Design Theory: Volume 2 (Encyclopedia of Mathematics and its Applications)

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS EDITED BY G.-C. ROTA Editorial Board B. Doran, M. Ismail, T.-Y. Lam. E. Lutwak Volume 78 Design Theory 27 N. H. Bingham, C. M. Goldie, and J. L. Teugels Regular Variation 28 P. P. Petrushev and V. A. Popov Rational Approximation of Real Functions 29 N. White (ed.) Combinatorial Geometries 30 M. Pohst and H. Zassenhaus Algorithmic Algebraic Number Theory 31 J. Aczel and J. Dhombres Functional Equations in Several Variables 32 M. Kuczma, B. Choczewski, and R. Ger Iterative Flinctional Equations 33 R. V. Ambartzumian Factorization Calculus and Geometric Probability 34 G. Gripenberg, S.-O. Londen, and O. Staffans Volterra Integral and Functional Equations 35 G. Gasper and M. Rahman Basic Hypergeometric Series 36 E. Torgersen Comparison of Statistical Experiments 37 A. Neumaier Imerval Methods for Systems of Equations 38 N. Korneichuk Exact Constallls in Approximation TheOlY 39 R. Brnaldi and H. Ryser Combinatorial Matrix Theory 40 N. White (ed). Matroid Applications 41 S. Sakai Operator Algebras in Dynamical Systems 42 W. Hodges Basic Model Theory 43 H. Stahl and V. Totik General Ol1hogonal Polynomials 44 R. Schneider Convex Bodies 45 G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions 46 A. Bjorner et al. Oriellfed Matroids 47 G. Edgar and L. Sucheston Stopping TImes and Directed Processes 48 C. Sims Computation with Finitely Presented Groups 49 T. Palmer Banach Algebras alld the General TheO/y of "-Algebras 50 F. Borceux Handbook of Categorical Algebra I 51 F. Borceux Handbook of Categorical Algebra II 52 F. Borceux Handbook of Categorical Algebra JlJ 54 A. Katok and B. Hasselblatt Introduction to the Modern Theory of Dynamical Systems 55 V. N. Sachkov Combinatorial Methods in Discrete Mathematics 56 V. N. Sachkov Probabilistic Methods ill Discrete Mathematics 57 P. M. Cohn Skew Fields 58 R. Gardner Geometric Topography 59 G. A. Baker Jr. and P. Graves-Morris Pade Approximants 60 J. Krajicek Bounded Arithmetic, Propositional Logic, and Complexity TheO/y 61 H. Groemer Geometric Applications of Fourier Series alld Spherical Hanl10llics 62 H. O. Fattorini Infinite Dimensional Optimization and Control Theory 63 A. C. Thompson Minkowski Geometry 64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications 65 K. Engel Spemer TheOf)' 66 D. Cverkovic, P. Rowlinson, S. Simic Eigenspaces of Graphs 67 F. Bergeron, O. Labelle, and P. Leroux Combinatorial Species and Tree-Like Structures 68 R. Goodman and N. Wallach Representations and in1'ariants of the Classical Groups

Design Theory Second Edition

Thomas Beth Universitiit Karlsruhe Dieter Jungnickel Universitiit Augsburg Hanfried Lenz Freie Universitiit Berlin

Volume II

CAMBRIDGE UNIVERSITY PRESS

Foreword Doch am vierten Tag im FeIsgesteine Hat ein Zollner ihm den Weg verwehrt: "Kostbarkeiten zu verzollen?" - "Keine." Und der Knabe, der den Ochsen fiihrte, sprach: "Er hat gelehrt." Und so war auch das erkliirt. (Brecht)

Although many excellent papers and some specialised monographs on different aspects of design theory had been published, prior to 1985 (when the first edition of this book appeared) there was no comprehensive monograph on the field. Thus it was our plan to cover the main concepts and ideas of modem design theory without being encyclopedic, but including a deeper study of several representative topics. As it turned out, these aims (which required a rather long book) seem to have been met by the first edition which has been quoted extensively in the research literature. A first draft of this book was obtained by merging different manuscripts of the authors. The first edition then grew from an iterative process of rewriting in which all the authors contributed their ideas to each of the parts. For reasons of time constraints and shifting research interests, the major part of the revision now presented has been done by the second author, drawing on his previous update Jungnickel (1989a) and surveys Jungnickel (1990a, 1992a); in particular, this holds for Chapter VI. Of course, there has still been considerable help from his co-authors (with Chapter XIII being the first author's contribution), but nevertheless he is willing to accept most of the blame for the mistakes introduced during the process of revision. As in 1985, it is again hoped that this book may prove useful in several ways: as most of the presentation is based on classroom experience, it should be suitable as both a textbook for (advanced) classroom use and as a reference book for private study. Although we usually do not require any previous knowledge beyond basic algebra (except for some more advanced parts of Chapter VI and for some of the applications studied in Chapter XIII), this text should also be attractive to the specialist. The proofs of several fundamental theorems have been simplified vii

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The Edinburgh Building, Cambridge CB2 2RU, UK www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarcon 13,28014 Madrid, Spain First edition

©

Bibliographisches Institut, Zurich, 1985 Cambridge University Press, 1993 Second edition © Cambridge University Press, 1999

©

This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United Kingdom at the University Press, Cambridge Typeset in Times Roman 10/13pt. in Jb.TEX2 e [TB] A catalogue recordfor this book is availablejrom the British Library Library of Congress Cataloguing in Publication data Beth, Thomas, 1949Design theory I Thomas Beth, Dieter Jungnickel, Hanfried Lenz. 2nded. p. cm. Includes bibliographical references and index. ISBN 0 521 772311 (hardbound) 1. Combinatorial designs and configurations. 1. Jungnickel, D. (Dieter), 1952- . II. Lenz, Hanfried. Ill. Title. QAI66.25.B47 1999 511'.6 - dc21 98-29508 CIP ISBN 0 521772311 hardback

In memoriam REINHOLD BAER

x

Foreword

Let us give a brief outline of the contents of this book. Chapter I is a leisurely introduction to the different topics of design theory. The fonnal treatment begins with Chapter II which is devoted to the combinatorial and algebraic analysis of incidence structures. In Chapters ill, IV, and V, we develop in depth the interaction between groups and designs. This part of the book is well suited to fonn the core of a special course on this topic. The same holds for Chapter VI, where difference sets are considered in much detail; these are equivalent to studying symmetric designs with a regular automorphism group, thus providing the link to representation theory. The five subsequent chapters emphasise the constructive aspects of design theory. The connection to the preceding part is provided by Chapters VII and Vill which contain constructions based mostly on abelian groups, including the asymptotic existence of difference families in Galois fields. The recursive methods presented in Chapter IX have their foundation in the preceding direct constructions. Thus we obtain several general existence theorems for block designs with small block sizes as well as for Steiner quadruple systems, illustrating the close interlacing of design-like incidence structures and Latin squares which are subject to a more detailed study in Chapter X. In Chapter XI all these methods converge in the proof of the asymptotic existence theorem for 2-designs and related structures. Chapter XII is devoted to characterising the classical examples among the families of designs we have studied. The final Chapter Xli discusses some known and new applications of design theory. The book concludes with an appendix of several tables of designs in a range feasible for applications, and an extensive bibliography. Items are numbered and quoted consecutively within each chapter as follows: "Item x.y" refers to the y'th item in §x of the given chapter; similarly, fonnulae are numbered with reference to the item where they occur, that is, in the fonn (x.y.a), (x.y.b), etc. If an item from another chapter needs to be quoted, the number of this chapter is added, resulting in quotes of the fonn "by item Z.x.y"; a similar convention holds for fonnulae. The symbol. indicates the end (or absence) of a proof; as mentioned before, we will have to state many re~ults without proofs. Finally, we would like to acknowledge those people and institutions, who by their mental or financial support throughout the last two decades made this work possible. Particular thanks are due to Klaus Metsch and Bernhard Schmidt who each provided us with drafts for some parts of the present text (namely Sections X.7 and some of the later parts of Chapter VI, respectively). Also the engagement of Detlef Zerfowski in preparing the completely new Chapter Xli must be acknowledged. We also thank the following colleagues for their comments:

;;

Foreword ;;

xi

A. E. Brouwer, P. J. Cameron, C. Chames, C. 1. Colbourn, 1. A. Davis, M. 1. de Resmini, R. H. F. Denniston, D. A. Drake, S. Furino, W. Geiselmann, 1. W. P. Hirschfeld, D. R. Hughes, J. ledwab, I. H. van Lint, H. Luneburg, W. H. Mills, R. C. Mullin, G. Pickert, F. C. Piper, A. Pott, D. K. Ray-Chaudhuri, 1. 1. Seidel, H. Siemon, N. J. A. Sloane, S. A. Vanstone, and V. Zinoviev. The institutions we are indebted to are: Deutscher Akademischer Austauschdienst, Deutsche Forschungsgemeinschaft, Heisenberg-Programm, National Science and Engineering Research Council (Canada), Istituto di Geometria dell' Universita di Bologna, Department of Mathematics of The Ohio State University, Mathematics Department of the University of Florida, Mathematics Department of Westfield College London, Department of Combinatorics and Optimization of the University of Waterloo, Institut fUr Mathematische Maschinen und Datenverarbeitung as well as Mathematisches Institut der Universitat Erlangen, Mathematisches Institut der Universitat GieSen, II. Mathematisches Institut der Freien Universitat Berlin, Institut fUr Algorithmen und Kognitive Systeme der Universitat Karlsruhe, Institut fur Mathematik der Universitat Augsburg, and, in particular, Mathematisches Forschungsinstitut Oberwolfach both for providing a multitude of stimulating contacts and for giving us the opportunity to design the final draft of the first edition of this book in its unique atmosphere.

Thomas Beth Dieter lungnicke1 Hanfried Lenz November, 1998

Neither art nor science can be achieved without studying. (Demokrit)

viii

Foreword

and many advanced results are presented in detail. Unfortunately, the explosive growth and advancing technical difficulty of many parts of design theory have made it impossible to include proofs for most of the new results obtained after our first edition appeared. To mention just two examples, the existence theory for designs is becoming more and more involved and nowadays requires a considerable amount of computer work; and the theory of difference sets has reached a stage where major new results can only be obtained by the use of algebraic number theory and lengthy and delicate computations. On the other hand, we still wanted to provide our readers with the present state of knowledge, instead of more or less only reprinting our first edition. Therefore, several important parts of the text have been completely rewritten (for instance, the chapter on difference sets and the appendix giving a collection of tables as well as the sections concerning constructions for sets of mutually orthogonal Latin squares of small order and the section on completing Bruck nets, to mention just a few) and a fair amount of new material has been added with a full exposition (for instance, a chapter on applications of design theory as well as sections on strongly regular graphs, Cayley graphs, codes of designs and nets, embedding designs, and Steiner quadruple systems). However, most new results can only be quoted without giving proofs - otherwise we would require three volumes. An extensive bibliography of well over 2000 titles, all quoted in the text, has been included. Thus the book may also be used as a reference, though it is - as already mentioned - not intended to meet encyclopedic standards. Last, but not least, in view of the central importance of discrete mathematics for a multitude of applications, the book will provide some ofthe necessary mathematical background for anyone working in communications engineering, optimisation, statistical planning, computer science, and signal processing. With these aims in mind we have tried to select and present the material not only according to our personal points of view: a large part of our work has been devoted to presenting appropriately the results of the different schools of combinatorics concerned with design theory. We have also tried to cover the different areas of this subject in such a manner that the reader may work with other related monographs and papers. We first list (in alphabetical order) some important books on different aspects of finite geometry, design theory and closely related areas, namely Assmus and Key (1992a), Bannai and Ito (1984), Batten and Beutelspacher (1993), Baumert (1971), Brouwer, Cohen and Neumaier (1989), Cameron (1976a), Cameron and van Lint (1991), Dembowski (1968), Denes and Keedwell (1974, 1991), Evans (1992b), Geramita and Seberry (1979), Godsil (1993), Haemers (1980), Hirschfeld (1985,1998), Hirschfeld and Thas (1991), Huppert (1967), Huppert and Blackburn (1982a, b), Kallaher (19al),

Foreword

ix

Lander (1983), Ltineburg (1965b, 1969, 1980), Pott (1995), Shrikhande and Sane (1991), Tsuzuku (1982), Wallis (1997) and Zieschang (1996a). There is also a direct link to books on related applications, for instance Beker and Piper (1982), Conway and Sloane (1993), Harwit and Sloane (1979), Jungnickel (1993a), Liineburg (1979), Mac Williams and Sloane (1978), Raghavarao (1971), Simmons (1992a) and Stinson (1995). We should also mention that there are by now five other books studying combinatorial designs, all of which are intended as introductory texts emphasising various aspects of the field; in order of appearance, these are Hughes and Piper (1985), Street and Street (1987), Tonchev (1988), Wallis (1988), and Anderson (1990). Design theoretic algorithms are surveyed in a collection edited by Colbourn (1985); a recent volume on computational and constructive aspects of design theory was edited by Wallis (1996). Surveys of applications of designs are given by Colbourn and van Oorschot (1989) and in Chapter V of Colboum and Dinitz (1996a); the latterreference basically is a vast compendium of tables. Recent expositions of designs from the point of view of statistics are due to Bailey (1989, 1996); for an upto-date treatment of the optimal design of experiments, see Pukelsheim (1993). The relevance of design theory to pure, applied, and even applicable mathematics has already been indicated. Nevertheless, we would like to quote some sentences from Lovasz (1979): The mathematical world had been attracted by the successes of algebra and analysis and only in recent years it has become clear, due largely to problems arising from economics. statistics, electrical engineering and other applied sciences, that combinatorics, the study of finite sets and finite structures, has its own problems and principles. These are independent of those in algebra and analysis but match them in difficulty, practical and theoretical interest and beauty. Yet the opinion of many first-class mathematicians about combinatorics is still in the pejorative. While accepting its interest and difficulty. they deny its depth. It is often forcefully stated that combinatorics is a collection of problems, which may be interesting in themselves but are not linked and do not constitute a theory .. " The above accusations are clearly characteristic of any field of science at an early stage of its development - at the stage of collecting data .... Those techniques whose absence has been disapproved of above await their discoverers. So nnderdevelopment is not a case against, but rather for, directing young scientists towards a given field. In my opinion, combinatorics is now growing out of this early stage.

In this spirit, we think that the development of design theory over the last few decades has clearly reached a critical point at which we now see at least the beginnings of a systematic treatment of the area leading to its codification into a mathematical theory, as opposed to a collection of individual results. We hope that this development can be discerned from the present text.

