C]. Most bounds obtained in this paper either come directly from a result in Table 1 or from such a result and one of the inequalities N (n2 + 1, n3 , 4) ≤ 2N (n2 , n3 , 4) ,

(2)

N (n2 , n3 + 1, 4) ≤ 3N (n2 , n3 , 4) .

(3)

By counting the marked bounds in Table 2, one can see that 48 entries have been improved since [1] appeared.

On Binary/Ternary Error-Correcting Codes with Minimum Distance 4

Table 1. Number of inequivalent (n2 , n3 , 4)M codes. n2 n3 M 3 3 4 5 5 6 6 6 6 7 7 7 7 7 7 7 8 9 2 2 3 3 4 4 4 5 6 1 2 3 3

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3

1 2 2 3 4 5 6 7 8 10 11 12 13 14 15 16 20 40 1 2 2 3 4 5 6 8 16 2 3 5 6

# 1 1 3 2 2 15 15 4 3 2103 732 309 101 41 15 7 2328 39720 1 1 3 1 10 1 1 50 702 1 2 12 4

n2 n3 M 4 4 4 5 5 5 6 0 0 1 2 2 3 3 3 4 4 4 4 5 5 5 6 0 0 0 1 1 1 1 1

3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5

9 10 11 18 19 20 35 2 3 4 7 8 13 14 15 25 26 27 28 49 50 51 97 4 5 6 8 9 10 11 12

# 1122 98 3 43751 916 21 0 1 1 1 9 2 582 50 1 12848 845 35 2 50 3 0 0 2 1 1 292 32 7 2 1

n2 n3 M 2 2 2 2 2 2 3 3 3 3 3 4 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 3 0

5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7

17 18 19 20 21 22 33 34 35 36 37 65 12 13 14 15 16 17 18 26 27 28 29 30 31 32 33 51 52 101 34

# 35124 4506 534 63 6 2 84372 1123 10 1 0 0 77 21 9 4 2 1 1 111087 21516 3890 675 122 23 6 2 1 0 0 0

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Table 2. Bounds on N (n2 , n3 , 4) for n2 + n3 ≤ 13, n3 ≥ 1. Key to Table: Unmarked [1]; a This paper (Table 1); b Eq. (2); c Eq. (3); d [6,7] and Eq. (1); ∗ Lower bound improved in this paper. n2 \n3 0 1 2 3 4 5 6 7 8 9 10 11 12 n2 \n3 0 1 2 3 4 5

1

2

3

1 1 1 1 1 2 2 3 2 6 4 8 8 16 16 26d 20 48–50d 40 96–100b 72d 192–200b 144b 384 256 8

4

1 3 2 4 3 8 6 15 11 28d ∗ 20 50a 34a 96a 64–68b 192b b 128–136 384b b 256–272 540–768b 400–544b

9 c

99 162–198b 324–396b 486–792b 891–1584b 1674–3168b

10 c

243–297 486–594b 729–1188b 1296–2376b 2484–4752b

c

5

6

7

9 12 22 ∗ 36a 58–64a 108–128b 208–256b 384–512b 768–1024b

18 33 51a 87–100a 144–200b 288–400b 576–800b 1152–1600b

33a 66b 108–132b 216–264b 360–528b 612–1056b 1152–2112b

11

12

13

729–891 1458–2561 4374–7029 8019–19682 972–1749 2916–4920 8019–14058 1944–3498 5589–9777 3726–6791

On Binary/Ternary Error-Correcting Codes with Minimum Distance 4

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In previous studies using the method discussed in this paper [6,7], only nonexistence bounds were improved. Therefore, it is a positive surprise that we are here able to find codes that improve on the existence bounds. Moreover, both of these cases are settled: N (5, 4, 4) = 50 and N (3, 5, 4) = 36. In particular, note that we have settled two entries for purely ternary codes. These codes lead to other improvements for ternary codes and therefore on the bounds in [1, Table I]. Ternary error-correcting codes were previously also considered in [8].

Appendix We here list the optimal codes with at least two words classified in this paper. Due to the large number of optimal codes for some parameters, the following codes are not listed but they can be obtained electronically from the author as mentioned in the main text: (8, 1, 4)20 , (9, 1, 4)40 , (5, 2, 4)8 , (6, 2, 4)16 , and (5, 3, 4)20 (there are, respectively, 2328, 39720, 50, 702, and 21 such codes). The codewords are compressed with the binary part in hexadecimal form and the ternary part in base 9. The binary and ternary parts are separately right-justified (so the length of the compressed words is n2 /4 + n3 /2). N (3, 1, 4) = 2: {00, 71}. N (4, 1, 4) = 2: {00, F2}, {00, F0}, {00, E2}. N (5, 1, 4) = 4: {000, 0F2, 161, 191}, {000, 0F0, 162, 192}. N (6, 1, 4) = 8: {000, 0F2, 161, 191, 251, 2A1, 330, 3C0}, {000, 0F2, 161, 191, 251, 2A1, 330, 3C2}, {000, 0F0, 162, 192, 252, 2A2, 330, 3C0}. N (7, 1, 4) = 16: {000, 0F2, 161, 191, 251, 2A1, 330, 3C0, 431, 4C1, 550, 5A0, 660, 690, 701, 7F1}, {000, 0F2, 161, 191, 251, 2A1, 330, 3C0, 431, 4C1, 550, 5A0, 660, 690, 702, 7F1}, {000, 0F2, 161, 191, 251, 2A1, 330, 3C0, 431, 4C1, 550, 5A0, 660, 692, 701, 7F1}, {000, 0F2, 161, 191, 251, 2A1, 330, 3C0, 431, 4C1, 550, 5A0, 662, 692, 701, 7F1}, {000, 0F2, 161, 191, 251, 2A1, 330, 3C0, 431, 4C1, 550, 5A2, 662, 692, 701, 7F1}, {000, 0F2, 161, 191, 251, 2A1, 330, 3C2, 431, 4C1, 550, 5A2, 660, 692, 701 , 7F1}, {000, 0F0, 162, 192, 252, 2A2, 330, 3C0, 432, 4C2, 550, 5A0, 660, 690, 702, 7F2}. N (2, 2, 4) = 2: {00, 34}. N (3, 2, 4) = 3: {00, 58, 64}. N (4, 2, 4) = 6: {00, 58, 64, 94, A8, F0}. N (1, 3, 4) = 2: {000, 114}. N (2, 3, 4) = 3: {000, 114, 328}, {000, 114, 326}. N (3, 3, 4) = 6: {000, 114, 328, 427, 605, 710}, {000, 114, 328, 427, 613, 701}, {000, 114, 326, 427, 613, 701}, {000, 114, 326, 425, 607, 712}. N (4, 3, 4) = 11: {000, 114, 326, 522, 604, 828, A13, B01, C11, D03, F18}, {000, 114, 326, 522, 604, 818, A21, B05, C23, D07, F10}, {000, 114, 326, 522, 605, 828, A13, B01, C11, D03, F18}. N (0, 4, 4) = 3: {00, 44, 88}. N (1, 4, 4) = 4: {000, 044, 156, 165}.

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¨ Patric R.J. Osterg˚ ard

N (2, 4, 4) = 8: {000, 044, 156, 165, 252, 267, 324, 370}, {000, 044, 156, 165, 238, 283, 322, 377}. N (3, 4, 4) = 15: {000, 044, 156, 165, 288, 307, 370, 427, 472, 513, 531, 605, 650, 748, 784}. N (4, 4, 4) = 28: {000, 044, 156, 165, 287, 313, 331, 476, 524, 542, 611, 655, 708, 780, 838, 883, 917, A04, A40, B22, B66, C15, C51, D33, D88, E26, E62, F74}, {000, 044, 156, 165, 283, 322, 377, 452, 467, 524, 570, 605, 646, 731, 825, 876, 981, A57, A62, B04, B40, C33, D02, D47, E20, E74, F55, F66}. N (5, 4, 4) = 50: {0000, 0044, 0156, 0165, 0288, 0307, 0342, 0481, 0518, 0605, 0646, 0720, 0774, 0852, 0867, 0911, 0A33, 0B25, 0B76, 0C13, 0D54, 0D60, 0E27, 0E72, 0F38, 1038, 1122, 1177, 1250, 1264, 1313, 1426, 1475, 1531, 1611, 1755, 1766, 1824, 1870, 1906 , 1945, 1A18, 1B57, 1B62, 1C02, 1C47, 1D88, 1E83, 1F04, 1F40}, {0000, 0044, 0156, 0165, 0288, 0307, 0342, 0481, 0518, 0605, 0646, 0720, 0774, 0852, 0867, 0911, 0A33, 0B25, 0B76, 0C13, 0D54, 0D60, 0E27, 0E72, 0F38, 1038, 1122, 1177, 1250, 1264, 1313, 142 6, 1475, 1533, 1611, 1757, 1762, 1824, 1870, 1906, 1945, 1A02, 1A47, 1B81, 1C31, 1D88, 1E55, 1E66, 1F04, 1F40}, {0000, 0044, 0156, 0165, 0288, 0307, 0342, 0481, 0518, 0605, 0646, 0720, 0774, 0852, 0867, 0911, 0A33, 0B25, 0B76, 0C13, 0D54, 0D60, 0E27, 0E 72, 0F38, 1038, 1122, 1177, 1250, 1264, 1313, 1426, 1475, 1533, 1611, 1757, 1762, 1824, 1870, 1906, 1945, 1A18, 1B81, 1C02, 1C47, 1D88, 1E55, 1E66, 1F04, 1F40}. N (0, 5, 4) = 6: {000, 044, 156, 165, 218, 281}. N (1, 5, 4) = 12: {0000, 0044, 0156, 0165, 0218, 0281, 1038, 1083, 1122, 1177, 1204, 1240}. N (2, 5, 4) = 22: {0000, 0044, 0183, 0227, 0272, 1056, 1065, 1122, 1177, 1231, 2025, 2076, 2111, 2138, 2250, 2264, 3007, 3042, 3154, 3160, 3213, 3288}, {0000, 0044, 0183, 0227, 0272, 1056, 1065, 1122, 1177, 1213, 1231, 2025, 2076, 2111, 2138, 2250, 2264, 3007, 3042, 3154, 3160, 3288}. N (3, 5, 4) = 36: {0000, 0044, 0187, 0218, 1085, 1113, 1251, 1266, 2072, 2125, 2131, 2256, 3027, 3033, 3148, 3180, 3202, 3274, 4052, 4076, 4111, 4165, 4223, 4237, 5004, 5128, 5130, 5272, 6008, 6084, 6143, 6260, 7041, 7167, 7216, 7255}. N (0, 6, 4) = 18: {000, 044, 156, 165, 218, 281, 338, 383, 422, 477, 504, 540, 627, 672, 713, 731, 855, 866}. N (1, 6, 4) = 33: {0000, 0044, 0183, 0227, 0272, 0356, 0365, 0422, 0477, 0531, 0618, 0704, 0740, 0855, 0866, 1025, 1076, 1111, 1138, 1250, 1264, 1307, 1342, 1454, 1460, 1513, 1588, 1633, 1681, 1726, 1775, 1802, 1847}, {0000, 0044, 0183, 0227, 0272, 0356, 0365, 0422, 0477, 0513, 0531, 0618, 0704, 0740, 0855, 0866, 1025, 1076, 1111, 1138, 1250, 1264, 1307, 1342, 1454, 1460, 1588, 1633, 1681, 1726, 1775, 1802, 1847}. N (2, 6, 4) = 51: {0000, 0044, 0156, 0165, 0218, 0281, 0338, 0383, 0422, 0477, 0504, 0540, 0627, 0672, 0713, 0731, 0855, 0866, 1124, 1142, 1237, 1273, 1315, 1351, 1526, 1562, 1646, 1664, 1708, 1780, 2107, 2170, 2223, 2232, 2316, 2361, 2557, 2575, 2605, 2650 , 2748, 2784, 3011, 3055, 3066, 3400, 3444, 3488, 3822, 3833, 3877}.