Contents

IX. Recursive constructions . . . . . . § 1. §2. §3. §4. §5. §6. §7. §S. §9. § 10. § 11. §12.

Product constructions . . . . . . Use of pairwise balanced designs Applications of divisible designs Applications of Hanani's lemmas Block designs of block size three and four Solution of Kirkman's schoolgirl problem The basis of a closed set . . . . . . . . Block designs with block size five .. . Divisible designs with small block sizes Steiner quadruple systems. . . . . . . . Embedding theorems for designs and partial designs Concluding remarks . . . . . . . . . . . . . . . . .

X. Transversal designs and nets § 1. §2. §3. §4. §5. §6. §7. §S. §9.

60S 60S 617 621 627 636 641 644

651 660 664 673 6Sl 690

A recursive construction Transversal designs with A > 1 A construction of Wilson . . . Six and more mutually orthogonal Latin squares The theorem of Chowla, Erdos and Straus Further bounds for transversal designs and orthogonal arrays . . . . . . . . . . . . . . . . . . . . . . . Completion theorems for Bruck nets . . . . . . . . . . Maximal nets with large deficiency . . . . . . . . . . . Translation nets and maximal nets with small deficiency xiii

690 693 696 703 706 70S 713 725 731

Contents

xiv

§1O. Completion results for f.L > I . . . . . . . . . . . . . . .. § 11. Extending symmetric nets. . . . . . . . . . . . . . . . ., §I2. Complete mappings, difference matrices and maximal nets. §13. Tarry's theorem. . . §14. Codes of Bruck nets. . .

XI. Asymptotic existence theory § 1. §2.

§3. §4. §5. §6. §7. §8.

§2. §3. §4. §5. §6.

Projective and affine spaces as linear spaces Characterisations of projective spaces . Characterisation of affine spaces Locally projective linear spaces . Good blocks . . . . Concluding remarks. .

XIII. Applications of designs . § 1.

§2. §3. §4. §5. §6. §7. §8. §9. § 1O.

Introduction...... Design of experiments Experiments with Latin squares and orthogonal arrays Application of designs in optics . Codes and designs . . . . . . . . . . . . . . . . . Discrete tomography . . . . . . . . . . . . . . . . Designs in data structures and computer algorithms. Designs in hardware . . . . . . . . . Difference sets rule matter and waves. No waves, no rules, but security.

Appendix. Tables . . . . . § 1. §2.

781

781 Preliminaries . . . . . . The existence of Steiner systems with v in given residue classes . . . . . . . . . . . . . . . . . . . . . . 783 The main theorem for Steiner systems S(2, k; v) 787 790 The eventual periodicity of closed sets The main theorem for ).. = 1 . . . . . . . . . . . 793 The main theorem for).. > 1 . . . . . . . . . . . 796 An existence theorem for resolvable block designs 801 Some results for t ::: 3 . . . . . . . . 805

XII. Characterisations of classical designs . § 1.

749 758 761 772 778

Block designs . . . Symmetric designs

806 806 808 821

828 833 841 852 852

856 874

880 892 926

930 937 946

956 971 . 971 . 981

!

I I

Contents

xv

§3. Abelian difference sets . . . . . §4. Small Steiner systems . . . . . . §S. Infinite series of Steiner systems §6. Remark on t-designs with t 2: 3 . §7. Orthogonal Latin squares

· 990 · 997 · 999 1001

Notation and symbols Bibliography . Index . . . . . . . . .

I

1002 100S 1013

1093

Contents of Volume I

I. Examples and basic definitions §l. §2. §3. §4. §5. §6. §7. §8. §9.

........ .

Incidence structures and incidence matrices . Block designs and examples from affine and projective geometry . . . . . . . . . . . . . . . . . . . . . . . . 6 t-designs, Steiner systems and configurations. 15 20 Isomorphisms, duality and correlations . . 24 Partitions of the block set and resolvability 32 Divisible incidence structures . 37 Transversal designs and nets 44 Subspaces . . . . . 50 Hadamard designs . . . . .

II. Combinatorial analysis of designs §l. §2. §3. §4. §5. §6.

Basics . . . . . . . . . . . . . Fisher's inequality for pairwise balanced designs Symmetric designs . . . . . . . . . . . . . . . The Bruck-Ryser-Chowla theorem . . . . . . . Balanced incidence structures with balanced duals Generalisations of Fisher's inequality and intersection numbers . . . . . . . . . . . . . . . . . . . . .. §7. Extensions of designs . §8. Mfine designs. . . . . . . §9. Strongly regular graphs . . §1O. The Hall-Connor theorem §11. Designs and codes . . . xvii

62 62 64 77 89

96 10 I III 123 136 146 152

Contents

§ 11. §12. §13. § 14. §15. §16. § 17.

Extended building sets and difference sets . . . . . . Constructions for Hadamard and Chen difference sets Some applications of algebraic number theory Further non-existence results . . Characters and cyclotomic fields . . . Schmidt's exponent bound . . . . . . Difference sets with Singer parameters

VII. Difference families .

§ 1. §2. §3. §4. §5. §6. §7. §8.

Basic facts .. Multipliers .. More examples Triple systems. Some difference families in Galois fields Blocks with evenly distributed differences Some more special block designs Proof of Wilson's theorem

VIII. Further direct constructions § 1. §2. §3. §4. §5. §6. §7. §8. §9. § 10.

Pure and mixed differences . . . . . . . . . . . . . . . .. Applications to the construction of resolvable block designs A difference construction for transversal designs Further constructions for transversal designs Some constructions using projective planes . . . t-designs constructed from graphs . . . . . . . The existence of t-designs for large values of A . Higher resolvability of t-designs Infinite t-designs . . . . . . . . Cyclic Steiner quadruple systems

Notation and symbols Bibliography . Index . . . . . . . . .

xix 382 388 410 . 4 19 . 435 441 455 468 . 468 .472 476 481 488 499 502 509 520 520 528 531 544 564 584 588 595 598 600 1005 1013 1093

Contents

§ 11. §12. §13. § 14. §15. §16. § 17.

Extended building sets and difference sets . . . . . . Constructions for Hadamard and Chen difference sets Some applications of algebraic number theory Further non-existence results . . Characters and cyclotomic fields . . . Schmidt's exponent bound . . . . . . Difference sets with Singer parameters

VII. Difference families .

§ 1. §2. §3. §4. §5. §6. §7. §8.

Basic facts .. Multipliers .. More examples Triple systems. Some difference families in Galois fields Blocks with evenly distributed differences Some more special block designs Proof of Wilson's theorem

VIII. Further direct constructions § 1. §2. §3. §4. §5. §6. §7. §8. §9. § 10.

Pure and mixed differences . . . . . . . . . . . . . . . .. Applications to the construction of resolvable block designs A difference construction for transversal designs Further constructions for transversal designs Some constructions using projective planes . . . t-designs constructed from graphs . . . . . . . The existence of t-designs for large values of A . Higher resolvability of t-designs Infinite t-designs . . . . . . . . Cyclic Steiner quadruple systems

Notation and symbols Bibliography . Index . . . . . . . . .

xix 382 388 410 . 4 19 . 435 441 455 468 . 468 .472 476 481 488 499 502 509 520 520 528 531 544 564 584 588 595 598 600 1005 1013 1093

§l. Product constructions

609

over G = {YI, ... , Yg}. Then the matrix A I8i D defined by au

~D

akl

+D

AI8iD=

(

is an OAu,(kk, gg').1

if E

is a (g, k, A.)-difference matrix, then E I8i D is a (g, kk, g)...)...I)-difference matrix. In particular, the Kronecker product of two generalised Hadamard matrices (VIII.3.4) is a generalised Hadamard matrix. •

1.3 Example. The existence of (3,3; 1)- and (3, 6; 2)-difference matrices (see Theorem VIII.3.14) implies that of a (3, 18; 6)-difference matrix over &::3 x &::6. More generally, we obtain (3,3),; A.)-difference matrices for all ). = 2i 3j with j :::: i-I. All these difference matrices are generalised Hadamard matrices. 1.4 Definition. Let (A, 0), (B, *) be quasigroups (see Definition VIII.4.lO). Their direct product is defined as the set A x B with the operation 0 defined by (lA.a)

Ca, b) 0 (a', b' ) := (a

0

a', b

* b').

1.5 Lemma. Let (A, 0) and (A, 0 ' ) be orthogonal quasigroups, and letCB, *). (B, *') be orthogonal quasigroups too. Then the direct products CA, 0) x (B, *) and (A, 0') x CB, *') are orthogonal quasigroups. if CA, 0) and (B, *) are idempotent, then (A, 0) x (B, *) is also idempotent. • The proof is straightforward. Again MacNeish's theorem (1.7.7.b) follows immediately; furthermore (1.5.a)

N*(gh) :::: min{N*(g), N*(h)}.

For the sake of completeness we mention the following connection between quasigroups and Steiner triple systems.

1

Note that this would be the Kronecker product of matrices if G were written multiplicatively.

§l. Product constructions

609

over G = {YI, ... , Yg}. Then the matrix A I8i D defined by au

~D

akl

+D

AI8iD=

(

is an OAu,(kk, gg').1

if E

is a (g, k, A.)-difference matrix, then E I8i D is a (g, kk, g)...)...I)-difference matrix. In particular, the Kronecker product of two generalised Hadamard matrices (VIII.3.4) is a generalised Hadamard matrix. •

1.3 Example. The existence of (3,3; 1)- and (3, 6; 2)-difference matrices (see Theorem VIII.3.14) implies that of a (3, 18; 6)-difference matrix over &::3 x &::6. More generally, we obtain (3,3),; A.)-difference matrices for all ). = 2i 3j with j :::: i-I. All these difference matrices are generalised Hadamard matrices. 1.4 Definition. Let (A, 0), (B, *) be quasigroups (see Definition VIII.4.lO). Their direct product is defined as the set A x B with the operation 0 defined by (lA.a)

Ca, b) 0 (a', b' ) := (a

0

a', b

* b').

1.5 Lemma. Let (A, 0) and (A, 0 ' ) be orthogonal quasigroups, and letCB, *). (B, *') be orthogonal quasigroups too. Then the direct products CA, 0) x (B, *) and (A, 0') x CB, *') are orthogonal quasigroups. if CA, 0) and (B, *) are idempotent, then (A, 0) x (B, *) is also idempotent. • The proof is straightforward. Again MacNeish's theorem (1.7.7.b) follows immediately; furthermore (1.5.a)

N*(gh) :::: min{N*(g), N*(h)}.

For the sake of completeness we mention the following connection between quasigroups and Steiner triple systems.

1

Note that this would be the Kronecker product of matrices if G were written multiplicatively.

611

§l. Product constructions

distinct points a, b, c, d are on a block iff f(a, b, c) = d.lf (V, f) and (W, g) are such ternary algebras, then their direct product, defined in the obvious way, has the same properties. Hence (l.9.f)

u,

V

E S(3, 4) =}

uv

E S(3, 4).

Note that 2 E S(3, 4) (two points, no blocks), and that a ternary operation 0 on a 2-set, satisfying (1.9.b), has all the properties of 1.9. Hence (1.9.£) specializes to (lo9.g)

2· S(3, 4) S; S(3, 4).

In view of Hanani's theorem stating that S(3, 4) = 2 N \ 6 N, (1.9.£) is a very weak result; see § 10 for a proof of Hanani' s theorem.

1.10 Remark. These examples may be generalised to the idea of applying universal algebra to combinatorial structures; cf. Ganter (1976a), Evans (1975), Quackenbush (1975). A nice introductory paper on this subject was given by Evans (1979). Next we shall use difference matrices and orthogonal arrays for product constructions of difference families, as well as the other way round.

1.11 Proposition. Let k be the order of an affine plane. Furthermore, let A = (aij) be an OA(k, k) over S = Moreover, let D = (D 1 ••..• Ds) and E = (EI,"" E t ) be difference families in the groups G and G' with parameters (v, k. A) and (v', k, A'), respectively, say Di = {do, ... , did and E j = {ejl, ... , ejk} for i E N~ and j E Ni. Then there is a (vv', k, U')difference family in G EB G'.

N1.

Proof. We may assume that the last k columns of A are (1, 1, ... , ll, ... , (k, k, ... , k)T. We omit these k columns and get a k x k(k -I)-matrix B = (b ij ). Now let the desired difference family F consist of the following base blocks: A' copies of {(do, 0), ... , (dik. O)} for each i E Nl;

A copies of {CO, ejd}, ... , (0, ejk)} for each j EN;; one copy of {(do, ejb,,,), ... , (dit. ejb,,)} for each n E and each pair (i, j) E N~

X

Nf- k

NI!.

The somewhat lengthy though not difficult verification of this construction is left to the reader; cf. Jungnickel (1978) . •

611

§l. Product constructions

distinct points a, b, c, d are on a block iff f(a, b, c) = d.lf (V, f) and (W, g) are such ternary algebras, then their direct product, defined in the obvious way, has the same properties. Hence (l.9.f)

u,

V

E S(3, 4) =}

uv

E S(3, 4).

Note that 2 E S(3, 4) (two points, no blocks), and that a ternary operation 0 on a 2-set, satisfying (1.9.b), has all the properties of 1.9. Hence (1.9.£) specializes to (lo9.g)

2· S(3, 4) S; S(3, 4).

In view of Hanani's theorem stating that S(3, 4) = 2 N \ 6 N, (1.9.£) is a very weak result; see § 10 for a proof of Hanani' s theorem.

1.10 Remark. These examples may be generalised to the idea of applying universal algebra to combinatorial structures; cf. Ganter (1976a), Evans (1975), Quackenbush (1975). A nice introductory paper on this subject was given by Evans (1979). Next we shall use difference matrices and orthogonal arrays for product constructions of difference families, as well as the other way round.

1.11 Proposition. Let k be the order of an affine plane. Furthermore, let A = (aij) be an OA(k, k) over S = Moreover, let D = (D 1 ••..• Ds) and E = (EI,"" E t ) be difference families in the groups G and G' with parameters (v, k. A) and (v', k, A'), respectively, say Di = {do, ... , did and E j = {ejl, ... , ejk} for i E N~ and j E Ni. Then there is a (vv', k, U')difference family in G EB G'.

N1.