On Binary/Ternary Error-Correcting Codes with Minimum Distance 4

481

Acknowledgment The research was supported by the Academy of Finland.

References ¨ 1. Brouwer, A.E., H¨ am¨ al¨ ainen, H.O., Osterg˚ ard, P.R.J., Sloane, N.J.A.: Bounds on mixed binary/ternary codes. IEEE Trans. Inform. Theory 44 (1998) 140–161 2. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, New York (1979) 3. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. NorthHolland, Amsterdam (1977) 4. McKay, B.D.: nauty user’s guide (version 1.5). Tech. Rep. TR-CS-90-02, Computer Science Department, Australian National University (1990) 5. McKay, B.D.: autoson – a distributed batch system for UNIX workstation networks (version 1.3). Tech. Rep. TR-CS-96-03, Computer Science Department, Australian National University (1996) ¨ 6. Osterg˚ ard, P.R.J.: Classification of small binary/ternary one-error-correcting codes. (submitted for publication) ¨ 7. Osterg˚ ard, P.R.J., Baicheva, T., Kolev, E.: Optimal binary one-error-correcting codes of length 10 have 72 codewords. IEEE Trans. Inform. Theory 45 (1999) 1229–1231 8. Vaessens, R.J.M., Aarts, E.H.L., van Lint, J.H.: Genetic algorithms in coding theory—a table for A3 (n, d). Discrete Appl. Math. 45 (1993) 71–87

The Euclidean Algorithm and Primitive Polynomials over Finite Fields James W. Bond1 , Stefen Hui2
, and Hank Schmidt3 1

Science Applications International Corp, San Diego, CA 92110, USA bond [email protected] 2 San Diego State University, San Diego, CA 92182, USA [email protected] 3 Technology Service Corporation, Silver Spring, MD 20910, USA [email protected]

Abstract. In this paper, we study the quotients that arise when the Euclidean algorithm is applied to a primitive polynomial and xs − 1. We analyze the asymptotic behavior of the the number of terms of the quotients as n → ∞. This problem comes from the study of Low-Density Parity-Check codes. We obtain two characterizations of primitive polynomials over the field with two elements that are based on the number of nonzero terms in two polynomials obtained via division and the Euclidean algorithm with polynomials of the form xs − 1. The analogous results do not hold for general finite fields but do restrict the order of the polynomials to a small set of positive integers with specific forms.

1

Introduction

The problems in finite fields we discuss in this paper arose from our study of sparse matrix codes (see Bond et al. [2]). In Section 2, we state the main results, which include two characterizations of primitive polynomials over the field with two elements that provide new criteria for the determination of the primitiveness of a polynomial over the field with two elements. In Section 3, we give a brief description of how the properties of primitive polynomials relate to the construction of error-correcting codes. Section 4 contains the proofs of the main results and other related results. Examples are given in Section 5 to illustrate the theorems and their hypotheses.

2

The Main Results

Let IFq be the finite field with q elements and let f(x) be a polynomial in IFq [x]. For each positive integer s, let gs (x) be the greatest common divisor of f(x) and xs − 1. By the Euclidean algorithm, there exist as (x) and bs (x) such that f(x)as (x) + (xs − 1)bs (x) = gs (x). 

All correspondences should be addressed to Stefen Hui.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 482–491, 1999. c Springer-Verlag Berlin Heidelberg 1999


(1)

The Euclidean Algorithm and Primitive Polynomials over Finite Fields

483

We assume that deg(as (x)) < s and deg(bs (x)) < deg(f(x)) have minimum degrees and are therefore unique. The question that interests us is the following: What is the asymptotic distribution of the number of nonzero terms in as (x) as s → ∞? If f(x) is a factor of xs − 1, then gs (x) = f(x), as (x) = 1, and bs (x) = 0. If we ignore this trivial case, we have the following theorem for primitive polynomials. Recall that a polynomial f(x) of degree d in IFq [x] is primitive if n = q d −1 is the smallest integer for which the polynomial divides xn − 1. The smallest integer n for which f(x) divides xn − 1 is called the order of f(x). It is known that the order of a polynomial of degree d in IFq [x] is at most q d − 1. We refer the reader to Lidl and Niederreiter [4] for the basic results on finite fields. Theorem 1. If f(x) is a primitive polynomial in IFq [x] of degree d, then for s ∈ IN such that f(x) is not a factor of xs − 1, Number of nonzero terms of as (x) q d−1 (q − 1) → as s → ∞. s qd − 1

(2)

An illustration of this theorem is given in Example 1 of Section 5. Theorem 5 of Section 4 gives a more technical result for polynomials that are not primitive. If f(x) is not primitive, then there may be more than one asymptotic value even if f(x) is irreducible. On the other hand, uniqueness of the asymptotic value does not even imply that f(x) is irreducible in general. Example 2 and Example 3 of Section 5 illustrate these situations. However, the converse is true for the finite field with 2 elements and there is a partial converse for other q’s. Theorem 2. Let f(x) be a polynomial in IFq [x] of degree d. If for s ∈ IN such that f(x) is not a factor of xs − 1 and the asymptotic condition (2) holds, then f(x) is primitive or the order of f(x) is m(q d −1)/(q −1), for m ∈ {1, . . . , q −1}. In particular, f(x) is primitive when q = 2. A fact concerning the quotient (xn − 1)/f(x) that we need for our proofs is the following theorem. This theorem is known but we include a proof for the convenience of the reader. Theorem 3. Let f(x) be a primitive polynomial in IFq [x] with degree d and let n = 2d − 1. Then for any nonzero h(x) ∈ IFq [x] with degree less than d, the product h(x)(xn − 1)/f(x) has q d−1 (q − 1) nonzero terms. There is a partial converse to Theorem 3 similar to the converse of Theorem 1. This partial converse characterizes primitive polynomials in the set of irreducible polynomials in IF2 [x] and almost does so in IFq [x]. Recall that if f(x) is irreducible, then the order of f(x) divides q d − 1 and so f(x) is a factor of d xq −1 − 1. d

Theorem 4. Let f(x) ∈ IFq [x] of degree d be irreducible. If (xq −1 −1)/f(x) has q d−1 (q − 1) nonzero terms, then f(x) is primitive or has order m(q d − 1)/(q − 1), where m is a divisor of q − 1. In particular, f(x) is primitive when q = 2.

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The converse of Theorem 4 is not true in general. It is possible for the order of f to be a multiple of (q d − 1)/(q − 1) and yet the number of nonzero terms in (xn − 1)/f(x) not to be q d−1 (q − 1). See Example 4 and Example 5 of Section 5.

3

Motivation

In linear parity-check codes, the encoded sequence c is obtained from the information sequence b, represented by a length n binary column vector, by a linear transformation c = Gb, where G is an (m + n) × n matrix known as the generator matrix. A parity-check matrix for the code generated by G is any matrix whose rows form a basis of the orthogonal complement of the columns of G. Clearly, if C is an m × m invertible matrix, C −1H is also a parity matrix. A parity-check matrix has the property that HG = 0, and so a sequence r is an encoded sequence only if Hr = 0. In general, decoding is accomplished by solving He = Hr, so that there are as few nonzero terms in e as possible. Then r − e is in the range of G and r decodes to be the inverse image of r − e under G. Sparse matrix parity-check codes were introduced by Gallager [3] and have attracted much attention recently. See MacKay [6], Luby et al [5], and Bond et al [1]. One approach to the construction of the code is to find a sparse IF2 parity-check matrix so that the number of ones in the rows and columns are preassigned constants. For the decoding algorithm to perform well, the bipartite graph associated with the matrix should have no 2 - or 4 - cycles. See [1] and [6]. The most common method of implementing the sparse
matrix method is to construct an m × (m + n) parity-check matrix H = R C , where R is m × n and C is m × m with the required properties and then put it into systematic form
  = C −1 H = C −1 R I . H It is easy to see that a generator matrix is then   I G= . C −1R Of course, there is no guarantee that the matrix C is invertible. Indeed, it is quite likely for C to be non-invertible. In our implementation, we decided to use a circulant matrix C. Recall that a square matrix is circulant if each row of the matrix is the cyclic shift of the previous row. It is easy to see that the inverse of a circulant matrix is again circulant. The use of a circulant matrix allows us to guarantee invertibility and to avoid 2 - and 4 - cycles. Assuming that the matrix is invertible in IF2 , the performance of the code depends on the number of ones in the rows of the inverse. The ideal proportion of ones in each row is 1/2. To see that, let p be the proportion of ones in each row of C −1 and let R be constructed randomly so that each column contains h nonzero entries. If the information sequence b has w ≥ 1 nonzero entries, then

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by elementary counting arguments, the expected proportion of nonzero entries in C −1 Rb is 1 − (1 − 2p)hw . 2 For p = 1/2, the expected proportion of ones in the parity-check portion of the coded sequence is 1/2 and is independent of h and w. However, we see that it is not crucial that the proportion be exactly 1/2 since the expected number of nonzero entries is very flat near p = 1/2. This led us to analyze the number of ones in the inverse of circulant matrices. The ring of n×n circulant matrices in Mn [IF2 ] is isomorphic to IF2 [x]/(xn −1) via the identification 
a0 a1 · · · an−1 ↔ a0 + a1 x + · · · + an−1 xn−1 ,
 where a0 a1 · · · an−1 is the first row of a circulant matrix. The number of ones in a row of a circulant matrix is the same as the number of nonzero coefficients in the polynomial associated with the first row. The problem of inverting a n × n matrix C in IF2 is equivalent to finding a(x), b(x) in IF2 [x] such that f(x)a(x) + (xn − 1)b(x) = 1, where f(x) is the polynomial associated with C. The coefficients of a(x) form the first row of C −1 and the other rows of C −1 are obtained by cyclic shifts. From Theorem 1, we see that we can always construct C so that the proportion of ones in its inverse is as close to 1/2 as desired by allowing more terms in the information sequence. It is also relatively easy to arrange the entries of C so it does not have any 2 - or 4-cycles.

4

Proofs of the Main Theorems and Additional Results

The following lemma is elementary. We include a proof for the convenience of the reader. Lemma 1. Let f(x) be a polynomial of degree d in IFq [x]. Then f(x) is primitive if and only if   d 0, 1, x, . . . , xq −2 mod f(x) = IFq [x] mod f(x), the set of polynomials of IFq [x] with degree less than d. Proof. Recall that f(x) is primitive if and only if the smallest n for which f(x) divides xn − 1 is n = q d − 1. There are q d distinct polynomials in   d 0, 1, x, . . . , xq −2 mod f(x) precisely when f(x) does not divide xm − xn for 0 < n < m ≤ q d − 2, which is equivalent to the fact n = q d − 1 is the smallest integer for which f(x) divides

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xn − 1. Since there are exactly q d distinct polynomials in IFq [x] of degree less than d, the two sets are equally if and only if f(x) is primitive. The following lemma is used in the counting of nonzero terms in quotients. Lemma 2. Let f(x) be a monic polynomial in IFq [x] of degree d ≥ 1. For s ≥ 0, let xs = f(x)gs (x) + rs (x), with deg rs < deg f, and let cs be the coefficient of xd−1 in rs (x). Then for s = 1, 2, . . ., gs (x) =

s−1 

ck xs−1−k .

k=0

Proof. We will use induction to verify the claim. Before proceeding with the induction, we first make some preliminary observations. First note that for s = 0, . . . , d − 1, gs (x) = 0 and rs (x) = xs . Therefore cs = 0 for 0 ≤ s < d − 1 and cd−1 = 1. For s = d,

xd = f(x) − f(x) − xd is the canonical decomposition and thus gd (x) = 1 and rd (x) = f(x) − xd . It follows that for s = 0, . . . , d − 1, gs+1 (x) = xgs (x) + cs . For s ≥ d, we have xs+1 = f(x)gs+1 (x) + rs+1 (x) and also xs+1 = xf(x)gs (x) + xrs (x) = xf(x)gs (x) + cs xd + x(rs (x) − cs xd−1 )
 = f(x) [xgs (x) + cs ] + cs rd (x) + x(rs (x) − csxd−1 ) . By uniqueness, we have gs+1(x) = xgs (x) + cs , for s = d, . . . also and therefore the recurrence relationship holds for s = 0, . . .. We now verify the formula for gs (x) by induction. For s = 1, we have from above that c0 = 1 if deg(f(x)) = 1 and c0 = 0 if deg(f(x)) > 1. If deg(f(x)) = 1, g1 (x) = 1 and c0 = 1 and so the formula holds. If deg(f(x)) > 1, g1 (x) = 0 and c0 = 0 and the formula also holds. So the formula holds for s = 1. Assume that the formula is true for s. Then gs+1(x) = xgs (x) + cs =

s−1  k=0

ck xs−k + cs =

s 

ck xs−k .

k=0

Thus the formula holds for s + 1 and by induction for all s ≥ 1. One interesting consequence of Lemma 2 is that the quotients of f(x) into xs are implicit in the standard residue class table for f(x). We will next prove Theorem 3 which is used in the proofs of the other theorems. Proof of Theorem 3. Without loss of generality, we can assume that f(x) is monic. Let n = q d − 1 and let g(x) = (xn − 1)/f(x). We first show that g(x) has q d−1 nonzero terms. Since xn = f(x)g(x) + 1, we have from Lemma 2 n−1 that g(x) = k=0 ck xn−1−k . Since f(x) is primitive, we can apply Lemma 1 to

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conclude that the number of nonzero terms of g(x) is the same as the number of residue classes of IFq [x] mod f(x) such that the coefficient of xd−1 is nonzero, which is easily seen to be q d−1 (q − 1). For the general case, let h(x) ∈ IFq [x] be nonzero with deg(h(x)) < d. Let c = deg(h(x)). By Lemma 1, there exists s, 0 ≤ s < q d−1 such that h(x) = xs − f(x)u(x). Since deg(h(x)) < deg(f(x)), deg(u(x)) = s − d < s < n. We have h(x)g(x) = xs g(x) − f(x)g(x)u(x) = xs g(x) − (xn − 1)u(x). Decompose g(x) = g1 (x) + g2 (x) with g1 (x) containing the terms with exponent less than n − s and g2 (x) containing the terms with exponent at least n − s. Then xs g1 (x) has degree less than n and all exponents of xs g2 (x) at least n. Since deg(h(x)g(x)) < deg(f(x)g(x)) = n, we have xs g2 (x) − xn u(x) = 0. It follows that u(x) = xs−n g2 (x) and thus h(x)g(x) = xs g1 (x) + xs−n g2 (x). The exponents of xs g1 (x) are at least s and deg(xs−n g2 (x)) = deg(u(x)) < s and so the two terms have no common exponents. We conclude that h(x)g(x) has the same number of nonzero terms as g(x) = g1 (x) + g2 (x) and the theorem is proved. From the proof of Theorem 3, we have the following corollary. Corollary 1. Let f(x) ∈ IFq [x] have order n. Then for all h(x) in the residue classes IFq [x] mod f(x), the functions h(x)(xn − 1)/f(x) have the same number of nonzero terms as (xn − 1)/f(x). We prove a more general version of Theorem 1 and use it to derive Theorem 1. Theorem 5. Let f(x) ∈ IFq [x] have order n and for 0 ≤ s < n, let Ns be the number of nonzero terms in bs (x)(xn − 1)/f(x). (See equation (1).) Then Number of nonzero terms in atn+s(x) Ns = . t→∞ tn + s n lim

Proof. Let m ≥ 1 and let n be the order of f. Write m = tn+s, where 0 ≤ s < n. Then xm − 1 = xs (xtn − 1) + xs − 1 = xs (xn − 1)

t−1 

xkn + xs − 1.

k=0 n

s

Since f(x) divides x − 1, it is easy to see that x − 1 and xm − 1 have the same greatest common divisor with f(x), say gs (x). Let f(x)as (x) + (xs − 1)bs (x) = gs (x) with deg(as (x)) < s and deg(bs (x)) < d. Let g(x) = (xn − 1)/f(x). Then

t−1  xkn + (xm − 1)bs (x) f(x) as (x) − bs (x)xs g(x) k=0

m

s

n

= f(x)as (x) + x − 1 − x (x − 1) = f(x)as (x) + (xs − 1)bs (x) = gs (x).

t−1  k=0

kn

x

bs (x)

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Observe that the degree of as (x) − bs (x)xs g(x)

t−1

kn k=0 x

is

(t − 1)n + s + deg(bs (x)) + deg(g(x)) = (t − 1)n + s + deg(bs (x)) + n − d, which is less than tn + s = m since deg(bs (x)) < d. We can conclude that s

am (x) = atn+s (x) = as (x) − bs (x)x g(x)

t−1 

xkn .