Proof. We may assume that the last k columns of A are (1, 1, ... , ll, ... , (k, k, ... , k)T. We omit these k columns and get a k x k(k -I)-matrix B = (b ij ). Now let the desired difference family F consist of the following base blocks: A' copies of {(do, 0), ... , (dik. O)} for each i E Nl;

A copies of {CO, ejd}, ... , (0, ejk)} for each j EN;; one copy of {(do, ejb,,,), ... , (dit. ejb,,)} for each n E and each pair (i, j) E N~

X

Nf- k

NI!.

The somewhat lengthy though not difficult verification of this construction is left to the reader; cf. Jungnickel (1978) . •

613

§1. Product construCtions

1.15 Examples. (a) Proposition 1.14 and Lemma VIII.3.10 yield alternative proofs for Examples 1.l3.(b), (c). An alternative proof of Example 1.13.(a) will be given a little later. (b) Using Proposition 1.14, Lemma VIII.3.10 and Wilson's Theorem VII.6.6, we obtain the existence of a (v, k, A)-difference family whenever v admits a prime power factorisation

1)

with A(qi large.

="

0 (mod k(k - 1», where the qi (i

= 1, ... , n) are sufficiently

(c) Assume that k or k - 1 divides 2A. Then there exists a (v, k, A)-difference family whenever each factor qi in the prime power factorisation of v satisfies the condition (1.15.b)

A(qi - 1)

== 0 (mod k(k

- 1».

This follows from Lemma 1.1 (with A' = 1), together with Lemma VIII.3.IO. Thus, for example, we obtain (compare with Examples VII.5.4) v E DI (3) n D2(4) n Ds(6) n D6(7),

(1.15.c)

= 1, ... , m;

whenever qi

== 1 (mod6) for i

whenever qi

== 1 (mod 10) for i

= 1, ... , n;

whenever qi

== 1 (mod 14) for i

= 1, ... , n;

== 1 (mod 18) for i

= 1, ... , n.

(1.15.f) whenever qi (d) Let t, n

E

(U5.g)

(4t - l)n

Nand 4t - 1 be a prime power. Then E

D t - l (2t - 1).

This result follows by using a Paley difference set (Theorem VI. 1.12), and is a special case of (c). (e) There is a (91, 10, I)-difference set in Z91 (Theorem VI. 1.10). Also there are (l3, 13; I)-difference matrices (Lemma VIII.3.1O), and (7, 14; 2)-difference

613

§1. Product construCtions

1.15 Examples. (a) Proposition 1.14 and Lemma VIII.3.10 yield alternative proofs for Examples 1.l3.(b), (c). An alternative proof of Example 1.13.(a) will be given a little later. (b) Using Proposition 1.14, Lemma VIII.3.10 and Wilson's Theorem VII.6.6, we obtain the existence of a (v, k, A)-difference family whenever v admits a prime power factorisation

1)

with A(qi large.

="

0 (mod k(k - 1», where the qi (i

= 1, ... , n) are sufficiently

(c) Assume that k or k - 1 divides 2A. Then there exists a (v, k, A)-difference family whenever each factor qi in the prime power factorisation of v satisfies the condition (1.15.b)

A(qi - 1)

== 0 (mod k(k

- 1».

This follows from Lemma 1.1 (with A' = 1), together with Lemma VIII.3.IO. Thus, for example, we obtain (compare with Examples VII.5.4) v E DI (3) n D2(4) n Ds(6) n D6(7),

(1.15.c)

= 1, ... , m;

whenever qi

== 1 (mod6) for i

whenever qi

== 1 (mod 10) for i

= 1, ... , n;

whenever qi

== 1 (mod 14) for i

= 1, ... , n;

== 1 (mod 18) for i

= 1, ... , n.

(1.15.f) whenever qi (d) Let t, n

E

(U5.g)

(4t - l)n

Nand 4t - 1 be a prime power. Then E

D t - l (2t - 1).

This result follows by using a Paley difference set (Theorem VI. 1.12), and is a special case of (c). (e) There is a (91, 10, I)-difference set in Z91 (Theorem VI. 1.10). Also there are (l3, 13; I)-difference matrices (Lemma VIII.3.1O), and (7, 14; 2)-difference

615

§l. Product constructions Note that s = ~ v, as it must be.

= A. g~g~\)' hence the column number of A is sg(g -

1) + A.

=

1.18 Examples. (a) Let q be a prime power and assume the existence of a TD[k; q + 1] with a parallel class. Then there exists a (q2 + q + 1, k; 1)difference matrix in Zq2+q+! by Singer's Theorem VI. 1. 10. (b) There are (15, 7; 3)-, (21,5; 1)-, (40, 13; 4)-, and (57, 8; I)-difference matrices in the cyclic groups of the respective orders. The reader is asked to check this. (c) Let q be a prime power and let G be a group of order q + 1. Then there is a (q + 1, q; q - I)-difference matrix over G. To see this, use the trivial (q + 1, q, q - I)-difference set G \ {OJ. Note that this yields non-abelian difference matrices too (though of moderate size; the maximum feasible value of k is q2 - 1 by Corollary VIII.3.7).

It is worthwile to state the following consequence of Proposition 1.17.

1.19 Corollary. Assume the existence of a (v, g, I)-difference family. Then (1.19.a)

g E TD(k) =? v E TD(k).

In particular, (1.19.b)

q

+ 1 E TD(k)

=? q2

+q + 1E

TD(k)

for prime powers q.

Thus (1.19.c)

21

(1.19.d)

57 E TD(9),

(1.19.e)

273 E TD(18).

E

TD(6),

Proof. Use Lemma 1.1 together with (I.7.15.a) and Corollary VIII.3.8. The particular examples follow from 1.18.(a). • The values given in (1.19.c, d, e) improve those following from MacNeish's Corollary 1.7.8 and give the best result for the numbers 57 and 273 known to date. For 21, the better result 21 E TD(7) is known, see Corollary VIII.3.17.

615

§l. Product constructions Note that s = ~ v, as it must be.

= A. g~g~\)' hence the column number of A is sg(g -

1) + A.

=

1.18 Examples. (a) Let q be a prime power and assume the existence of a TD[k; q + 1] with a parallel class. Then there exists a (q2 + q + 1, k; 1)difference matrix in Zq2+q+! by Singer's Theorem VI. 1. 10. (b) There are (15, 7; 3)-, (21,5; 1)-, (40, 13; 4)-, and (57, 8; I)-difference matrices in the cyclic groups of the respective orders. The reader is asked to check this. (c) Let q be a prime power and let G be a group of order q + 1. Then there is a (q + 1, q; q - I)-difference matrix over G. To see this, use the trivial (q + 1, q, q - I)-difference set G \ {OJ. Note that this yields non-abelian difference matrices too (though of moderate size; the maximum feasible value of k is q2 - 1 by Corollary VIII.3.7).

It is worthwile to state the following consequence of Proposition 1.17.

1.19 Corollary. Assume the existence of a (v, g, I)-difference family. Then (1.19.a)

g E TD(k) =? v E TD(k).

In particular, (1.19.b)

q

+ 1 E TD(k)

=? q2

+q + 1E

TD(k)

for prime powers q.

Thus (1.19.c)

21

(1.19.d)

57 E TD(9),

(1.19.e)

273 E TD(18).

E

TD(6),

Proof. Use Lemma 1.1 together with (I.7.15.a) and Corollary VIII.3.8. The particular examples follow from 1.18.(a). • The values given in (1.19.c, d, e) improve those following from MacNeish's Corollary 1.7.8 and give the best result for the numbers 57 and 273 known to date. For 21, the better result 21 E TD(7) is known, see Corollary VIII.3.17.

617

§2. Use of pairwise balanced designs

§IV.ll.4 of Colboum and Dinitz (l996a). Further recursive constructions for difference matrices and difference families are due to Buratti (1998a). In this section we have seen how closely related the concepts of difference families and difference matrices are. The corresponding interaction between pairwise balanced designs and transversal designs will be of paramount importance in the recursive constructions of designs.

§2. Use of Pairwise Balanced Designs We shall present some important recursive constructions, in particular for PBD's and TD's. We use Ranani's notation (cf. 1.2.19).

2.1 Lemma. Let g

o ::s gi ::s g for i = (2.l.a)

kg

E TD(k + n) and k, k 1, ... , n. Then

+ 1, ... , k + n

+ g, + ... + gn E GD(K, {g, g" ~

E K; also, let

... , gnD

B(K U {g, g" ... , gnD.

Proof. Delete g - g" ... , g - gn points from the last n point classes of a TD[k +n; g] . •

2.2 Lemma. Let g gn + 1 E K; 0 < gi (2.2.a)

kg

E TD(k+n) and k, k+

1, ... , k+n, g + 1, g,

+ 1, ... ,

::s g for i = 1, ... ,n. Then

+ g, + ... + gn + 1 E

B(K).

Proof. Proceed as in Lemma 2.1 and use (I.6.5.b) . •

2.3 Theorem. ifv

E

B(L) and L ~ B(K), then v

E

B(K).

Proof. Let D = (V, {B" ... , Bb}, E) be an S(2, L; v). By hypothesis, there are PBD's Di = (B i , {Ci!, ... , Cib,}, E) with ICiJ-t1 E K for all i, f.L. Then

is the desired PBD. Wilson (l972a) calls this procedure "breaking up blocks" .

•

617

§2. Use of pairwise balanced designs

§IV.ll.4 of Colboum and Dinitz (l996a). Further recursive constructions for difference matrices and difference families are due to Buratti (1998a). In this section we have seen how closely related the concepts of difference families and difference matrices are. The corresponding interaction between pairwise balanced designs and transversal designs will be of paramount importance in the recursive constructions of designs.

§2. Use of Pairwise Balanced Designs We shall present some important recursive constructions, in particular for PBD's and TD's. We use Ranani's notation (cf. 1.2.19).

2.1 Lemma. Let g

o ::s gi ::s g for i = (2.l.a)

kg

E TD(k + n) and k, k 1, ... , n. Then

+ 1, ... , k + n

+ g, + ... + gn E GD(K, {g, g" ~

E K; also, let

... , gnD

B(K U {g, g" ... , gnD.

Proof. Delete g - g" ... , g - gn points from the last n point classes of a TD[k +n; g] . •

2.2 Lemma. Let g gn + 1 E K; 0 < gi (2.2.a)

kg

E TD(k+n) and k, k+

1, ... , k+n, g + 1, g,

+ 1, ... ,

::s g for i = 1, ... ,n. Then

+ g, + ... + gn + 1 E

B(K).

Proof. Proceed as in Lemma 2.1 and use (I.6.5.b) . •

2.3 Theorem. ifv

E

B(L) and L ~ B(K), then v

E

B(K).

Proof. Let D = (V, {B" ... , Bb}, E) be an S(2, L; v). By hypothesis, there are PBD's Di = (B i , {Ci!, ... , Cib,}, E) with ICiJ-t1 E K for all i, f.L. Then

is the desired PBD. Wilson (l972a) calls this procedure "breaking up blocks" .

•

§2. Use ojpairwise balanced designs

619

is a closure operator; i.e. it has the properties (2.7.b)

K

~

B(K),

(2.7.c)

K

~

B(L) :::} B(K)

~

B(L),

and hence (2.7.d)

= B(L)

B(B(L»

for all K, L ~ N. A subset K

~

N is called closed if B(K) = K or, in case of ambiguity, B-closed.

The concept of closed subsets of N was introduced by Wilson (1 972a, b). It is a very important tool which considerably simplified previous constructions of Ranani and other authors.

2.8 Examples. (a) For each K

~

N and A E N, the set B(K, A) is closed.

(b) The sets 6N + {I, 3}, 12N + {I, 4}, 20N + {I, 5} are closed. The next lemma generalises this example. ~

2.9 Lemma. Let K

N be given and define a, f3 by

(2.9.a)

a := gcd{k - 1 : k E K},

(2.9.b)

f3

:= gcd{k(k - 1) : k E K}.

Then

== 0 (mod a), I) == 0 (modf3)

(2.9.c)

A(V - 1)

(2.9.d)

AV(V -

are necessary conditions jor the existence ojan SA (2, K; v). Let L be the set (!f all v E N satisfying these two conditions. Then L is closed. Proof. The necessity of (2.9.c) and (2.9.d)follows by counting the flags (p, B) with P =F C and c, p I B for a given point c, and the triples (x, y, B) with x =F y and x, y lB. Note that Corollary I.2.11 is the special case K = {k}.

§2. Use ojpairwise balanced designs

619

is a closure operator; i.e. it has the properties (2.7.b)

K

~

B(K),

(2.7.c)

K

~

B(L) :::} B(K)

~

B(L),

and hence (2.7.d)

= B(L)

B(B(L»

for all K, L ~ N. A subset K

~

N is called closed if B(K) = K or, in case of ambiguity, B-closed.

The concept of closed subsets of N was introduced by Wilson (1 972a, b). It is a very important tool which considerably simplified previous constructions of Ranani and other authors.

2.8 Examples. (a) For each K

~

N and A E N, the set B(K, A) is closed.

(b) The sets 6N + {I, 3}, 12N + {I, 4}, 20N + {I, 5} are closed. The next lemma generalises this example. ~

2.9 Lemma. Let K

N be given and define a, f3 by

(2.9.a)

a := gcd{k - 1 : k E K},

(2.9.b)

f3

:= gcd{k(k - 1) : k E K}.

Then

== 0 (mod a), I) == 0 (modf3)

(2.9.c)

A(V - 1)

(2.9.d)

AV(V -

are necessary conditions jor the existence ojan SA (2, K; v). Let L be the set (!f all v E N satisfying these two conditions. Then L is closed. Proof. The necessity of (2.9.c) and (2.9.d)follows by counting the flags (p, B) with P =F C and c, p I B for a given point c, and the triples (x, y, B) with x =F y and x, y lB. Note that Corollary I.2.11 is the special case K = {k}.

§3. Applications of divisible designs

621

2.12 Examples. We use Theorem 2.11 and Lemma 2.1 with n = 1, 2. Thus 50 = 7·7 + 1

E

54 = 7·7 + 5

E B({5,

70 = 7·9 + 7

E

57 253

E

B({7, 8}) ~ TD*(7),

7, 8}) ~ TD*(5),

B({7, 8, 9}) ~ TD*(7),

B(8) ~ TD*(8),

E B({16,

13})

~ TD*(11),

cf. (VIII.5.6.b) with q = 4.

2.13 Proposition. (2.13.a) Proof. 6

B({3,4}) = (3 No

+ {l, 3}) \

{6}.

'1. B({3, 4}) follows from (I.8.4.a). "~" now follows from (2.9.c, d).