k=0

Since deg(bs (x)gs (x)) < n, we see from the above formula that if we increase t to t + 1, the number of nonzero terms in a(t+1)n+s(x) is increased by the number of nonzero terms in bs (x)g(x), say Ns . Note that the factor xs does not affect the number of nonzero terms. Thus for an increase of n in the index, the number of nonzero terms in am is increased by Ns . Therefore lim

t→∞

Number of nonzero terms in atn+s (x) Ns = . tn + s n

The proof of Theorem 5 is complete. Theorem 1 and Theorem 2 are now simple consequences of Theorem 5. Proof of Theorem 1. Let f(x) have degree d and let n = q d − 1 be the order of f(x). By Theorem 3, the number of nonzero terms in bs (x)(xn − 1)/f(x) is q d−1 (q − 1) if bs (x) is not the zero polynomial. The polynomial bs (x) is zero if and only if f(x) is a factor of xs − 1. It follows from Theorem 5 that for s such that f(x) is not a factor of xs − 1, Number of nonzero terms of as (x) q d−1 (q − 1) → as s → ∞. s qd − 1 Theorem 1 is proved. Proof of Theorem 2. Let n be the order of f(x). Then 1 ≤ n ≤ q d −1. Suppose that for s such that f(x) is not a factor of xs − 1, Number of nonzero terms of as (x) q d−1 (q − 1) → as s → ∞. s qd − 1 By Theorem 5, the limit must also have the form N/n for some N ≥ 1. Therefore q d−1 (q − 1)n = (q d − 1)N. Since q d−1 and q d − 1 are relatively prime, q d − 1 divides (q − 1)n. If q d − 1 divides n, then n = q d − 1 and f(x) is primitive. Otherwise, n < q d − 1 and we must have (q − 1)n = m(q d − 1) or n = m(q d − 1)/(q − 1), for m ∈ {1, . . . , q − 1}. When q = 2, then n = q d − 1 in both cases and so f(x) is always primitive when q = 2.

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Proof of Theorem 4. Let f(x) ∈ IFq [x] be irreducible and have degree d and d order n. Since f(x) is irreducible, d|q d − 1. Let q d − 1 = te. Then xq −1 − 1 =  t−1 (xe − 1) k=0 xke . Let N be the number of nonzero terms in (xe − 1)/f(x). Since the degree of (xe − 1)/f(x) is less than n, the number of nonzero terms in (xn − 1)/f(x) is tN . Hence q d − 1 = tn and q d−1 (q − 1) = tN. We conclude as in the proof of Theorem 2 that n = q d −1 or n = m(q d −1)/(q−1). Since n divides q d − 1 in this case, we must have m a factor of q − 1. Thus f(x) is either primitive or has order m(q d − 1)/(q − 1) with m a factor of q − 1 and the proof is complete.

5

Examples

In this section, we give examples to illustrate the theorems we presented. Example 1. This example illustrates Theorem 1. Let f(x) = 1 + x + x4 . Then f(x) is primitive in IF2 [x]. Therefore by Theorem 1, the only nonzero asymptotic value is 23 /(24 − 1) = 8/15. In fact, all primitive polynomials in IF2 [x] of degree 4 have the same asymptotic value. The number of nonzero terms in as (x) for s from 1 to 200 is shown in Figure 1. The horizontal line of height 8/15 is also shown.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

20

40

60

80

100

120

140

160

180

200

Fig. 1. Distribution of Asymptotic Values for 1 + x + x4 .

We illustrate Theorem 5 with the following examples. Example 2. Let f(x) = 1 + x + x2 + x3 + x4 . Then f(x) is irreducible but not primitive IF2 [x]. Clearly, f(x) has order 5. For s = 0, 1, 2, 3, 4, 5, we computed as (x) and bs (x) such that f(x)as (x) + (xs − 1)bs (x) = gcd(f(x), xs − 1).

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Let g(x) = (x5 − 1)/f(x) = 1 + x. The polynomials as (x), bs (x), and g(x)bs (x) are given in the following table. Table: The Polynomials related to 1 + x + x2 + x3 + x4 . s as (x) bs (x) g(x)bs (x) 0 1 0 0 1 1 x3 + x x4 + x2 + x3 + x 2 1 x2 + x x3 + x 2 3 x x +x+1 x3 + 1 2 3 2 4 x + 1 1 + x + x x + x4 + 1 + x3 5 1 0 0

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

20

40

60

80

100

120

140

160

180

200

Fig. 2. Distribution of Asymptotic Values for 1 + x + x2 + x3 + x4 . When f(x) is not a factor of xs −1, g(x)bs (x) has either 2 or 4 nonzero terms. Therefore, the asymptotic values are 2/5 and 4/5 by Theorem 5. When f(x) is a factor of xs − 1, the asymptotic value is 0. The number of nonzero terms of as (x) for s from 1 to 200 are given in Figure 2. The horizontal lines with height 2/5 and 4/5 are also shown. Example 3. Let f(x) = 1 + x + x5 . In IF2 [x],



1 + x + x5 = x2 + x + 1 x3 + x2 + 1 . One can easily check that f(x) has order 21 and that g(x) = (x21 − 1)/f(x) factors into





g(x) = x6 + x4 + x2 + x + 1 (x + 1) x6 + x5 + x4 + x2 + 1 1 + x + x3 By direct computation, one can show that g(x)bs (x) has 10 nonzero terms when f(x) is not a factor of xs − 1. It follows that the only nonzero asymptotic value is 10/21. The number of nonzero terms of as (x) for s from 1 to 200 are given in Figure 3. The horizontal line with height 10/21 is also shown.

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1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

20

40

60

80

100

120

140

160

180

200

Fig. 3. Distribution of Asymptotic Values for 1 + x + x5 . The next example shows that for q = 2, it is possible for f(x) to have order m(q d − 1)/(q − 1), where m is a divisor of q − 1, and (xn − 1)/f(x) to have q d−1 (q − 1) nonzero terms as stated in Theorem 4. This example also shows that for q = 2, it is possible to have nonzero asymptotic value q d (q − 1)/(q d − 1) for a polynomial that is not primitive. Example 4. Let f1 (x) = 1+2x+x3 and f2 (x) = 4+2x+x3 . Then f1 (x) and f2 (x) 3 are irreducible in IF5 [x]. One can check that (x5 −1 −1)/fi (x) has 100 = 52 (5−1) nonzero terms for i = 1, 2. The order of f1 (x) is 31 = (53 − 1)/(5 − 1) and that of f2 (x) is 62 = 2(53 − 1)/(5 − 1). The nonzero asymptotic values of both f1 (x) and f2 (x) are 52 (5 − 1)/(53 − 1) = 100/124. The next example shows that the converse of Theorem 4 is not true. Example 5. Let f(x) = 1 + x + x2 + x4 . Then f(x) is irreducible but not primitive in IF3 [x]. The order of f(x) is 40 = (34 − 1)/(3 − 1) but there are 48 nonzero terms in (x80 − 1)/f(x). Note that q d−1 (q − 1) = 33 2 = 54.

References 1. Bond, J. W., Hui, S. and Schmidt, H. (1997) Low density parity check codes based on sparse matrices with no small cycles, Proceedings of the IMA. 2. Bond, J. W., Hui, S. and Schmidt, H. (1998) Sparse matrix parity-check codes and circulant matrices, In preparation. 3. Gallager, R. G. (1963) Low Density Parity-Check Codes, MIT Press. 4. Lidl, R. and Niederreiter, H. (1994) Introduction to Finite Fields and their Applications, Cambridge University Press. 5. Luby, M., Mitzenmacher, M., Shokrollahi, M. A., Spielman, D. (1998) Analysis of Low Density Codes and Improved Designs using Irregular Graphs, Preprint. 6. MacKay, D. J. C. (1999) Good error-correcting codes based on very sparse matrices, IEEE Trans. on Information Theory, vol 45, p. 399.

On the Computational Hardness of Testing Square-Freeness of Sparse Polynomials Marek Karpinski1 and Igor Shparlinski2 1

Dept. of Computer Science, University of Bonn R¨ omerstrasse 164, 53117 Bonn, Germany [email protected] 2 Dept. of Computing, Macquarie University Sydney, NSW 2109, Australia [email protected]

Abstract. We show that deciding square-freeness of a sparse univariate polynomial over ZZ and over the algebraic closure of a finite field IFp of p elements is NP-hard. We also discuss some related open problems about sparse polynomials.

1

Introduction

In this paper we extend the class of problems on sparse polynomials which are known to be NP-hard. We recall that a polynomial f ∈ R[X] over a ring R is called t-sparse if it is of the form f(X) =

t


ai X ni ,

(1)

i=1

with some a1 , . . . , at ∈ R and some integers 0 ≤ n1 < . . . < nt . For a sparse polynomial f ∈ ZZ[X], given by (1), the input size S(f) of f is defined as t
S(f) = log (|ai |ni + 2) i=1

where log z denotes the binary logarithm. Let p be a prime number. Denote by Ωp the algebraic closure of the finite field IFp of p elements. Similarly, for a sparse polynomial f ∈ Ωp [X] given by (1), the input size S(f) of f is defined as t
log (qni + 2) . S(f) = i=1

where IFq ⊂ Ωp is the smallest subfield of Ωp containing all coefficients of f. Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 492–497, 1999. c Springer-Verlag Berlin Heidelberg 1999


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We recall that a polynomial f ∈ R[X] over the unique factorization domain R is called square-free if it is not divisible by a square of a non-constant polynomial. We also refer to [7] for formal description of NP-hard and other related complexity classes. Since the pioneering papers [17,18,19] complexity problems on sparse polynomials have been studied quite extensively [6,8,9,10,11,12,13,15,16]. Nevertheless many natural questions about such polynomials remain open. Here we prove that testing square-freeness of sparse polynomials over ZZ and over Ωp is NP-hard. Besides just being a natural problem, this question has also been motivated by several other possible links and applications. First of all we mention the problem of deciding whether a given sparse polynomial over ZZ has a real root. The existence of multiple roots is a major obstacle in obtaining efficient algorithms for this problem, see [3]. Another well-known related problem is sparse polynomial divisibility. That is, given two sparse polynomials f, g ∈ ZZ[X], decide whether g|f. It has recently been proved [11] that under the Extended Riemann Hypothesis this problem belongs to the class co-NP, that is, there exists a short proof of the property f /| g. We also discuss such possible applications and mention several new related problems.

2

Main Results

We consider the following two problems: Sparse Square-Free: Given a t-sparse polynomial f ∈ R[X], decide whether f is square-free and Sparse GCD: Given two t-sparse polynomials f, g ∈ R[X], decide whether deg gcd(f, g) > 0. First of all we consider the case R = ZZ. Theorem 1. Over ZZ, Sparse Square-Free and Sparse GCD are equivalent under randomized polynomial time reduction. Proof. It is easy to see that Sparse Square-Free is deterministic polynomial time reducible to Sparse GCD. Indeed, f is square-free if and only if f and f  are relatively prime. It remains to show that Sparse GCD can be reduced to Sparse SquareFree. Denote by M(s, t) the set of all t-sparse polynomials over ZZ of size at most s. Obviously |M(s, t)| ≤ 22ts . We show that for all, but at most 210st , pairs a, b ∈ ZZ the polynomials f + ag and f + bg are square-free for all relatively prime pairs f, g ∈ M(s, t).

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Let us fix a pair f, g ∈ M(s, t) of relatively prime polynomials. The discriminant Df,g (Y ) of the polynomial f(X) + Y g(X) is a polynomial in Y of degree at most max{deg f, deg g} ≤ 2s . We remark that, because f and g are relatively prime, the bivariate polynomial f(X) + Y g(X) ∈ ZZ[X, Y ] is irreducible over Q. Therefore the polynomials f(X) + Y g(X) and f  (X) + Y g  (X) are relatively prime. Recalling the resultant representation Df,g (Y ) = ResX (f(X) + Y g(X), f  (X) + Y g  (X)) we conclude that Df,g (Y ) is not identical to zero and thus has at most 2s zeros. Considering all possible pairs f, g ∈ M(s, t) we see that there are at most   22ts−1 22ts − 1 2s < 25ts values of y which are roots of the discriminant Df,g (Y ) for at least one relatively prime pair f, g ∈ M(s, t). Thus the number of pairs a, b ∈ ZZ such that they are not roots of all discriminants DX (Y ) corresponding to all relatively prime pairs f, g ∈ M(s, t) does not exceed 210st . Now to test whether f, g ∈ M(s, t) are relatively prime we select a random pair a, b of integers a and b with 0 ≤ a < b ≤ 26ts and test if F = (f + ag)(f + bg) is square-free. Indeed, if f and g are not relatively prime then, obviously, F is not squarefree. If f and g are relatively prime then it is easy to verify that f + ag and f + bg are relatively prime as well. Because of the choice of a and b we conclude that f + ag and f + bg are square-free with probability at least 1 + O(2−2ts ) and thus F is square-free. It is also easy to check that the size of F is polynomially bounded in terms of S(f) and S(g).

It has been shown in [19] that over ZZ Sparse GCD is NP-hard, see also [17], [18]. Therefore, from Theorem 1 we obtain the following statement. Corollary 1. Over ZZ, Sparse Square-Free is NP-hard. Now we turn to the case R = Ωp . Theorem 2. Over Ωp , Sparse Square-Free and Sparse GCD are equivalent under randomized polynomial time reduction. Proof. As before, only the reduction of Sparse GCD to Sparse Square-Free is non-trivial.

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495

Denote by Mq (s, t) the set of all t-sparse polynomials over IFq of size at most s. Obviously |Mq (s, t)| ≤ q t 2ts . Using the algorithm of [21] (or one of previously known less efficient algorithms) in probabilistic polynomial time we construct an extension of IFq of degree N = 6st. As in the proof of Theorem 1, we see that for all, but at most q 2t 23st , pairs a, b ∈ IFqN , the polynomials f + ag and f + bg are square-free for all relatively prime pairs f, g ∈ Mq (s, t). Now to test whether f, g ∈ Mq (s, t) are relatively prime we select a random pair a, b ∈ IFqN and test if F = (f + ag)(f + bg)

(2)

is square-free. Indeed, if f and g are not relatively prime then, obviously, F is not squarefree. If f and g are relatively prime then it is easy to verify that f + ag and f + bg are relatively prime as well. Because of the choice of a and b we conclude that f + ag and f + bg are square-free with probability at least q N − q 2t 23st = 1 + O(2−s ) qN and thus F is square-free. It is also easy to check that the size of F is polynomially bounded in terms of S(f) and S(g).

It follows from the chain of reductions of [10], which has been used to show #P-hardness of the counting of rational points on a sparse plane curve over a finite field, that over Ωp the problem Sparse GCD is NP-hard. Therefore, from Theorem 2 we obtain the following statement. Corollary 2. Over Ωp , Sparse Square-Free is NP-hard.

3

Remarks

There are several more possible extensions of our results. First of all the reduction we describe in Theorems 1 and 2 can be applied to polynomials given by straightline programs and to multivariate sparse polynomials. Our reduction in Theorem 2 uses an extension of the ground field IFq . It would be interesting to find a reduction over the same field. For polynomial given by straight-line programs this can be done via considering the norm of the polynomial (2) Ψ(X) = Norm IFqN

:IF q F (X) =

N   i=1

N    i i f(X) + aq g(X) f(X) + bq g(X) . i=1

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Marek Karpinski and Igor Shparlinski

We see that if f and g are given by straight-line programs of polynomial size then Ψ also has a straight-line program of polynomial size. On the other hand, unfortunately Ψ contains a superpolynomial number of monomials. Indeed, it is easy to show that Ψ(X) is T -sparse with T ≤ scp logp t , where p is the characteristic of IFq and c > 0 is an absolute constant. If p and t are both fixed then both the sparsity T and the S(Ψ) are polynomial in S(f) and S(g). However, for sparse polynomial with fixed number of monomials we do not have the corresponding NP-hardness result for computing gcd(f, g). In both works [10] and [19] the sparsity grows together with the input size, and thus the final link is missing. Another interesting related question which probably can be studied by the method of this paper is deciding irreducibility of sparse polynomials. Unfortunately for irreducibility there is no analogue of the discriminant characterization of square-freeness. Nevertheless, it is possible that effective versions [1,2,4,5,14], [20,22] of the Hilbert Irreducibility Theorem (or their improvements) can help to approach this problem. Unfortunately we do not know any nontrivial upper bounds for the aforementioned problems. For example, it will be interesting to show that testing squarefreeness of sparse univariate polynomials over ZZ can be done in PSPACE. Finally, it is very interesting to study similar questions for sparse integers, that is, for integers of the form f(2), where f is a sparse polynomial. Several results have been obtained in [18,19] but many more natural questions remain open. Acknowledgments The authors would like to thank Joachim von zur Gathen for a number of fruitful discussions, in particular for the suggestion to use resultants in the proofs of Theorems 1 and 2.

References 1. S. D. Cohen, ‘The distribution of Galois groups and Hilbert’s irreducibility theorem’, Proc. London Math. Soc., 43 (1981), 227–250. 2. P. Corvaja and U. Zannier, ‘Diophantine equations with power sums and universal Hilbert sets’, Indag. Math., 9 (1998), 317–332. 3. F. Cucker, P. Koiran and S. Smale, ‘A polynomial time algorithm for Diophantine equations in one variable’, J. Symb. Comp., 27 (1999), 21–29. 4. P. D´ebes, ‘Hilbert subsets and S-integral points’, Manuscr. Math., 89 (1996), 107–137. 5. P. D´ebes, ‘Density results on Hilbert subsets’, Preprint, 1996, 1–25. 6. A. Diaz and E. Kaltofen, ‘On computing greatest common divisors with polynomials given by black boxes for their evaluations’, Proc. Intern. Symp. on Symb. and Algebraic Comp., 1995, 232–239.