If g E K := B({3, 4}) and g E TD(4), gl E K U {O}, and gl :::0 g, then 3g + gl E K. This follows with Lemma 2.1, since K is closed. Note that 7 E B(3) c K, and that g E TD(4) if g =1= 2 (mod 4), by (I.7.8.a). The table

yields some small values of K, but leaves the gaps 18, 19, and33. But 19,33 E K by (I.6.11.a), and 18 E K by (1.5. 13.a). Now it is easily seen that each x == 0 or 1 (mod 3) with x > 52 has a representation x = 3g + gl with g, gl == 0 or 1 (mod 3), g =1= 2 (mod 4) and 7 :::0 gl :::0 g. Hence the assertion follows by induction. • The proof would be even shorter if one used result (1.7.9.b).

§3. Applications of Divisible Designs 3.1 Definition. A GD)..[K, G] is a divisible design D with parameter A, block sizes in K ~ N and point class sizes in G, and a GD)..[K] is a GD)..[K, NJ. Without loss of generality, we may assume that the point set is {I, 2, ... , v} c N, and that the (non-empty) point classes are G 1 , G2, ... , G s , where (3.1.a)

x

E

Gi ,

Y E Gj

and

i < j

=}

x < y.

§3. Applications of divisible designs

621

2.12 Examples. We use Theorem 2.11 and Lemma 2.1 with n = 1, 2. Thus 50 = 7·7 + 1

E

54 = 7·7 + 5

E B({5,

70 = 7·9 + 7

E

57 253

E

B({7, 8}) ~ TD*(7),

7, 8}) ~ TD*(5),

B({7, 8, 9}) ~ TD*(7),

B(8) ~ TD*(8),

E B({16,

13})

~ TD*(11),

cf. (VIII.5.6.b) with q = 4.

2.13 Proposition. (2.13.a) Proof. 6

B({3,4}) = (3 No

+ {l, 3}) \

{6}.

'1. B({3, 4}) follows from (I.8.4.a). "~" now follows from (2.9.c, d).

If g E K := B({3, 4}) and g E TD(4), gl E K U {O}, and gl :::0 g, then 3g + gl E K. This follows with Lemma 2.1, since K is closed. Note that 7 E B(3) c K, and that g E TD(4) if g =1= 2 (mod 4), by (I.7.8.a). The table

yields some small values of K, but leaves the gaps 18, 19, and33. But 19,33 E K by (I.6.11.a), and 18 E K by (1.5. 13.a). Now it is easily seen that each x == 0 or 1 (mod 3) with x > 52 has a representation x = 3g + gl with g, gl == 0 or 1 (mod 3), g =1= 2 (mod 4) and 7 :::0 gl :::0 g. Hence the assertion follows by induction. • The proof would be even shorter if one used result (1.7.9.b).

§3. Applications of Divisible Designs 3.1 Definition. A GD)..[K, G] is a divisible design D with parameter A, block sizes in K ~ N and point class sizes in G, and a GD)..[K] is a GD)..[K, NJ. Without loss of generality, we may assume that the point set is {I, 2, ... , v} c N, and that the (non-empty) point classes are G 1 , G2, ... , G s , where (3.1.a)

x

E

Gi ,

Y E Gj

and

i < j

=}

x < y.

623

§3. Applications of divisible designs

where each M(Xi, B) is a W(Xi) x bB-matrix such that M(Xi, B)M(Xi, B)T is diagonal and M (Xi, B)M (X j, Bl = J11 fori =I' j. If W(Xi) = 0, then the matrix M (Xi, B) is omitted. Now replace each I = axB in the incidence matrix (ax, B) of D by the auxiliary matrix M(x, B) defined by (3.2.b), and each 0= axB by a w(x) x bB-zero-matrix. In this case write 00 .. M(x,B)=

(

,0)

........

.

00 ... 0

Then the matrix

describes the desired GD).p.[K]. • 3.3 Remark. We may w.l.o.g. assume that each M (x, B) =I' 0 has a I in the upper left comer. If A. = J1 = 1, then E has a subspace which is isomorphic to D. (Consider only the upper left comers of all auxiliary matrices.) 3.4 Corollary. IfD is an S)..(2, H; v), thenE has class type (W(XI), ... , w(xv»· •

3.5 Corollary. Let D be a GD)..[H, G; v] (see I.6.1) and suppose that mk E GD/L(K, m) for each k E H. Then there is a GD)../L[K, mG; mv], say E. Here mG := {mx : X E G) ~ N.

Proof. Use Theorem 3.2 with w(x) = m for each x E V. Or directly: replace the l's in each column of the incidence matrix of D by the partial matrices Ml, ... , Mk (see Proposition 1.6.2) of the incidence matrix of the respective GD/L[K, m; mk] . •

3.6 Corollary. Let D be a GD)..[K, G; v] and suppose that (3.6.a)

m E TD/L(l)

for I := max K.

Then there is a GD)../L[k, mG; mv].

Proof. Condition (3.6.a) is equivalent to requiring that km GD/L(K, m) for each k E K. Now apply Corollary 3.5 . •

E

GD/L(k, m)

~

623

§3. Applications of divisible designs

where each M(Xi, B) is a W(Xi) x bB-matrix such that M(Xi, B)M(Xi, B)T is diagonal and M (Xi, B)M (X j, Bl = J11 fori =I' j. If W(Xi) = 0, then the matrix M (Xi, B) is omitted. Now replace each I = axB in the incidence matrix (ax, B) of D by the auxiliary matrix M(x, B) defined by (3.2.b), and each 0= axB by a w(x) x bB-zero-matrix. In this case write 00 .. M(x,B)=

(

,0)

........

.

00 ... 0

Then the matrix

describes the desired GD).p.[K]. • 3.3 Remark. We may w.l.o.g. assume that each M (x, B) =I' 0 has a I in the upper left comer. If A. = J1 = 1, then E has a subspace which is isomorphic to D. (Consider only the upper left comers of all auxiliary matrices.) 3.4 Corollary. IfD is an S)..(2, H; v), thenE has class type (W(XI), ... , w(xv»· •

3.5 Corollary. Let D be a GD)..[H, G; v] (see I.6.1) and suppose that mk E GD/L(K, m) for each k E H. Then there is a GD)../L[K, mG; mv], say E. Here mG := {mx : X E G) ~ N.

Proof. Use Theorem 3.2 with w(x) = m for each x E V. Or directly: replace the l's in each column of the incidence matrix of D by the partial matrices Ml, ... , Mk (see Proposition 1.6.2) of the incidence matrix of the respective GD/L[K, m; mk] . •

3.6 Corollary. Let D be a GD)..[K, G; v] and suppose that (3.6.a)

m E TD/L(l)

for I := max K.

Then there is a GD)../L[k, mG; mv].

Proof. Condition (3.6.a) is equivalent to requiring that km GD/L(K, m) for each k E K. Now apply Corollary 3.5 . •

E

GD/L(k, m)

~

§3. Applications of divisible designs

625

Proof. Put H:= R'lI:. w By hypothesis, mH = GD1.JK, m). Now apply Corollary 3.5 . • These theorems and lemmas are fairly general, and hence they need some explanation by examples. The following results are due to Wilson (1972b).

3.10 Corollary. The sets R'lI:,).. are closed. In particular the sets B(K, A) and Rk are closed. Proof. In the Main Lemma take G = {I}, A = 1. Then

By (3.7.b),

3.11 Lemma. Let K '" {I} be a non-empty subset of N. Put (3.1 1. a)

L := B(K, A),

and suppose that the positive integer m has the property

(3.11.b)

x=l(modm)

forallxEL.

Define Y S; No by

(3.11.c)

L

=

B(K, A)

= mY + 1.

Then, with the convention 0 E GD)..(X, Y)for all X, YEN,

(3.11.d)

GD(R'lI:.).., Y)

= Y.

Proof. By the Main Lemma 3.9, m GD(R~.).., Y) = GD)..(K, my) = GD)..(K, B(K, A) - 1).

With (2.6.b) and (3.11.c) the assertion follows, as trivially Y S; GD(R'K.).., Y) .

•

Note that the case m

= 1 is already contained in (2.6.b) and (2.S.a).

§3. Applications of divisible designs

625

Proof. Put H:= R'lI:. w By hypothesis, mH = GD1.JK, m). Now apply Corollary 3.5 . • These theorems and lemmas are fairly general, and hence they need some explanation by examples. The following results are due to Wilson (1972b).

3.10 Corollary. The sets R'lI:,).. are closed. In particular the sets B(K, A) and Rk are closed. Proof. In the Main Lemma take G = {I}, A = 1. Then

By (3.7.b),

3.11 Lemma. Let K '" {I} be a non-empty subset of N. Put (3.1 1. a)

L := B(K, A),

and suppose that the positive integer m has the property

(3.11.b)

x=l(modm)

forallxEL.

Define Y S; No by

(3.11.c)

L

=

B(K, A)

= mY + 1.

Then, with the convention 0 E GD)..(X, Y)for all X, YEN,

(3.11.d)

GD(R'lI:.).., Y)

= Y.

Proof. By the Main Lemma 3.9, m GD(R~.).., Y) = GD)..(K, my) = GD)..(K, B(K, A) - 1).

With (2.6.b) and (3.11.c) the assertion follows, as trivially Y S; GD(R'K.).., Y) .

•

Note that the case m

= 1 is already contained in (2.6.b) and (2.S.a).

627

§4. Applications of Hanani's lemmas

3.15 Notation. Let F K(U) denote the set of positive integers for which there exists an S(2, K; v) with a subspace of order u. For instance h(O) = FK(1) = B(K), Fk(k) = B(k),

h(u)=C/J

ifu 632, by (4.3.h). Choose g, mEN such that

n=7g+m, 7:::0 m :::0 76, and g E B(PIT), which is possible by (f). Since n :::: 608 we also have g :::: 76 :::: m,

n

1M7 g+ 76 E 1'I7g+7 '

and n E L by (4.3.d),

a contradiction. • As a corollary, we obtain the following theorem on the existence of five MOLS which is essentially due to Hanani (1970). For the small values not covered by the preceding arguments, see Table 7.1 in the appendix; with the exception of 52 and 55, constructions for all these values were given in Chapter VIII.

4.4 Theorem. One has (4.4.a)

P7 U{12, 21, 24, 28, 33, 35, 36}UN:gUN~~ UNnUN~ S; TD(7). 6I

The following result is due to Wilson (1974b) and Brouwer (1979b).

4.5 Proposition. One has (4.5.a)

B({4, 7)) ;2 (3No

+ 1) \

{lO, 19, 31}.

and 10,19 r:f. B{(4, 7)).

Proof. Use notation as in 3.11: K 3Y + 1. The assertion is (4.5.b)

Y

=

{4, 7),

J....

=

I

== No \ {3, 6, IOJ.

By (3.11.d) and (3.7.c), we have (4.5.c)

Ri,

=

GD(Ri, 1)

S;

GD(Ri, Y)

=

Y.

1, m

=

3, L

= B(K) =

633

§4. Applications of Hanani's lemmas

Using this for g = 4, 5, 7, and 8, together with the previous results, we obtain (4.5.n)

NjQ \ {3, 6, 1O}

c

Y.

Attaching two or seven "infinite" points to a resolvable S(2, 4; 40) yields (4.5.0)

42,47 E GD({4, 5}, {1, 2, 7}) S; Y,

cf. (VII.7.5.a). Similarly, 54,66,78,90 (4.5.p)

E

Y. By (4.5.g),

58,67,70,82 E Y.

Using (4.S.m) with g = 9,11,12,13,16,17, 19,21,23,25,27, 6n± l(n :::: 5) yields the assertion. • Brouwer (1979b) used refined methods to get the following sharper result. If v = 12n + 7 or v = 12n - 2 (n > 1), then there is an S(2, {4, 7}; v) with exactly one line of size 7. In particular, 31 E B({4, 7}).

4.6 Proposition. (4.6.a)

L := B({4, 5, 8, 9, 12}) = 4N + {I, 4}.

Proof. First we note 13, 28 E B(4) S; L; cf. (VII.3.2.a). Also, 29 = 28 + 1 E GD(4, 7)

+ 1 S; B({4, 8}) c

L.

Again we use Lemma 2.1; here (4.6.b)

4g

+ u E L,

if gEL n TD(5), U E L U {O}, and u :'S g. Using this for g = 4,5,8,9, 12, 12n + 1, 12n + 4, 12n + 5, 12n + 8, and 12n + 13 = 12(n + 1) + 1 proves the assertion. •

4.7 Lemma. Let P be the set of prime powers, and put K := Pt. Then (4.7.a)

B(K) = N \ ({2, 3,6, 10, 12, 14, 15, 18} U X),

where X S; {26, 30, 38, 42}.

Proof. Put L := B(K). Again, (4.6.b) holds. Since we already know (4.3.a), only the values in N \ (Pt! U Nti}) are in doubt. By (4.6.b), we get

633

§4. Applications of Hanani's lemmas

Using this for g = 4, 5, 7, and 8, together with the previous results, we obtain (4.5.n)

NjQ \ {3, 6, 1O}

c

Y.

Attaching two or seven "infinite" points to a resolvable S(2, 4; 40) yields (4.5.0)

42,47 E GD({4, 5}, {1, 2, 7}) S; Y,

cf. (VII.7.5.a). Similarly, 54,66,78,90 (4.5.p)

E

Y. By (4.5.g),

58,67,70,82 E Y.

Using (4.S.m) with g = 9,11,12,13,16,17, 19,21,23,25,27, 6n± l(n :::: 5) yields the assertion. • Brouwer (1979b) used refined methods to get the following sharper result. If v = 12n + 7 or v = 12n - 2 (n > 1), then there is an S(2, {4, 7}; v) with exactly one line of size 7. In particular, 31 E B({4, 7}).

4.6 Proposition. (4.6.a)

L := B({4, 5, 8, 9, 12}) = 4N + {I, 4}.

Proof. First we note 13, 28 E B(4) S; L; cf. (VII.3.2.a). Also, 29 = 28 + 1 E GD(4, 7)

+ 1 S; B({4, 8}) c

L.

Again we use Lemma 2.1; here (4.6.b)

4g

+ u E L,

if gEL n TD(5), U E L U {O}, and u :'S g. Using this for g = 4,5,8,9, 12, 12n + 1, 12n + 4, 12n + 5, 12n + 8, and 12n + 13 = 12(n + 1) + 1 proves the assertion. •

4.7 Lemma. Let P be the set of prime powers, and put K := Pt. Then (4.7.a)

B(K) = N \ ({2, 3,6, 10, 12, 14, 15, 18} U X),

where X S; {26, 30, 38, 42}.