Computational Hardness of Testing Square-Freeness of Sparse Polynomials

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7. M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. Feeman, NY, 1979. 8. J. von zur Gathen, ‘Irreducibility of multivariate polynomials’, J. Comp. and Syst. Sci., 31 (1985), 225–264. 9. J. von zur Gathen and E. Kaltofen, ‘Factoring sparse multivariate polynomials’, J. Comp. and Syst. Sci., 31 (1985), 265–287. 10. J. von zur Gathen, M. Karpinski and I. E. Shparlinski, ‘Counting points on curves over finite fields’, Comp. Compl., 6 (1997), 64–99. 11. D. Grigoriev, M. Karpinski and A. M. Odlyzko, ‘Short proof of nondivisibility of sparse polynomials under the Extended Riemann Hypothesis’, Fundamenta Informaticae, 28 (1996), 297–301. 12. D. Grigoriev, M. Karpinski and M. Singer, ‘Fast parallel algorithm for sparse multivariate polynomials over finite fields’, SIAM J. Comput., 19 (1990), 1059– 1063. 13. D. Grigoriev, M. Karpinski and M. Singer, ‘Computational complexity of sparse rational interpolation’, SIAM J. Comput., 23 (1994), 1–11. 14. M.-D.A. Huang and Y.-C. Wong, ‘Extended Hilbert irreducibility and its applications’, Proc. 9-th Annual ACM-SIAM Symp. on Discr. Algorithms, ACM, NY, 1998, 50–58. 15. E. Kaltofen and B. M. Trager, ‘Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separations of nominators and denominators’, J. Symb. Comp., 9 (1990), 301–320. 16. M. Karpinski and I. E. Shparlinski, ‘On some approximation problems concerning sparse polynomials over finite fields’, Theor. Comp.. Sci., 157 (1996), 259–266. 17. D. A. Plaisted, ‘Sparse complex polynomials and polynomial reducibility’, J. Comp. Sys. Sci., 14 (1977), 210–221. 18. D. A. Plaisted, ‘Some polynomial and integer divisibility problems are NP-hard’, SIAM J. Comput., 7 (1978), 458–464. 19. D. A. Plaisted, ‘New NP-hard and NP-complete polynomial and integer divisibility problems’, Theor. Comp.. Sci., 31 (1984), 125–138. 20. A. Schinzel and U. Zannier, ‘The least admissible value of the parameter in Hilbert’s Irreducibility Theorem’, Acta Arithm., 69 (1995), 293–302. 21. V. Shoup, ‘Fast construction of irreducible polynomials over finite fields’, J. Symb. Comp., 17 (1994), 371–391. 22. U. Zannier, ‘Note on dense universal Hilbert sets’, C.R. Acad. Sci. Paris, Ser.I , 322 (1996), 703–706.

Mastrovito Multiplier for General Irreducible Polynomials




A. Halbutoˇ gulları1 and C ¸ .K. Ko¸c2 1

2

i2 Technologies, 565 Technology Square, 9th Floor, Cambridge, MA 02139 Electrical & Computer Engineering, Oregon State University, Corvallis, OR 97331

Abstract. We present a new formulation of the Mastrovito multiplication matrix and an architecture for the multiplication operation in the field GF (2m ) generated by an arbitrary irreducible polynomial. We study in detail several specific types of irreducible polynomials, e.g., trinomials, all-one-polynomials, and equally-spaced-polynomials, and obtain the time and space complexity of these designs. Particular examples, illustrating the properties of the proposed architecture, are also given. The complexity results established in this paper match the best complexity results known to date. The most important new result is the space complexity of the Mastrovito multiplier for an equally-spaced-polynomial, which is found as (m2 − ∆) XOR gates and m2 AND gates, where ∆ is the spacing factor.

1

Introduction

Efficient hardware implementations of the arithmetic operations in the Galois field GF (2m ) are frequently desired in coding theory, computer algebra, and public-key cryptography [10,9,6]. The measure of efficiency is the number of gates (XOR and AND) and also the total gate delay of the circuit. The representation of the field elements have crucial role in the efficiency of the architectures for the arithmetic operations. For example, the well-known Massey-Omura [11] algorithm uses the normal basis representation, where the squaring of a field element is equivalent to a cyclic shift in its binary representation. Efficient bit-parallel algorithms for the multiplication operation in the canonical basis representation, which have much less space and time complexity than the Massey-Omura multiplier, have also been proposed. The standard (polynomial) basis multiplication requires a polynomial modular multiplication followed by a modular reduction. In practice, these two steps can be combined. A novel method of multiplication is proposed by Mastrovito [7,8], where a matrix product representation of the multiplication operation is used. The Mastrovito multiplier using the special generating trinomial xm +x+1 is shown to require (m2 −1) XOR gates and m2 AND gates [7,8,12,13]. It has been conjectured [7] that the space complexity of the Mastrovito multiplier would also be the same for all trinomials of the form xm + xn + 1 for n = 1, 2, . . . , m − 1. 

This research is supported in part by Secured Information Technology, Inc.

Marc Fossorier et al. (Eds.): AAECC-13, LNCS 1719, pp. 498–507, 1999. c Springer-Verlag Berlin Heidelberg 1999


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This conjecture was shown to be false by Sunar and Ko¸c [14] for the case of m = 2n. The architecture proposed in [14] requires (m2 − 1) XOR gates and m2 AND gates, when m = 2n. However, the required number of XOR gates is m m 2 + 1 where m is even. reduced to (m2 − m 2 ) for the trinomial x + x In this study, we generalize the approach of [14] in several different ways. We describe a method of construction for the Mastrovito multiplier for a general irreducible polynomial. We give detailed space and time analysis of the proposed method for several types of irreducible polynomials. In each case, the method proposed in this paper gives complexity results matching the best known results up to date. The detailed analyzes are given in the full version of this paper [2]. In this paper, we give the general approach and summarize the findings. The most important result of the this study is in the case equally-spacedpolynomial (ESP), i.e., a polynomial of the form p(x) = xk∆ + x(k−1)∆ + · · · + x∆ + 1 ,

(1)

where k∆ = m. The proposed Mastrovito multiplier for an ESP requires (m2 −∆) XOR gates and m2 AND gates. For k = 2, the ESP reduces to the equally-spacedm trinomial (EST) xm +x 2 +1, and for ∆ = 1, it reduces to the all-one-polynomial 2 (AOP). Our method requires (m2 − m 2 ) XOR and m AND gates for the trinomial m m of the form x + x 2 + 1 for an even m, matching the result in [14]. Furthermore, our proposed architecture requires (m2 − 1) XOR gates and m2 AND gates when the irreducible polynomial is an AOP. This result matches the best known space complexity result to date for the canonical basis multiplication based on an irreducible AOP, as given in [5]. Their architecture requires (m2 − 1) XOR gates and m2 AND gates, and has the lowest space complexity among similar bit-parallel multipliers [7,4,3]. We introduce the fundamentals of the Mastrovito multiplier and the notation of this paper in §2. The architecture of the Mastrovito multiplier for a general irreducible polynomial is described in §3. We also give detailed complexity analysis in §4. The full version of this paper [2] contains the architectural details of the multipliers based on binomials, trinomials, ESPs, and AOPs. For each case, a detailed complexity analysis and a design example are given in [2]. The conclusions of this study are summarized in §5.

2

Notation and Preliminaries

Let p(x) be the irreducible polynomial generating the Galois field GF (2m ). In order to compute the multiplication c(x) = a(x)b(x) mod p(x) in GF (2m ), where a(x), b(x), c(x) ∈ GF (2m ), we need to first compute the product polynomial d(x) = a(x)b(x) =


m−1  i=0

i

ai x


m−1  i=0

 i

bi x

(2)

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and then reduce d(x) using p(x) compute coefficients of d(x) using    a0 d0 0  d1   a1 a 0     d2   a2 a1      ..  .. ..   .  . .     dm−2   am−2 am−3     dm−1  =  am−1 am−2     dm   0 am−1     dm+1   0 0      .  .. ..   ..  . .     d2m−3   0 0 d2m−2 0 0

to find the result c(x) ∈ GF (2m ). We can following matrix-vector product:  0 ··· 0 0 0 ··· 0 0   a0 ··· 0 0    b0 .. .. ..  ..  . . . .   b1    am−4 · · · a0 0     b2   am−3 · · · a1 a0   .  . (3)  ..  am−2 · · · a2 a1     bm−2  am−1 · · · a3 a2   .. .. ..  bm−1 .. . . . .   0 · · · am−1 am−2  0 ··· 0 am−1

We will denote this multiplication matrix by M and its rows denoted by Mi , where i = 0, 1, . . . , 2m−2. Note that the entries of M solely consist of coefficients of a(x). We also define the m × m submatrix U(0) of M as the first m rows of M, and the (m − 1) × m submatrix L(0) of M as the last (m − 1) rows of M, i.e.,   0 0 ··· 0 0 a0  a1 a0 0 ··· 0 0     a2 a1 a0 ··· 0 0    (0) U = . (4) .. .. .. ..  , ..  .. . . . . .     am−2 am−3 am−4 · · · a0 0  am−1 am−2 am−3 · · · a1 a0   a1 0 am−1 am−2 · · · a2 0 0 am−1 · · · a3 a2     .. . . . ..  . (0) . .. .. .. .. L = . (5) .     0 0 0 · · · am−1 am−2 0 0 0 ··· 0 am−1 We use the superscripts to denote the step numbers during the reduction process, for example, L(0) , L(1) , etc. The superscript f indicates the final form of the matrix, for example, L(f ) . The rows in the submatrix L(0) of matrix M is reduced using the irreducible polynomial p(x), so that, at the end of reduction, L(f ) becomes the zero matrix. During the reduction process, the rows of L are added to the rows with lower indices according to the irreducible polynomial. During the reduction of a single row, this row is added to certain other rows. We call all of these rows the children of the reduced row. The final submatrix U(f ) is equal to the so-called Mastrovito matrix Z, which is multiplied by the column vector b to produce the result as c = Zb. We use  to represent the concatenation of two vectors. For example, the vectors V = [vn , vn−1 , . . . , v0 ] and W = [wn, wn−1 , . . . , w0 ] can be concatenated

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to form a new vector of length 2(n + 1) as follows: V  W = [ vn vn−1 · · · v1 v0 wn wn−1 · · · w1 w0 ] . In general, two vectors need not be of equal length in order to be concatenated. Also during the reduction, the vectors are shifted to the right or left, where the empty locations are filled with zeros. We use the right and left arrows to represent the right and left shifts. For example, (V → 3) and (V ← 2) represent right and left shifts of the vector V by 3 and 2 positions, respectively, which are explicitly given as (V → 3) = [ 0 0 0 vn vn−1 · · · v6 v5 v4 v3 ] , (V ← 2) = [ vn−2 vn−3 vn−4 vn−5 vn−6 · · · v1 v0 0 0 ] . Furthermore, at certain steps of the reduction, vectors are used to form matrices. For example, to form a matrix using the last (n − 1) entries of the above vectors, the following notation is adopted:     vn−2 vn−3 vn−4 vn−5 · · · v3 v2 v1 v0 V  (V → 3)  vn vn−1 vn−2 · · · v6 v5 v4 v3  . =  0 vn−4 vn−5 vn−6 vn−7 · · · v2 v1 0 0 (V ← 2) 3×(n−1) As seen above, although the original vectors are longer, only the last (n − 1) entries are used, and the rest is discarded. During the reduction operation, we frequently encounter certain special matrices. A particular matrix type is the Toeplitz matrix, which is a matrix whose entries are constant along each diagonal. It is well-known that the sum of two Toeplitz matrices is also a Toeplitz matrix [1]. This property will be used to establish a recursion. Finally, we note that the gates used in the proposed design are 2-input AND and XOR gates, whose delays are denoted by TA and TX , respectively.

3

General Polynomials

We start with the most general form an irreducible polynomial as p(x) = xnk + xnk−1 + · · · + xn1 + xn0 ,

(6)

where ni for i = 0, 1, 2, . . . , k are positive integers with the property m = nk > nk−1 > · · · > n1 > n0 = 0 . The difference between the highest two orders, i.e., nk − nk−1 = m − nk−1 will be denoted by ∆. In the following, we first summarize the general outline of the reduction process, and then, propose a method to obtain the same result more efficiently. When the irreducible polynomial (6) is used to reduce the rows of L(0) , each row will have k children. The one corresponding to the constant term xn0 = 1 is

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guaranteed to be added to a row in U, but the others might be added to the rows of L, and will need to be reduced further. To simplify the notation and observe the regularity, we use k additional matrices. The children produced due to the reductions corresponding to the xni term will be added to the m × m matrix Xi(1) , for i = 0, 1, . . . , (k − 1). The children that fall back into the submatrix L are stored in L(1) , which are to be reduced later. By introducing the Xi matrices, we preserve the matrix U(0) during the reduction. At the end of the first step, i.e., when every row of matrix L(0) is reduced exactly once, the following matrices will be produced: (1)

U(1) = U(0) + X0(1) + X1(1) + · · · + X(k − 1) where

Xi(1)



··· .. .

0 0 0  .. .. ..  . . .  0 0 0  =  0 am−1 am−2 0 0 am−1   . . .. ..  .. . 0 0 0

0 .. .

··· 0 · · · ani · · · ani +1 .. .. . . · · · am−1

··· .. .

0 .. .

0 .. .

··· ··· ··· .. .

0 a2 a3 .. .

0 a1 a2 .. .

· · · am−ni +1 am−ni

for i = 0, 1, . . . , (k − 1). The part of matrix after the first step, will be  (1) (1) 0 · · · 0 lm−1 lm−2 · · · (1)  0 ··· 0 0 lm−1 · · ·   . . . .. .. ..  . . . . . .  . . .   0 ··· 0 (1) 0 0 · · · L =  0 ··· 0 0 0 ···   0 ··· 0 0 0 ···   . . . . . .. .. ..  .. .. .. . 0 ··· 0

0

0

···

,

(7)



0  ..  .   ni − 1   ni   ni + 1   ..  . m−1

(8)

M, which is to be further reduced (1)

(1)

l∆+2 l∆+1 (1) (1) l∆+3 l∆+2 .. .. . . (1) (1) lm−1 lm−2 (1) 0 lm−1 0 0 .. .. . . 0

0

 0  1   ..  .    n(k−1) − 3 .   n(k−1) − 2   n(k−1)−1  ..  .  m−2

(9)

As seen above, the new matrix L(1) which will be reduced in the next step is also triangular. This means the new children will also be in the same form, except that they will contain more zero terms at the beginning. Thus, it is clear that if the same procedure is recursively applied, the submatrix U(0) will never change, and the forms of the matrices Xi and L will remain the same after every step, i.e., Xi matrices will be trapezoidal and L will be triangular. The entries which are zero and outside the indicated geometric regions after the first iteration will always remain zero. Only the values inside these regions will be changed during the rest of the reduction. The number of nonzero rows of L after step j, i.e., the number of nonzero rows in L(j) is given as rj = (m − 1) − j (m − n(k−1) ) = (m − 1) − j ∆ ,

(10)

Mastrovito Multiplier for General Irreducible Polynomials

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since there are (m − 1) nonzero rows initially, i.e., r0 = (m − 1), and the number is reduced by ∆ = (m − n(k−1)) after each step. Thus, it will take

N [m, ∆] =

m−1 ∆

 (11)

steps to reduce the whole matrix L(0) . This number is also equal to the number of nonzero terms in the row L0 at the end of step j, i.e., the number of nonzero (j) terms in the row L0 for j = 1, 2, . . . , N[m, ∆] − 1. Note that the range of j does not include j = N [m, ∆], as the number of nonzero rows becomes zero after step N [m, ∆], but the number rj will be negative for j = N [m, ∆]. First the matrix M is divided into the upper and lower submatrices U(0) and (0) L . The matrix L(0) is reduced into k matrices using the irreducible polynomial, while U(0) is kept unchanged. The upper and lower parts of the new matrices are then separated. The upper parts form the Xi(1) matrices and the lower parts are accumulated into the matrix L(1) . The merged lower parts are to be further reduced in the next step. This procedure is repeated until the matrix L becomes the zero matrix, i.e., all rows are reduced. The total reduction process is illustrated in Figure 4. The sum of the matrices in the last row, i.e., U(0) and Xi(f ) for i = 0, 1,. . .,(k − 1), yields the Mastrovito matrix Z. A close inspection shows that all submatrices formed by the nonzero rows of the matrices produced after the first iteration are Toeplitz matrices. When the matrix L(1) is reduced further, the children will be added to the nonzero rows of the matrices Xi(1) and L(2) , which are Toeplitz submatrices. Since the sum of two Toeplitz matrices is also a Toeplitz matrix [1], the submatrices formed by the nonzero rows will all be Toeplitz submatrices. Furthermore, these matrices are special Toeplitz matrices, computing only the first nonzero rows of Xi(f ) is sufficient to reconstruct them. Similarly, the matrix M and the submatrices U(0) and L(0) can be constructed using only the row Mm−1 . Furthermore, since all first nonzero rows of the matrices Xi(f ) are identical, we only need to compute (f ) one of them. Thus, it suffices to work on X00 whose final value is computed as follows: N [m,∆]−1  (j) (0) (1) (N [m,∆]−1) L0 = L0 + L0 + · · · + L0 . (12) j=0

This will be used with U(0) to construct the final matrix Z. First, the rows Zni are constructed by adding the corresponding rows of the matrix U(0) to ) Xi(f ni for i = 0, 1, . . . , k − 1. Then, they are extended to larger vectors Yi by concatenating the necessary parts of U(0) to their beginning so that the shifts of Yi produce the rows below them up to the row ni+1 , or up to the row (m − 1) if i = (k − 1). This will simplify the construction of the matrix Z. To further simplify the representations, the first nonzero rows of Xi, which are all identical, represented by the vector V, will be used. Instead of referring to U(0) and L(0) , we will use the original multiplication matrix M or its entries ai . The summary of the proposed method is given below:

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1. First, we compute V given as N [m,∆]−1



V =

(j)

L0

= [ 0 vm−1 vm−2 · · · v3 v2 v1 ]

(13)

j=0

using the recursive definition of 

(j)

L0 =

(j−1)

(L0

→ (m − ni ))

(14)

(k − 1) ≥ i ≥ 0 r(j−1) > (m − ni ) for 1 ≤ j ≤ N [m, ∆] − 1 to reduce everything to the sum of shifts of the row (0) L0 or equivalently to the sum of rows in M. The above summation means (j−1) that the row L0 is shifted (m − n(k−1) ) times, (m − nk−2 ) times, etc., (j)

until all entries become zero. Then, these are all added to form L0 . Here we note that V is not computed until it is completely reduced to the sum of rows of M, since there might be cancellations. This fact will be taken into account during the complexity analysis. 2. Then, we compute Zni for i = 0, 1, . . . , (k − 1) using the following recursive relations: Z0 = [a0 ]  [V]1×(m−1) = [ a0 vm−1 vm−2 · · · v3 v2 v1 ] , (15)

Zni = [Mm+n(i−1) ]1×∆i  [Zn(i−1) → ∆i ]1×(m−∆i ) + V , (16) where ∆i = (ni − n(i−1) ) for i = 1, 2, . . . , (k − 1). Thus, if V and Zn(i−1) are given as V = [ 0 vm−1 vm−2 · · · v3 v2 v1 ] , Zn(i−1) = [ an(i−1) wm−1 wm−2 . . . w3 w2 w1 ] , then Zni is obtained as Zni = [ ani (ani −1 + vm−1 ) · · · (an(i−1) + vm−∆i ) (wm−1 + vm−1−∆i ) · · · · · · (w∆i +1 + v1 ) ] . 3. By extending Zni , we find Yi = [ Mm+ni ]1×(∆(i+1) −1)  Zni ] for i = 0, 1, . . . , (k − 1) as follows: Yi = [ an(i+1) −1 · · · ani +1 ani (ani −1 + vm−1 ) · · · · · · (a0 + vm−ni ) (wm−1 + vm−ni −1 ) · · · (wni +1 + v1 ) ].(17)

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4. Finally, the whole Z matrix is constructed as follows: 

Y0 Y0 → 1 .. .