Proof. Put L := B(K). Again, (4.6.b) holds. Since we already know (4.3.a), only the values in N \ (Pt! U Nti}) are in doubt. By (4.6.b), we get

§4. Applications of Hanani 's lemmas

635

4.9 Theorem. For every n 9'. {2, 3, 6}, there are two idempotent MOLS of order n, andfor n 9'. {2, 6}, there are two MOLS of order n . • 4.10 Remark. In order to prove Theorem 4.9, one may avoid the cumbersome constructions of self-orthogonal Latin squares given in Example VIll.4.9 by using the MOLS constructed in Examples Vlll.4.17.(b) and (d).

4.11 Remark. There is a wealth of results similar to Propositions 4.1 and 4.2. For instance, Gronau, Mullin and Pietsch (1995) determined the closure of all subsets of (3, 4, ... , lO} which include 3. Other important references include Drake and Larson (1983), Lenz (1984), Colbourn and Ling (1997), Ling, Zhu et a1. (1997) and Mullin, Linget a1. (1997). A big list of closures can be found in Table 1ll.3.18 of Colbourn and Dinitz (1996a). In what follows, we report on some results related to Lemma 4.3 and Theorem 4.4, where we considered the PBD-closure B(L) of the set L = {q: q :::: 7 a prime power}. Lemma 4.3 was rediscovered by Bennett (1987a); it is not too difficult to see that one actually has equality in this result. We now list some results concerning the closure of certain interesting subsets of the set of all prime powers. 4.12 Theorem. Let Pq and OPq denote the set ofprime powers ::::q and of odd prime powers ::::q, respectively. Then one has: (4. 12.a)

B(P3) = N \ {2, 6}

(4.12.b)

B(P4 ) ;2 N \ {2, 3, 6,10, 12, 14, 15, 18,26, 30}

(4.l2.c)

B(Ps) ;2 N \ {2, 3, 6,10, 12, 14, 15, 18,20,22,24,26,28,

(4. 12.d)

B(P7) = p~l U {50, 56, 57, 58} U N6'3

(4. 12.e)

B(OPs ) S; (2N + 1) \ {IS, 33, 39, 51, 75,87,183, 195} . •

30,33,34,38, 39,42,44,46,51, 52, 60}

Here (4.12.a) is immediate from Proposition 4.2, while (4.12.b, d) just restate Lemmas 4.3 and 4.7. Result (4.12.e) essentially is due to Mullin and Stinson (1984), with some improvements by Bennett (1990a) who also proved (4.12.c). Similarly, Bennett (1987b) proved that B(Ps) contains every v :::: 2207, with 675 possible exceptions below this bound. Mullin and Stinson (1990) studied B (0 P7) and proved that this set contains all odd v :::: 2129, with 103 possible exceptions below this bound. These results can be somewhat improved using work of Greig (1998c), see Table III.3.18 in Colbourn and Dinitz (1996a).

§4. Applications of Hanani 's lemmas

635

4.9 Theorem. For every n 9'. {2, 3, 6}, there are two idempotent MOLS of order n, andfor n 9'. {2, 6}, there are two MOLS of order n . • 4.10 Remark. In order to prove Theorem 4.9, one may avoid the cumbersome constructions of self-orthogonal Latin squares given in Example VIll.4.9 by using the MOLS constructed in Examples Vlll.4.17.(b) and (d).

4.11 Remark. There is a wealth of results similar to Propositions 4.1 and 4.2. For instance, Gronau, Mullin and Pietsch (1995) determined the closure of all subsets of (3, 4, ... , lO} which include 3. Other important references include Drake and Larson (1983), Lenz (1984), Colbourn and Ling (1997), Ling, Zhu et a1. (1997) and Mullin, Linget a1. (1997). A big list of closures can be found in Table 1ll.3.18 of Colbourn and Dinitz (1996a). In what follows, we report on some results related to Lemma 4.3 and Theorem 4.4, where we considered the PBD-closure B(L) of the set L = {q: q :::: 7 a prime power}. Lemma 4.3 was rediscovered by Bennett (1987a); it is not too difficult to see that one actually has equality in this result. We now list some results concerning the closure of certain interesting subsets of the set of all prime powers. 4.12 Theorem. Let Pq and OPq denote the set ofprime powers ::::q and of odd prime powers ::::q, respectively. Then one has: (4. 12.a)

B(P3) = N \ {2, 6}

(4.12.b)

B(P4 ) ;2 N \ {2, 3, 6,10, 12, 14, 15, 18,26, 30}

(4.l2.c)

B(Ps) ;2 N \ {2, 3, 6,10, 12, 14, 15, 18,20,22,24,26,28,

(4. 12.d)

B(P7) = p~l U {50, 56, 57, 58} U N6'3

(4. 12.e)

B(OPs ) S; (2N + 1) \ {IS, 33, 39, 51, 75,87,183, 195} . •

30,33,34,38, 39,42,44,46,51, 52, 60}

Here (4.12.a) is immediate from Proposition 4.2, while (4.12.b, d) just restate Lemmas 4.3 and 4.7. Result (4.12.e) essentially is due to Mullin and Stinson (1984), with some improvements by Bennett (1990a) who also proved (4.12.c). Similarly, Bennett (1987b) proved that B(Ps) contains every v :::: 2207, with 675 possible exceptions below this bound. Mullin and Stinson (1990) studied B (0 P7) and proved that this set contains all odd v :::: 2129, with 103 possible exceptions below this bound. These results can be somewhat improved using work of Greig (1998c), see Table III.3.18 in Colbourn and Dinitz (1996a).

§5. Block designs of block size three and four

637

5.3 Lemma. (5.3.a)

B(3,2) = 3No + to, I}.

Proof. Trivially 3, 4

6

E

B(3, 2). By (I.5.14.a),

E B(3, 2).

Since B(3, 2) is closed, the assertion follows with (2.l3.iIl) •

5.4 Lemma. (5.4.a)

B(3, 6) = N \ {2}.

Proof. Obviously B(3, 6) ::2 B(3, 2) U B(3, 3). Hence (5.4.b)

3,4,5,6

E

B(3, 6).

Since B(3, 6) is closed, (4.2.a) yields B(3, 6) ::2 N \ {2, 8}, and the trivial S6(2, 3; 8) takes care of the value 8. •

5.5 Theorem. The necessary existence conditions for an S).(2, 3; v) (Ire sufficient. Proof. As in VII.4.l2. • 5.6 Lemma. (S.6.a)

R4 = 4No + {I, 4}

(S.6.b)

B(4) = 12No + {I, 4}.

or, equivalently,

Proof. We know that 13, 16, 2S, 28, 37 E B(4); cf. (VII.3.2.a). Hence (S.6.c)

4, S, 8, 9, 12 E R4 .

Since R4 is closed, the assertion follows by (4.6.a) . •

5.7 Lemma. (S.7.a)

B(4, 2) = 3No + l.

§5. Block designs of block size three and four

637

5.3 Lemma. (5.3.a)

B(3,2) = 3No + to, I}.

Proof. Trivially 3, 4

6

E

B(3, 2). By (I.5.14.a),

E B(3, 2).

Since B(3, 2) is closed, the assertion follows with (2.l3.iIl) •

5.4 Lemma. (5.4.a)

B(3, 6) = N \ {2}.

Proof. Obviously B(3, 6) ::2 B(3, 2) U B(3, 3). Hence (5.4.b)

3,4,5,6

E

B(3, 6).

Since B(3, 6) is closed, (4.2.a) yields B(3, 6) ::2 N \ {2, 8}, and the trivial S6(2, 3; 8) takes care of the value 8. •

5.5 Theorem. The necessary existence conditions for an S).(2, 3; v) (Ire sufficient. Proof. As in VII.4.l2. • 5.6 Lemma. (S.6.a)

R4 = 4No + {I, 4}

(S.6.b)

B(4) = 12No + {I, 4}.

or, equivalently,

Proof. We know that 13, 16, 2S, 28, 37 E B(4); cf. (VII.3.2.a). Hence (S.6.c)

4, S, 8, 9, 12 E R4 .

Since R4 is closed, the assertion follows by (4.6.a) . •

5.7 Lemma. (S.7.a)

B(4, 2) = 3No + l.

639

§5. Block designs of block size three and four

in Zv-l U (oo} determine an S6(2, 4; v). In particular 6, 12, 14,18,26,30,38,42 By (5.7.b), we know that 10

E

E

B(4, 6).

B(4, 6).

(d) In Z15 put A := to, 2, 3, ll}, B := to, 1,5, 1O}, C := (A, A, A, A, A, B, C) is a (15, 4; 6)-difference family. •

to, 3, 5,

1O}. Then

The preceding results allow us to prove the following theorem due to Hanani (1961, 1975).

5.10 Theorem. The necessary existence conditions v i= 2,3, J..(v -1) =

o(mod 3),

and

J..v(v -1) =

o(mod 12)

for an S')., (2, 4; v) are sufficient.

Proof. Note that the necessary existence conditions for J.. and gcd(J.., 6) are identical. For J.. I 6 the theorem is true by the previous lemmas. Hence it follows in general by (1.5.3.b) . • 5.11 Remark. We emphasise that Hanani's existence theorems 5.5 and 5.10 allow repeated blocks. The existence problem for simple designs is more difficult. We ask the reader to show that the necessary conditions for the existence of a simple design S').,(2, k; v) are as follows: (5.Il.a)

J..(v - 1) =0 (modk -1), J..v(v-I)=O(modk(k-I»

and

J..::sG=~).

By a result of Dehon (1983), these conditions are also sufficient for k = 3.

5.12 Theorem. A simple triple system S),,(2, 3; v) exists if and only if (5.I2.a)

J..(v -1) =0(mod2),

J..v(v -1) =0(mod6)

and

J..::sv - 2. II

Another proof of Theorem 5.12 was given by Sarvate (1986). A simple direct construction (similar to the one given in Theorem VIIA.9, but using four infinite points) for simple S2 (2,3, v)' s was provided by Bhat-Nayak and Kane (1988). We also have the following strengthening of Theorem 5.12, namely an analogue

639

§5. Block designs of block size three and four

in Zv-l U (oo} determine an S6(2, 4; v). In particular 6, 12, 14,18,26,30,38,42 By (5.7.b), we know that 10

E

E

B(4, 6).

B(4, 6).

(d) In Z15 put A := to, 2, 3, ll}, B := to, 1,5, 1O}, C := (A, A, A, A, A, B, C) is a (15, 4; 6)-difference family. •

to, 3, 5,

1O}. Then

The preceding results allow us to prove the following theorem due to Hanani (1961, 1975).

5.10 Theorem. The necessary existence conditions v i= 2,3, J..(v -1) =

o(mod 3),

and

J..v(v -1) =

o(mod 12)

for an S')., (2, 4; v) are sufficient.

Proof. Note that the necessary existence conditions for J.. and gcd(J.., 6) are identical. For J.. I 6 the theorem is true by the previous lemmas. Hence it follows in general by (1.5.3.b) . • 5.11 Remark. We emphasise that Hanani's existence theorems 5.5 and 5.10 allow repeated blocks. The existence problem for simple designs is more difficult. We ask the reader to show that the necessary conditions for the existence of a simple design S').,(2, k; v) are as follows: (5.Il.a)

J..(v - 1) =0 (modk -1), J..v(v-I)=O(modk(k-I»

and

J..::sG=~).

By a result of Dehon (1983), these conditions are also sufficient for k = 3.

5.12 Theorem. A simple triple system S),,(2, 3; v) exists if and only if (5.I2.a)

J..(v -1) =0(mod2),

J..v(v -1) =0(mod6)

and

J..::sv - 2. II

Another proof of Theorem 5.12 was given by Sarvate (1986). A simple direct construction (similar to the one given in Theorem VIIA.9, but using four infinite points) for simple S2 (2,3, v)' s was provided by Bhat-Nayak and Kane (1988). We also have the following strengthening of Theorem 5.12, namely an analogue

§6. Solution of Kirkman's schoolgirl problem

641

(b) A topic which is somewhat related to the construction of simple designs is the existence question for indecomposable designs; here an SA (2, k; v) is called indecomposable if it is not the union of two designs SA (2, k; v) and Sv(2, k; v) with A = J1- + v. Many of the references cited above actually consider indecomposable simple designs. For a construction of indecomposable simple triple systems SA (2, 3; v) for all values.of Aand for all sufficiently large values ofv (given A), see Archdeacon and Dinitz (1993). Griittmiiller (1996) gave computational results on the numbers of indecomposable block designs with small parameters.

§6. Solution of Kirkman's Schoolgirl Problem Ray-Chaudhuri and Wilson (1971) proved RB(3) = 6N - 3, i.e. for every v == 3 (mod 6) there is a Kirkman design, i.e. a resolvable STS. The proof will need several steps.

6.1 Observation. If q (6. 1. a)

2q

==

1 (mod 6) is a prime power, then

+ 1, 3q E RB(3).

This was proved in Theorem VIII.2.1 and Corollary VIll.2.3. In particular, (6.l.b)

15,21,39,63 E RB(3)

and, of course, (6.l.c)

9 E RB(3).

6.2 Notation. (6.2.a)

RZ := {x EN: (k - l)x

+ 1 E RB(k)}.

Ray-Chaudhuri and Wilson (1971) proved that the sets R;; are closed. This enabled them to determine RB(3) and later RB(4) completely; cf. Ranani, RayChaudhuri and Wilson (1972). The sets R;; and RB(k) for k > 4 are unknown. If we know that Rj is closed, then the assertion (6.2.b)

RB(3)

= 6N -

3

follows immediately with (6.1.b, c), which imply 4,7, 10, 19,31 and (4.5.a).

E

Rj,

§6. Solution of Kirkman's schoolgirl problem

641

(b) A topic which is somewhat related to the construction of simple designs is the existence question for indecomposable designs; here an SA (2, k; v) is called indecomposable if it is not the union of two designs SA (2, k; v) and Sv(2, k; v) with A = J1- + v. Many of the references cited above actually consider indecomposable simple designs. For a construction of indecomposable simple triple systems SA (2, 3; v) for all values.of Aand for all sufficiently large values ofv (given A), see Archdeacon and Dinitz (1993). Griittmiiller (1996) gave computational results on the numbers of indecomposable block designs with small parameters.

§6. Solution of Kirkman's Schoolgirl Problem Ray-Chaudhuri and Wilson (1971) proved RB(3) = 6N - 3, i.e. for every v == 3 (mod 6) there is a Kirkman design, i.e. a resolvable STS. The proof will need several steps.

6.1 Observation. If q (6. 1. a)

2q

==

1 (mod 6) is a prime power, then

+ 1, 3q E RB(3).

This was proved in Theorem VIII.2.1 and Corollary VIll.2.3. In particular, (6.l.b)

15,21,39,63 E RB(3)

and, of course, (6.l.c)

9 E RB(3).

6.2 Notation. (6.2.a)

RZ := {x EN: (k - l)x

+ 1 E RB(k)}.

Ray-Chaudhuri and Wilson (1971) proved that the sets R;; are closed. This enabled them to determine RB(3) and later RB(4) completely; cf. Ranani, RayChaudhuri and Wilson (1972). The sets R;; and RB(k) for k > 4 are unknown. If we know that Rj is closed, then the assertion (6.2.b)

RB(3)

= 6N -

3

follows immediately with (6.1.b, c), which imply 4,7, 10, 19,31 and (4.5.a).

E

Rj,

§6. Solution of Kirkman's schoolgirl problem

643

Here the k-row submatrices 1 (6.4.b)

o

(i = 1, ... , r)

M' I

o satisfy the equations Mi MJ = J for i =I j, see 1.6.2.

= (aij) be the incidence matrix of an S(2, Rk;v) with block sizes (j = 1, ... , b). For any column with rj =: r entries 1 we take the submatrices M; of (6.4.a) as auxiliary matrices, that is we replace these l's by

Now let A rj E

Rk

the matrices M; in any order, and the O's in the same column by zero matrices of the same size. We get a vk-row large matrix N' which splits into k-row matrices N;, ... , N~; that is

....~.; ..... N='~ . .. ...... ... .

N'v Now put the column (1 0 0 ...

of in front of each submatrix N; and get

Ni=UQ~ :

N[

fori=I, ... ,v,

o

N~6d Then N is the incidence matrix of a GD[{k + 1, v}, k; kv] which actually contains lines of sizes v and k + 1. The reader is asked to check this. Hence, by Lemma 6.3, v E Rk, and Lemma 6.4 is proved. • As noted before, this finishes the proof of the following theorem of RayChaudhuri and Wilson (1971).

§6. Solution of Kirkman's schoolgirl problem

643

Here the k-row submatrices 1 (6.4.b)

o

(i = 1, ... , r)

M' I

o satisfy the equations Mi MJ = J for i =I j, see 1.6.2.

= (aij) be the incidence matrix of an S(2, Rk;v) with block sizes (j = 1, ... , b). For any column with rj =: r entries 1 we take the submatrices M; of (6.4.a) as auxiliary matrices, that is we replace these l's by

Now let A rj E

Rk

the matrices M; in any order, and the O's in the same column by zero matrices of the same size. We get a vk-row large matrix N' which splits into k-row matrices N;, ... , N~; that is

....~.; ..... N='~ . .. ...... ... .

N'v Now put the column (1 0 0 ...

of in front of each submatrix N; and get

Ni=UQ~ :

N[

fori=I, ... ,v,

o

N~6d Then N is the incidence matrix of a GD[{k + 1, v}, k; kv] which actually contains lines of sizes v and k + 1. The reader is asked to check this. Hence, by Lemma 6.3, v E Rk, and Lemma 6.4 is proved. • As noted before, this finishes the proof of the following theorem of RayChaudhuri and Wilson (1971).

§7. The basis of a closed set

645

7.1 Problem and Examples. For a given closed set L 0. If we again use the difference triples of Example 1 or 2, respectively, and proceed exactly as before we fail, since the resulting graph reD) has degree 6h + 5 which cannot be the order of an STS. We deal with this problem by adding further lines to D in such a way that the new (smaller) graph remains cyclic but has a suitable (smaller) degree. This requires another case distinction, depending on the congruence class of u modulo 6. Case 4a: u

== 3 (mod 6), say u = {i, 2m

6h

+ i, 4m + i}

+ 3. We introduce 2m new lines (i = 0,1, ... , 2m - 1).

IX. Recursive constructions

676

This removes all edges of the form {i, 2m +i} from reD). Hence the resulting graph is cyclic of degree u = 6h + 3 with 0 < h :::: s. Case 4b: u == 1 (mod 6), say u = 6h + 1. In this case, we select (2t -1, 2t, 4t - 1) or (2t -1, 2t + 1, 4t + 1), respectively, as one of the h difference triples which are deleted in the construction of D. Then we introduce the following 4m new lines (with i

= 0, 1, ... , m -

1):

+ i, 2m + i}; {4m + i, 5m + i, i};

{i, m

+ i, 3m + i, 4m + i}; {m + i, 3m + i, 5m + i}.

{2m

The resulting RPTS is not cyclic, but its graph is cyclic, as any pair of the form {x, m + x} or {x, 2m + x} is contained in exactly one of the new lines. The degree of the new graph is u = 6h + 1 with 0< h:::: s. This finishes Step 2. (The reader should check that our case distinction is indeed complete.) As already noted, Steps 3, 4, and 5 are now obvious . • The Doyen-Wilson theorem is a seminal result which prompted much research on the analogous problem for larger values of k and on related questions. We now discuss the most important results in this direction, beginning with the classical subspace problem already mentioned in Remark 1.8.12. The combined efforts of Rees and Stinson (1989a, b) and Wei and Zhu (1987,1989) settle the case k = 4. Thus we have the following result.

11.4 Theorem. The necessary conditions (1 1. 1. a) for the existence of an S(2,4; v) with a sub-S(2, 4; u) are sufficient. • Some results on the number of maximal subsystems in an S (2, 4; v) were given by Zeitler (1993). Bennett and Yin (1995) considered the existence problem for S(2, 5; v) with a sub-S(2, 5; u) and proved that the necessary conditions are sufficient in the case u == v (mod 20) whenever v 2: 5w - 4, with the possible exception of the pair (425, 65). We now discuss a problem related to the subspace problem, namely the existence oflinear spaces where all lines but one (which is of size u) have size k; we shall denote such a PBD by S(2, {k, u*}; v). In the special case where there exists an S(2, k; u), this problem is clearly equivalent to the existence problem for an S(2, k; v) with a sub-S(2, k; u) just discussed. We also note that the existence

§11. Embedding theorems

677

of a resolvable design RS(2, k; v) is equivalent to that of an

S(2, {k + 1, r*}; v + r),

where r = (v - l)/(k - 1),

by Proposition 1.8.7. Rees and Stinson (1989b) and Mills (1989) considered the case k = 4 and proved the following result.

11.5 Theorem. An S(2, {4, u*}; v) exists if and only if either (1l.1.a) holds (with k = 4) or ifv :::: 3u + 1 and u, v == 7 or 10 (mod 12). • The preceding problem has also been studied for k = 5, though considerably less is known in this case. Assaf (1994), Hamel, Mills et al. (1993) and Bennett and Yin (1995) prove that the necessary conditions for the existence of an S(2, {5, u*}; v) are sufficient for the values u' E {9, 13, 17,21,25,29, 33}, with 26 possible exceptions. Hartman and Heinrich (1993) and Chee, Colbourn et al. (1997) completely settle the existence question for PBDs with one block of prescribed size u and all other blocks of size at least 3, that is (in self-explanatory notation) for S(2, {[3, 00], u*}; v). Note that there may be more than one block of size u in such a linear space. Colboum, Haddad and Linek (1996) show the existence of S(2, {{3, 5, 7, ... }, u*}; v) for all odd integers v and u* satisfying v:::: 2k + 1. Bennett, Du andZhu (1990) settle the existence problem for S(2, {4, 5, 11 *}; v) up to 33 possible exceptions. Theorem 11.5 deals with PBD's with two block sizes where both these values have to occur. More generally, a mandatory representation design MR(K; v) is a PBD on v points with line sizes from K in which for each k E K there actually is a line of size k in the design. The study of these designs was initiated by Mendelsohn and Rees (1988) who considered the special case where 3 E K. In particular, these authors determined the spectrum for MR({3, k}; v) for any odd integer k. Similar questions have been studied for designs with arbitrary A. Here the analogue of the Doyen-Wilson theorem for arbitrary triple systems is due to Stem (1979); see also Bigelow and Colboum (1993) for a generalisation of this result where "embeddings" of triple systems into systems with a larger A-value are considered.

11.6 Theorem. An SA (2,3; v) with a sub-SA (2, 3; v) exists if and only if both u and v are admissible (that is, satisfy the necessary arithmetic conditions for the existence of the respective triple system) and v :::: 2u + 1. •

678

IX. Recursive constructions

The analogous result for the case k = 4 is due to Wei (1991), Rees and Rodger (1993) and Kong and Zhu (1994), and an analogue of Theorem 11.5 for the case A. > 1 was given by Kong and Wei (1993). We only state the first of these results.

11.7 Theorem. An S).(2, 4; v) with a sub-S).(2, 4; v) exists ifand only ifboth u and v are admissible (that is, satisfy the necessary arithmetic conditions for the existence of the respective design) and v 2: 3u + 1. • The analogous problem for Steiner quadruple systems if far from being solved; the best known results are due to Granville and Hartman (1991). Of course, one may also consider subdesigns while imposing extra structural conditions. For example, cyclic STS with cyclic sub-STS were considered by Phelps, Rosa and Mendelsohn (1989). Some results on Kirkman triple systems with sub-KTS's were already discussed at the end of §6. Some general existence results for resolvable Steiner systems RS(2, 4; v) with resolvable sub-systems RS(2, 4; u) were obtained by Cai (1990). Another related problem concerns the "embedding" of a "partial" design. Here a partial design PS).(t, k; v) is defined in the obvious way as an incidence structure on v points with blocks of size k for which any t points are contained in at most A. blocks. Note that this definition is not essentially different from the one we gave for the special case of partial triple systems in Definition 11.2 above: we just have to omit all lines of size 2. The problem of embedding a PSl.(t, k; v) into some design S).(t, k; v*) by adding further points and blocks has been studied extensively for various types of designs. We emphasise that v* may be much larger than the original v, though of course embeddings with small values of v* are of particular interest. We refer the reader to Lindner (1980) for a good survey for the special case of partial triple systems and only state a few major theorems. The seminal result in this area is the theorem of Treash (1971) which states that every PTS can be embedded into a finite STS. This is a special case of the following more general result due to Ganter (1976a).

11.8 Theorem. Any partial Steiner system PS(2, k; u) can always be embedded into an S(2, k; v) (jor a suitable value of v) . • Regarding Steiner systems with t 2: 3, one has the following analogous result of Ganter (1974) for quadruple systems.

11.9 Theorem. Any partial Steiner quadruple system can be embedded into a Steiner quadruple system SQS( v) (jor a suitable value of v). •

§1l. Embedding theorems

679

As mentioned above, embeddings with small values of v are of particular interest. For partial Steiner triple systems, the best known lower bound is due to Andersen, Hilton and Mendelsohn (1980). The corresponding result for partial triple systems with arbitrary index A is due to Rodger and Stubbs (1987) and Hilton and Rodger (1991).ltis easy to see that v :::: 2u + 1 is always a necessary condition; thus the known results we quote now are quite good.

11.10 Theorem. Any partial Steiner triple system on u points can be embedded into an STS( v) for all v satisfying v :::: 4u + I and v == 1 or 3 (mod 6). • 11.11 Theorem. Any partial design S).'-(2, 3, v) for all admissible v satisfying v :::: 4u + I.lf A is a multiple of 4, the bound on v can be improved to v :::: 2u + 1. • There has also been considerable interest in the problem of embedding designs into designs with a larger block size. We now report on some results in this direction, beginning with the rather special (but popular) problem of "nesting". Here we let D = (V, B) bean S>-(2, 3; v) and assume the existence ofa mapping f:B -+ V such that (V, {B U feB): B E B}) is an SJl.(2, 4, v); then fis called a nesting of D. The combined efforts of Colboum and Colboum (1983) and Stinson (1985) have settled this problem. The reader may check as an exercise that the condition given for this to be parametrically feasible below is indeed necessary. For a similar construction in the case of Steiner triple systems with v == 3 (mod 6), see Lindner and Rodger (1987).

11.12 Theorem. An S>-(2, 3; v) which is nested into an SJl.(2, 4, v) exists ifand only if

(11.12.a)

A(V - 1)

== 0 (mod 6)

and

/h = 2A . •

11.13 Example. As an example for nestings, we give a particularly simple construction due to Stinson (1985) which covers the case A = 6. Thus let v :::: 4 and assume v # 6. (The case v = 6 needs special treatment and will not be considered here.) By Theorem 4.9, there exists a pair of orthogonal Latin squares of order v and hence an orthogonal array OA(4, v), say A, which we assume to be in normal form, see Observation VIll.4.5. Discard the v columns of the form (i, i, i, i) from A, and use the remaining columns as blocks. Clearly, the result is an Sd2, 4, v), say D. Now omit from every block of D the symbol that came from the last row of A. This yields a triple system S6(2, 3; v) which is nested into D.

680

IX. Recursive constructions

The final topic which We will discuss in this section, namely embedding Steiner systems into projective planes, has generated much interest for geometric reasons. The most important case concerns a subset S in a projective plane P such that the lines of P induce on S a Steiner system. Obvious examples include AG(2, q) in PG(2, q), PG1(d, q) in PG1(e, q) or AGl(d, q) in AG1 (e, q) for e > d as well as subplanes and subgeometries of projective or affine planes and spaces, respectively. A series ofless obvious examples is provided by AG(2, 3) in PG(2, q) for q == 1 (mod 3), a result due to Ostrom and Sherk (1964); see also Rigby (1965), Vedder (1981) and Abduhl-Elah, Al-Dhahir and Jungnicke1 (1987), where the number of affine planes of order 3 in PG(2, q) is determined. Two further large classes of examples are given by the maximal arcs and by the unitals in PG(2, q2) (and other projective planes) which were discussed in §VIII.5. It is conjectured that every Steiner system (and, more generally, every linear space) admits an embedding into a finite projective plane. For some very general results in this direction, we refer the reader to Beutelspacher and Metsch (1986, 1987), Metsch (1991a) and the monographs Batten and Beutelspacher (1993) and Metsch (1991b). As an example of what type of results can be obtained, we state a theorem due to Metsch (1991a).

11.14 Theorem. Let D be a linear space with b lines, and assume the existence of an integer n 2:: 2 such that every point is on exactly n + I lines. Denote the smallest and the largest line size in D by n + 1 - A and n + 1 - a, respectively. Then there is an absolute constant c such that D can be extended to a projective plane of order n whenever n 2:: ca 3 • • An explicit value for the constant c occurring in Theorem 11.14 may be determined in every particular case. As an example, we mention the following consequence of Theorem 11.14 which characterises the complements of maximal k-arcs in projective planes; this is also due to Metsch (1991a).