      Y0 → (∆1 − 1)   Y1   Y1 → 1  ..   .   ..  .   Y(i − 1) → (∆i − 1) Z =   Yi   Yi →1   ..  .   ..  .   Y(k − 2) → (∆ (k−1) − 1)   Y(k − 1)   Y(k − 1) → 1   ..  . Y(k − 1) → (∆ − 1)



0 1 .. .

                                    

(n1 − 1) n1 (n1 + 1) .. . .. . (ni − 1) ni (ni + 1) .. . .. . (n(k−1) − 1) n(k−1) (n(k−1) + 1) .. . m×m

(18)

(m − 1)

We note that while the vectors Yi or their shifted versions are of different length, we take the last m elements of these vectors to form the m × m matrix Z.

4

Complexity Analysis

The formula of the vector V in Equation (13) includes the shifted versions of L(j) , (0) which finally reduce to the shifted versions of the row L0 when the recursive (0) formula is used. Since all right-shifted versions of the row L0 = Mm are present in the original multiplication matrix M, it is possible to represent the vector V (0) as a sum of rows of this matrix. Except for the row L0 itself, the minimum shift is equal to ∆. Thus, after cancellations, the indices of the rows will be a subset S of the set of indices m, (m + ∆), (m + ∆ + 1), . . . , (2m − 3), (2m − 2) . The first row with the smallest index can be used as a base for the addition, and the rest of the rows will be added to this row. Thus, the actual subset to be added to this base vector is (S − min S), which will be called as S ∗ . Since the row Mj has exactly (2m − 1 − j) nonzero terms for 2m − 2 ≥ j ≥ m, and adding each nonzero term requires a single XOR gate, the total number

506

A. Halbutoˇ gulları and C ¸ .K. Ko¸c

of XOR  gates required to compute the first form of V will be equal to the sum j∈S ∗ (2m − 1 − j). The delay in the computation of the vector V can be minimized when the binary tree method is used to compute the summation in each entry. Since there are at most |S| items to be added to compute any entry, the delay of the computation will be log2 |S| TX , where |S| denotes the order of the set S. When the recursive relations in Equations (15) and (16) are used, the construction of Z0 requires only rewiring. We then construct Zni by adding the vector V to the vector formed by concatenation, which requires (m − 1) XOR gates since the vector V has only (m − 1) nonzero terms. Thus, we need (k − 1)(m − 1) XOR gates to compute all Zni for i = 1, 2, . . . , (k − 1). Since the time needed to compute a single row Zni is TX , the total delay to compute all rows is (k − 1)TX . The vectors Y0 and Y1 are then found using Equation (17) by rewiring. The construction of Z in Equation (18) is also performed using rewiring since it consists of shifts. To find the final result c(x) via the product c = Zb, we also need m2 AND gates and m(m − 1) XOR gates. Each coefficient of the final result c(x) can be computed independently via the product ci = Zi b. All multiplications can be done in one level, and the m terms can be added using the binary tree method in log2 m TX time. Therefore, the entire computation for the general case requires m2 AND gates and  (2m − 1 − j) (19) (m − 1)(m + k − 1) + j∈S ∗

XOR gates. The total delay of the circuit is given as TA + (log2 |S| + (k − 1) + log2 m) TX .

5

(20)

Conclusions

In this paper, we present a new architecture for the Mastrovito multiplication and rigorous analysis of the complexity for a general irreducible polynomial. In this paper, we give a rigorous analysis of the Mastrovito multiplier for a general irreducible polynomial, and show that it requires m2 AND gates and  (m − 1)(m + k − 1) + j∈S ∗ (2m − 1 − j) XOR gates, where S ∗ is defined in §4. In the full version of this paper [2], we extend this analysis to certain types of irreducible polynomials. These results are summarized in Table 1. Table 1: The XOR complexity results for the Mastrovito multiplier. Polynomial XOR Complexity Reference 2 Trinomial m −1 [7] [8] [12] [13] [14] [2] EST m2 − m [14] [2] 2 AOP m2 − 1 [5] [2]  General (m − 1)(m + k − 1) + j∈S ∗ (2m − 1 − j) §4 ESP m2 − ∆ [2]

Mastrovito Multiplier for General Irreducible Polynomials

507

References 1. G. H. Golub and C. F. van Loan. Matrix Computations. Baltimore, MD: The Johns Hopkins University Press, 3rd edition, 1996. 2. A. Halbutoˇ gulları and C ¸ . K. Ko¸c. Mastrovito multiplier for general irreducible polynomials. Submitted for publication in IEEE Transactions on Computers, June 1999. 3. M. A. Hasan, M. Z. Wang, and V. K. Bhargava. Modular construction of low complexity parallel multipliers for a class of finite fields GF (2m ). IEEE Transactions on Computers, 41(8):962–971, August 1992. 4. T. Itoh and S. Tsujii. Structure of parallel multipliers for a class of finite fields GF (2m ). Information and Computation, 83:21–40, 1989. 5. C ¸ . K. Ko¸c and B. Sunar. Low-complexity bit-parallel canonical and normal basis multipliers for a class of finite fields. IEEE Transactions on Computers, 47(3):353– 356, March 1998. 6. R. Lidl and H. Niederreiter. Introduction to Finite Fields and Their Applications. New York, NY: Cambridge University Press, 1994. 7. E. D. Mastrovito. VLSI architectures for multiplication over finite field GF(2m ). In T. Mora, editor, Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, 6th International Conference, AAECC-6, Lecture Notes in Computer Science, No. 357, pages 297–309, Rome, Italy, July 1988. New York, NY: SpringerVerlag. 8. E. D. Mastrovito. VLSI Architectures for Computation in Galois Fields. PhD thesis, Link¨ oping University, Department of Electrical Engineering, Link¨ oping, Sweden, 1991. 9. A. J. Menezes. Elliptic Curve Public Key Cryptosystems. Boston, MA: Kluwer Academic Publishers, 1993. 10. A. J. Menezes, I. F. Blake, X. Gao, R. C. Mullen, S. A. Vanstone, and T. Yaghoobian. Applications of Finite Fields. Boston, MA: Kluwer Academic Publishers, 1993. 11. J. Omura and J. Massey. Computational method and apparatus for finite field arithmetic. U.S. Patent Number 4,587,627, May 1986. 12. C. Paar. Efficient VLSI Architectures for Bit Parallel Computation in Galois Fields. PhD thesis, Universit¨ at GH Essen, VDI Verlag, 1994. 13. C. Paar. A new architecture for a Parallel finite field multiplier with low complexity based on composite fields. IEEE Transactions on Computers, 45(7):856–861, July 1996. 14. B. Sunar and C ¸ . K. Ko¸c. Mastrovito multiplier for all trinomials. IEEE Transactions on Computers, 48(5):522–527, May 1999.

Author Index

Beth, T. 148, 231, 392 Betsumiya, K. 462 Bhandari, M.C. 170, 334 Blake, I.F. 55 Blass, U. 142 Bond, J.W. 482 Bossert, M. 181, 252 Bozta¸s, S. 410 Carlet, C. 94 Chanduka, K.K.P. Chen, W. 131 Clarkson, L. 104 Clausen, M. 29 Cohen, G. 454 Couselo, E. 211

334

Encheva, S. 454 Engdahl, K. 77 Fekri, F. 355 Forney, Jr. G.D. 1 Fossorier, M.P.C. 282 Fujiwara, T. 201 Gathen, von zur J. 404 Geiselmann, W. 231 Gennero, M.C. 444 Gonzalez, S. 211 Grassl, M. 231 Greferath, M. 160 Grießer, H. 181 Griffin, F. 87 Guillot, P. 94 Gulliver, T.A. 462 Gupta, M.K. 170 Halbutoˇ gulları, A. 498 Harada, M. 462 Heegard, C. 314 Hisamatsu, K. 221 Høholdt, T. 260 Honkala, I. 142 Horadam, K.J. 121 Hughes, G. 346 Hui, S. 482

Imai, H. Jin, H.

282 10

Kaneko, T. 221 Karpinski, M. 492 Kasami, T. 43, 201 Kennedy, R.A. 191 Kløve, T. 131 Ko¸c, C ¸ .K. 498 Kurakin, V.L. 365 Kurihara, M. 302 Kuzmin, A.S. 365 Lal, A.K. 170, 334 Le Ruyet, D. 324 Lentmaier, M. 77 Litsyn, S. 142 Liu, S. 294 Markov, V.T. 211, 365 Matsumoto, R. 271 Maucher, J. 181, 252 McEliece, R.J. 10 McLaughlin, S.W. 355 Mersereau, R.M. 355 Mihaljevi´c, M.J. 282 Mikhalev, A.V. 365 Miller, E. 19 Miura, S. 271 Motani, M. 314 Mourrain, B. 430 M¨ uller, M. 29 M¨ uller–Quade, J. 392 Nagy, Z. 245 Nechaev, A.A. 211, 365 Niederreiter, H. 87 Nielsen, R.R. 260 ¨ Osterg˚ ard, P.R.J. Parker, M.G. 420 Poli, A. 444 P¨ uschel, M. 148

472

510

Author Index

R¨ otteler, M.

148

Sakata, S. 302 Schafer, R.W. 355 Schmidt, H. 482 Schmidt, S.E. 160 Shokrollahi, M.A. 65 Shparlinski, I.E. 87, 404, 492 Sidorenko, V. 252 Sturmfels, B. 19 Sun, H. 324 Tanaka, H. 221 Tang, Y. 201

Thien, H.V.

324

Vaughan, I.

104

Wang, Q. 191 Wang, Y. 294 Wei, L. 191 Xin, D.

444

Zeger, K. 245 Zigangirov, K.S. 77 Zyablov, V.V. 181