11.15 Theorem. Let D be a linear space, and let nand k be positive integers for whichk divides n. Assume that every pointofD lies on n/ k lines of degree n+ 1 andonn + I-(n/ k) linesofsizen + 1 - k.lfonehas3n > (k-I)(8k 2 -2k-4), then L can be obtained as the complement of a maximal k-arc in a projective plane of order n. • Finally, we mention a completely different approach to embedding Steiner systems into a projective plane P. Jungnickel and Vanstone (1991) considered

§12. Concluding remarks

681

the existence of a Steiner triple system on the point set of P such that each block of the STS is a triangle of P. They proved the following result.

11.16 Theorem. Let G be a cyclic Singer group for the Desarguesian projective plane PG(2, q), as in Theorem Ill.6.2. Then there exists an STS(q2 +q + 1) on the point set ofPG(2, q) which is invariant under G and only contains blocks which are triangles in PG(2, q). • More generally, one might ask to embed an S(2, k; v), say D, with v q2 + q + 1 in such a way into PG(2, q) that the blocks of D are arcs of PG(2, q) (and, possibly, such that D is invariant under a Singer group or some other nice type of automorphism group of PG(2, If one allows "non-collinear embeddings" in the sense just discussed, one might even find interesting embeddings of designs with A#-1 into a projectiveplane. We mention the following example due to Bruen (1989). A similar result concerning the spherical 3-designs was already mentioned in Remark III.6.lD.

q».

11.17 Theorem. Let q ::: 5 be an odd prime power. Then there exist a set S of16 points and a set C of 16 conics in PG(2, q) such that the 16 intersections S n C (with C E C) form a biplane, that is an S2(2, 6; 16). •

§12. Concluding Remarks

In this final section of Chapter IX, we discuss some further general existence results beyond those dealt with in more detail in the preceding sections. In particular, we consider 2-designs, resolvable 2-designs, quadruple systems and some related structures, for instance near-resolvable designs, coverings and packings.

Let us begin by discussing the existence problem for (resolvable) 2-designs. As we shall see in Chapter XI, the necessary existence conditions are asymptotically sufficient in this case. However, the resulting explicit bounds are astronomic, leaving a lot of difficult work if one wants to completely settle the existence question for a particular pair k and A. We have already experienced this phenomenon when we settled the cases with k :::: 5, cf. Theorems 5.5, 5.lD and 8.14, Similarly, for resolvable 2-designs, the cases with k :::: 4 and A = I have also been settled completely, see Theorems 6.5 and 7.11, respectively. Beyond these ranges, the existence problem becomes more and more involved and requires a considerable amount of computer work; in fact, the existence problem for Steiner systems S(2, k; v) with k :::: 6 and for resolvable Steiner

682

IX. Recursive constructions

systems RS(2, k; v) with k ::: 5 has not been settled completely for any value of k. Presenting proofs for the present state of knowledge is clearly outside the scope of this book, even though the methods are more or less the same as those we used in the previous sections. We first summarise the state of the art regarding 2-designs with k = 6. As usual, the most difficult subcase is the case A = 1. In fact, for A ::: 2, the complete answer has been known for a long time, by the following result of Hanani (1975).

12.1 Theorem. The necessary conditions for the existence of an S)..(2, 6; v) with A::: 2 are sufficient, except for the non-existent S2(2, 6; 21). • The case A = 1 is indeed much more difficult and is, in spite of considerable effort, still open. A series of papers by Mills (1984), Mullin, Hoffman and Lindner (1987),Zhu, Du and Yin (1987), Mullin (1989a), Abel and Mills (1995), Abel (1996), Abel and Greig (1997) and Mills (1997) reduce the open cases to the 33 values of v listed in Table 5.2 of the appendix; here the most recent updates are taken from Colbourn and Dinitz (1998). Summarising, we have the following result.

12.2 Theorem. The necessary conditions v == 1 or 6 (mod 15) for the existence of a Steiner system S (2, 6; v) are sufficient for v ::: 811. • An outline of the proof of Theorem 12.2 (still with 55 open values) was given by Mills (1996). We next discuss the case k = 7. The most difficult subcase A = 1 was first systematically studied by Zhang (1990); by recent work of Greig (1998a), Abel and Greig (1998) and Janko and Tonchev (1998), the necessary conditions are now known to be sufficient except for the non-existent S(2, 7, 43) and the 22 open cases listed in Table 5.3 in the appendix; again, the most recent updates are taken from Colbourn and Dinitz (1998). For the subcase A ::: 2, the pioneering work was done by Hanani (1989); subsequently, considerable progress was made in papers of Greig (1990), Zhu and Wu (1990), Zhu (1992), Yin and Wu (1993), Abel (1994, 1996), Greig (1998b) and Abel and Greig (1998) so that this case is now almost settled, with only six open cases for A = 2. We summarise the known results as follows:

12.3 Theorem. The necessary conditions for the existence of an S)..(2, 7; v) are sufficient, except for the non-existent designs S(2, 7, 43) and S2(2, 7; 22)

§12. Concluding remarks

683

and, possibly, the following undecided cases: A = 1 and 22 values of v with v ::::: 2605;

A = 2 and v E {274, 358, 574, 694, 988, 994} . •

Greig (1990), Du and Zhu (1990), Du (1994), Greig (1998a) and Abel (1996) obtained similar, powerful results for Steiner systems S(2, k; v) with k = 8 or 9; there are just 38 and 94 undecided cases, respectively. The open cases for k = 8 are listed in Table 5.4 in the appendix. It should be noted that the construction of an S(2, 9; 153) given in Hall (1986) contains an error which apparently cannot be corrected; this also invalidates some recursive constructions in the liteni.ture making use of this design. For this and for a listing of the undecided cases with k = 9, see Section 1.2.1 of Colboum and Dinitz (1996a) and Colboum and Dinitz (1998).

12.4 Theorem. The necessary conditions for the existence of an S(2, k; v) are sufficient for k = 8 and v 2:: 3760 as well asfor k = 9 and v 2:: 16561. • Regarding higher values of A, one knows that the necessary existence conditions for S),.(2, k; v) are sufficient for the pair (k, A) = (8,7), see Yin (1994). With a few possible exceptions, they are also sufficient for the pairs (k, A) = (8,4) and (9, 8); this follows from unpublished work of Malcolm Greig, see Section 1.2.1 of Colbourn and Dinitz (1996a). The existence problem for simple designs S)"(2, k; v) with k ::::: 4 has been discussed in §5. Similar results for k = 5 and various values of A were obtained by Ding and Shen (1993) and Ding (1994). We next consider the status of the existence problem for resolvable 2-designs, beginning with Kirkman systems, that is with A= 1. The existence of RS(2, 5; v) is considered by Chen and Zhu (1987), Zhu, Chen and Du (1987) and Zhu, Du and Zhang (1991) and Abel and Greig (1997) who made considerable progress in settling this problem; there are only five undecided cases which are listed below cf. Colboum and Dinitz (1998). Similarly, the existence of RS(2, 8; v) is considered by Greig and Abel (1997); there are just 66 undecided cases which are listed in Section 1.6.6 of Colboum and Dinitz (1996a).

12.5 Theorem. The necessary conditions for the existence of an RS(2, k; v) are sufficient for k = 5, except possibly for v E {45, 225, 345, 465, 645}, and for k = 8 and v 2:: 24536. •

684

IX. Recursive constructions

Regarding higher values of A, no really general result (in the sense that it would apply to all values of A for a specific value of k) has been established up to now. However, there are a few pairs (k, A) for which the necessary conditions are known to be sufficient; the following results are due to Hanani (1974a), Baker (1975), Rokowska (1981), Baker and Wilson (1983) and Baker (1983).

12.6 Theorem. The necessary conditions for the existence of a RS).(2, k; v) are sufficient for the pairs (k, A) E {(3, 2), (4, 3), (6, 1O)} except for the nonexistent RS2 (2, 3; 6). • Moreover, there are a few more pairs (k, A) for which the necessary conditions are known to be almost sufficient; the following results are due to Rokowska (1984), Miao (1994), Miao and Zhu (1995), Furino, Greig et al' (1994), Lee (1995), Abel (1996) and Greig and Abel (1997), see also Section 1.6 ofColboum and Dinitz (1996a, 1998).

12.7 Theorem. The necessary conditions for the existence of an RS).(2, k; v) are sufficient for the following pairs (k, A) as soon as v > vo(k, A), with the number of possible exceptions for v listed: k }..

vo(k, }..)

exceptions

5 2 405

5 4 195

5 594

7 6 33936

13

10

10

?

6

8 7 1488 22

12 11 108768 see Lee (1995)

The existence question for simple resolvable designs RS).(2, k; v) with k = 3, A = 2 and k = 4, A ::: 5 is considered and almost completely settled by Shen (1988) and Shen and Wu (1990); there are only two undecided cases for k = 4, A = 11.

12.8 Remark. We note that a Kirkman system RS(2, k; v) can also be defined in terms of divisible designs as either an RGD[k, 1; v] (if we consider the singletons as trivial point classes) or an RGD[k, k; v] (if we consider the blocks in a fixed parallel class as point classes). In this context, we report briefly on nearly Kirkman systems which may be defined in terms of GDD's as RGD[k, k - I; w)'s. However, we prefer the following interpretation: an RGD[k, k - I; v-I] is, of course, equivalent to an S(2, k; v) where one omits a fixed point and considers the blocks through this point as point classes and where the remaining blocks are partitioned into parallel classes; accordingly, we

§12. Concluding remarks

685

shall call such an S(2, k; v) nearly resolvable and denote it as an NRS(2, k; v). The necessary condition for the existence of anNRS(2, k; v) is easily seen to be (12.8.a)

v == 1 (mod k(k - 1».

We have the following existence results for k = 3 due to Kotzig and Rosa (1974), Baker and Wilson (1977), Brouwer (1978a) and Guy, see Huang, Mendelsohn and Rosa (1983), and for k = 4 due to Shen (1992b):

12.9 Theorem. The necessary condition v == 1 (mod 6) for the existence of an NRS(2, 3; v) is sufficient for v ::: 18. The necessary condition v == 1 (mod 12) for the existence ofanNRS(2, 4; v) is sufficientfor v ::: 901 (with 16 undecided cases below this bound). • 12.10 Remark and Definition. There is an different notion of nearly resolvable designs in the literature: An NRB( v, k) is a design D whose block set can be partitioned into v classes such that each class contains every point of D but one, and each point is missing from exactly one class. It is easily seen that D has to have "A = k - 1 and that the necessary existence condition is v == I (mod k). We refer the interested reader to Yin and Miao (1993), Furino (1990, 1995), Furino and Vanstone (1993), Lee (1995) and the monograph by Furino, Miao and Yin (1996) for the following result. For the cases 6 ::; k ::; 8, see Table 6.26 in Colbourn and Dinitz (1996a, 1998). 12.11 Theorem. The necessary condition v == 1 (mod k) for the existence of an NRB(v, k) is always sufficient for k ::; 5. • One may also consider a-resolvable designs for arbitrary a. The necessary conditions here are easily seen to be"A(v-l) == 0 (moda(k-l» and"Av(v-l) == o (mod k(k - 1». The case k = 3 has been completely settled by Jungnickel, Mullin and Vanstone (1991); Furino (1990) and Furino and Mullin (1993) considered the case k = 4 and solved it almost completely.

12.12 Theorem. The necessary conditions (12.12.a)

"A(v - 1)

== 0 (mod a(k-I»

and "Av(v-I)

== 0 (mod k(k-l»

for the existence of an a-resolvable SA (2, k; v) are sufficient for k = 3, v oF 6 and for k = 4, except possibly for 11 undecided cases. •

§ 12. Concluding remarks

687

12.15 Theorem. The necessary conditions for the existence of an S),. (3, 4; v) are sufficient. • Regarding simplicity, the only case that has been settled is J.... = 3, where the necessary condition is sufficient, cf. Theorem VIII.6.5. Hanani (1979) gave the following recursive constructions for 3-designs. Only recently progress towards a general theory of recursive constructions of 3-designs (similar to Wilson's methods discussed in the present chapter) was made by Hartman (1994). 12.16 Theorem. Letq be a prime power. The existence of S),.(3, q + 1; v + 1) implies that of S),.(3, q + 1; qv + 1). Moreover, the existence of S),.(3, 6; v + 1) implies that of S),. (3,6; 4v + 2). • Hanani (1979) also obtained the only other known general existence theorem for t-designs with t :::: 3: 12.17 Theorem. An S30(3, 5; v) exists for every v :::: 5. • Regarding resolvable designs .with t :::: 3, Hartman (1987) almost settled the existence problem for resolvable Steiner quadruple systems; there are just 23 undecided cases listed in Table 5.5 of the appendix. 12.18 Theorem. The necessary condition v ==40r8 (mod 12)fortheexistence ofanRS(3,4; v) issufficientforv:::: 6364. • The necessary condition v == 0 (mod 4) for designs RS3b(3, 4, v)'s is also sufficient, see Theorem Vill.6.6. We conclude this section with a brief discussion of some further generalisations of t-designs. 12.19 Definition. Let t, k, v and J.... be positive integers with 2 ::: t < k < v. A t-(v, k, J....) covering is an incidence structure (V, B) on v points with blocks of size k such that any t-subset of points is contained in at least J.... blocks in B; the covering problem asks for the determination of the covering number C),.(v, k, t), that is, of the minimal number of blocks in a t-(v, k, J....) covering. Similarly, a t-(v, k, J....) packing is an incidence structure (V, B) on v points with blocks of size k such that any t -subset of points is contained in at most J.... blocks in B; the packing problem asks for the determination of the packing number D),.(v, k, t), that is, of the maximal number of blocks in a t-(v, k, J....) packing.

§ 12. Concluding remarks

687

12.15 Theorem. The necessary conditions for the existence of an S),. (3, 4; v) are sufficient. • Regarding simplicity, the only case that has been settled is J.... = 3, where the necessary condition is sufficient, cf. Theorem VIII.6.5. Hanani (1979) gave the following recursive constructions for 3-designs. Only recently progress towards a general theory of recursive constructions of 3-designs (similar to Wilson's methods discussed in the present chapter) was made by Hartman (1994). 12.16 Theorem. Letq be a prime power. The existence of S),.(3, q + 1; v + 1) implies that of S),.(3, q + 1; qv + 1). Moreover, the existence of S),.(3, 6; v + 1) implies that of S),. (3,6; 4v + 2). • Hanani (1979) also obtained the only other known general existence theorem for t-designs with t :::: 3: 12.17 Theorem. An S30(3, 5; v) exists for every v :::: 5. • Regarding resolvable designs .with t :::: 3, Hartman (1987) almost settled the existence problem for resolvable Steiner quadruple systems; there are just 23 undecided cases listed in Table 5.5 of the appendix. 12.18 Theorem. The necessary condition v ==40r8 (mod 12)fortheexistence ofanRS(3,4; v) issufficientforv:::: 6364. • The necessary condition v == 0 (mod 4) for designs RS3b(3, 4, v)'s is also sufficient, see Theorem Vill.6.6. We conclude this section with a brief discussion of some further generalisations of t-designs. 12.19 Definition. Let t, k, v and J.... be positive integers with 2 ::: t < k < v. A t-(v, k, J....) covering is an incidence structure (V, B) on v points with blocks of size k such that any t-subset of points is contained in at least J.... blocks in B; the covering problem asks for the determination of the covering number C),.(v, k, t), that is, of the minimal number of blocks in a t-(v, k, J....) covering. Similarly, a t-(v, k, J....) packing is an incidence structure (V, B) on v points with blocks of size k such that any t -subset of points is contained in at most J.... blocks in B; the packing problem asks for the determination of the packing number D),.(v, k, t), that is, of the maximal number of blocks in a t-(v, k, J....) packing.

§12. Concluding remarks

689

quoted there. The best studied case is that of triple systems; here we recommend the survey by Colbourn and Rosa (1992). Finally, we mention ternary and, more generally, n-ary designs, where -loosely speaking - points can have multiplicities; ordinary designs are, in this terminology, "binary". See Francel and Sarvate (1994) for possible definitions and references.

§12. Concluding remarks

689

quoted there. The best studied case is that of triple systems; here we recommend the survey by Colbourn and Rosa (1992). Finally, we mention ternary and, more generally, n-ary designs, where -loosely speaking - points can have multiplicities; ordinary designs are, in this terminology, "binary". See Francel and Sarvate (1994) for possible definitions and references.

§1. A recursive construction

691

Proof. Let D be a GD}..[TD*(k), TD},.(k); u] with point set N~, block set A, and point class set G. We shall construct a TD).[k; u], say E, with point set V := N~ x Nj' and point classes {i} x Nj, i E N1. By hypothesis, the following TD's exist. For each G E G there is a TD).[k; IGI], say D G , with

point set point classes

N~ x G, {i} x G, i E N7,

and block set

BG•

For each A EA there is a TD[k; IAIl, say D A, with point set point classes

N1 x (A), {i} x (A), i E N7,

and block set CA , w.l.o.g. with the parallel class P A = {N7 x {x} : x E (A)}. Now set

Let the block list B of E be defined by

B= LB GeG

G

+ LC~, AeA

We claim that E with block list B and point classes {i} x Ni, i EN~, is the desired TD).[k; u]. Let (i, x), (j, y) be two points. If i = j they are in a point class of E, but not in a block of E.lndeed, if x and yare in a point class G (resp. a block A) of D, then (i, x) and (i, y) are in a point class of DG (resp. DA)' If i # j and x = y, then the two points x, y of D are joined by exactly).. blocks of B G , where G E G is the point class containing x. Note that (i, x), (j, x) EPA if x E (A); hence (i, x) and (j, x) are not joined by a block in any C~. Now assume i # j and x # y. If x and yare in a point class G E G, then (i, x) and (j, y) are in exactly ).. blocks of B G , but not in any block of aBA, A EA. Otherwise x and y are in exactly).. blocks AI, ... , A). of D, and for each h E {I, ... , )..} the points (i, x) and (j, y) are in exactly one block of C Ah • Hence E is the desired TD).[k; u].

§1. A recursive construction

691

Proof. Let D be a GD}..[TD*(k), TD},.(k); u] with point set N~, block set A, and point class set G. We shall construct a TD).[k; u], say E, with point set V := N~ x Nj' and point classes {i} x Nj, i E N1. By hypothesis, the following TD's exist. For each G E G there is a TD).[k; IGI], say D G , with

point set point classes

N~ x G, {i} x G, i E N7,

and block set

BG•

For each A EA there is a TD[k; IAIl, say D A, with point set point classes

N1 x (A), {i} x (A), i E N7,

and block set CA , w.l.o.g. with the parallel class P A = {N7 x {x} : x E (A)}. Now set

Let the block list B of E be defined by

B= LB GeG

G

+ LC~, AeA

We claim that E with block list B and point classes {i} x Ni, i EN~, is the desired TD).[k; u]. Let (i, x), (j, y) be two points. If i = j they are in a point class of E, but not in a block of E.lndeed, if x and yare in a point class G (resp. a block A) of D, then (i, x) and (i, y) are in a point class of DG (resp. DA)' If i # j and x = y, then the two points x, y of D are joined by exactly).. blocks of B G , where G E G is the point class containing x. Note that (i, x), (j, x) EPA if x E (A); hence (i, x) and (j, x) are not joined by a block in any C~. Now assume i # j and x # y. If x and yare in a point class G E G, then (i, x) and (j, y) are in exactly ).. blocks of B G , but not in any block of aBA, A EA. Otherwise x and y are in exactly).. blocks AI, ... , A). of D, and for each h E {I, ... , )..} the points (i, x) and (j, y) are in exactly one block of C Ah • Hence E is the desired TD).[k; u].

§2. Transversal designs with A > 1

693

(e) Removing one point from an affine plane of order q yields q2 -1 EGD(q,q -1),

and thus (1.3.f)

N(q2 - 1)::: N(q - 1)

for prime powers q;

in particular (1.3.g)

N(24)::: 3, N(80)::: 7.

(f) Removing three collinear points from a projective plane of order 7 yields

54E GD({7, 8}, {I, 5}) S; TD(6); i.e. (1.3.h)

N(54) :::4.

1.4 Exercise. Give an alternative proof of 30 E TD 2 (7), using an S2(2, 9; 37); cf. Example VI.2.13. §2. Transversal Designs with oX > 1 We shall give a slightly simplified proof of the following theorem of Ranani (1974b, 1975). 2.1 Theorem. (2. La)

TDJ,.(7)=f' 1.

Proof. The proof needs several steps. Trivially 1 E TDJ,. (7). For g > 1 we shall show g E TD2(7) n TD3(7). Then, reasoning as in Proposition 1.5.3, we get g E TDJ,.(7) for all A > 1. Also we use MacNeish's Theorem 1.7.7 as well as 12 E TD(7) (see Lemma Vill.3.l3) and Theorem IXAA. Hence it suffices to prove Theorem 2.1 for A = 2, 3 and for g = 2, 3,4,5,6,10, 15,20,30. This will be done below. Recall that the existence of a (g, k; A)-difference matrix implies that of a TDJ,.[k + 1; g]; see Corollary VIll.3.8 . • 2.2 Lemma. (2.2.a)

2 E TD)..(7) for all A > 1.

§2. Transversal designs with A > 1

693

(e) Removing one point from an affine plane of order q yields q2 -1 EGD(q,q -1),

and thus (1.3.f)

N(q2 - 1)::: N(q - 1)

for prime powers q;

in particular (1.3.g)

N(24)::: 3, N(80)::: 7.

(f) Removing three collinear points from a projective plane of order 7 yields

54E GD({7, 8}, {I, 5}) S; TD(6); i.e. (1.3.h)

N(54) :::4.

1.4 Exercise. Give an alternative proof of 30 E TD 2 (7), using an S2(2, 9; 37); cf. Example VI.2.13. §2. Transversal Designs with oX > 1 We shall give a slightly simplified proof of the following theorem of Ranani (1974b, 1975). 2.1 Theorem. (2. La)

TDJ,.(7)=f' 1.

Proof. The proof needs several steps. Trivially 1 E TDJ,. (7). For g > 1 we shall show g E TD2(7) n TD3(7). Then, reasoning as in Proposition 1.5.3, we get g E TDJ,.(7) for all A > 1. Also we use MacNeish's Theorem 1.7.7 as well as 12 E TD(7) (see Lemma Vill.3.l3) and Theorem IXAA. Hence it suffices to prove Theorem 2.1 for A = 2, 3 and for g = 2, 3,4,5,6,10, 15,20,30. This will be done below. Recall that the existence of a (g, k; A)-difference matrix implies that of a TDJ,.[k + 1; g]; see Corollary VIll.3.8 . • 2.2 Lemma. (2.2.a)

2 E TD)..(7) for all A > 1.

§2. Transversal designs with A > 1

695

Now the exceptional values 10, 15, 20, 30 are covered by (VillA.33.c, d, e, f), (X.1.3.c, e), and Exercise X.1.4, proving Theorem 2.1. 2.8 Corollary. (a) If A > 1, k (b) If A > 1, k ~7, and r

~ 7,

and r E B(k, A), then kr E B(k, A).

+ 1 E B(k, A), then kr + 1 E B(k, A).

Proof. (a) k E GD(k, 1) and r E TD;.(k) imply kr E GD)..(k, r) S; B(k, A), by Corollary IX.3.6 and (IX.2.6.a). (b) k E GD(k, 1) andr E TD)..(k) imply kr E GD)..(k, r), kr + 1 E B(k, A), using (IX.2.6.b) . • 2.9 Examples.

(2.9.a)

36 E B(6, A)

for all A > 1,

(2.9.b)

43 E B(7, A)

for all A > 1.

(2.9.c)

91,96 E B(6, 2),

using Corollary II.3.16.

(2.9.d)

99, 105 E B(7, 3),

using Proposition 1.2.16.

2.10 Corollary. For k

(2.1O.a)

B(k) S;

~ 7,

A > 1, and all g EN

Rf)..·

Proof. Exercise. • 2.11 Exercise and Remark. (a) Show (2.11.a)

2ETD)..(11)

(2.11.b)

2u E TD)..(11)

for all 1..:::3,

for A::: 3 and U E B(P{f) (with the notation of Lemma IXA.3). (b) The question whether 2 E TD)..(4A - 1) for all A EN depends on the existence conjecture for Hadamard matrices. Colbourn (1996b) studied the spectrum of transversal designs with A > 2 and k 8 or 9. He obtained most of the following results, the proofs of which are considerably more involved than that of Hanani's Theorem 2.1; a few further

=

§2. Transversal designs with A > 1

695

Now the exceptional values 10, 15, 20, 30 are covered by (VillA.33.c, d, e, f), (X.1.3.c, e), and Exercise X.1.4, proving Theorem 2.1. 2.8 Corollary. (a) If A > 1, k (b) If A > 1, k ~7, and r

~ 7,

and r E B(k, A), then kr E B(k, A).

+ 1 E B(k, A), then kr + 1 E B(k, A).

Proof. (a) k E GD(k, 1) and r E TD;.(k) imply kr E GD)..(k, r) S; B(k, A), by Corollary IX.3.6 and (IX.2.6.a). (b) k E GD(k, 1) andr E TD)..(k) imply kr E GD)..(k, r), kr + 1 E B(k, A), using (IX.2.6.b) . • 2.9 Examples.

(2.9.a)

36 E B(6, A)

for all A > 1,

(2.9.b)

43 E B(7, A)

for all A > 1.

(2.9.c)

91,96 E B(6, 2),

using Corollary II.3.16.

(2.9.d)

99, 105 E B(7, 3),

using Proposition 1.2.16.

2.10 Corollary. For k

(2.1O.a)

B(k) S;

~ 7,

A > 1, and all g EN

Rf)..·

Proof. Exercise. • 2.11 Exercise and Remark. (a) Show (2.11.a)

2ETD)..(11)

(2.11.b)

2u E TD)..(11)

for all 1..:::3,

for A::: 3 and U E B(P{f) (with the notation of Lemma IXA.3). (b) The question whether 2 E TD)..(4A - 1) for all A EN depends on the existence conjecture for Hadamard matrices. Colbourn (1996b) studied the spectrum of transversal designs with A > 2 and k 8 or 9. He obtained most of the following results, the proofs of which are considerably more involved than that of Hanani's Theorem 2.1; a few further

=

§3. A construction of Wilson

697

As point classes in the point set V*, define the subsets (3. 1.d)

G1 := (Gi

x M) U ({i} x S)

for i E 1, ... , k,

and put G* := {Gi, .. ·, Gk}. Next we construct blocks in V* from any given block A EA in V. By (3.l.b) there is a TD[k; m + UA], say T A , with point set PA := (Ao x M) U (Nt x A'), point classes [(Ao n Gi) x M] U ({i} x A'), i = 1, ... , k, and block

setB A . Since T A has

UA

disjoint blocks, we may assume that, w.l.o.g.,

N1 x {Z},

ZEA',

are blocks of B A. (The case A' = 0 is not excluded.) Let B~ denote the set of the remaining (m + UA)2 - UA blocks of B A , and put (3.1.e)

B:=

U B~. AeA

Finally, we use (3.l.a) to construct further blocks. For each j E N~ there is a TD[k; h j ], say T j , with

:=N1

Pj x (SnHj ), {i} x (SnHj ),iE N

point set point classes

1,

and block set

Cj

•

Put (3.1.f)

A* :=BUC1 U ... UC/.

Note that the case S n Hj = 0, Cj = 0 is not excluded. We claim that (V*, A *, G*) is the desired TD[k; mg + s].

(b) The verification First we show that two points of a point class Gi are not joined by a block. Let x, y E G i and m, m' E M. The points (x, m) and (y, m') are not in a Pj , and hence not in a block of C := C 1 U ... U C/. If x =I y they are not both in a set PA , hence not in a block of B. If x = y the points (x, m), (x, m') are in a point class of each TA with x E Ao n G i , and hence in no block of a B A . Now consider two points (i, p), (i, q) with p, q E S. If p and q are in distinct point classes of T, they are joined by a unique A, and they are in a point class of T A.

§3. A construction of Wilson

697

As point classes in the point set V*, define the subsets (3. 1.d)

G1 := (Gi

x M) U ({i} x S)

for i E 1, ... , k,

and put G* := {Gi, .. ·, Gk}. Next we construct blocks in V* from any given block A EA in V. By (3.l.b) there is a TD[k; m + UA], say T A , with point set PA := (Ao x M) U (Nt x A'), point classes [(Ao n Gi) x M] U ({i} x A'), i = 1, ... , k, and block

setB A . Since T A has

UA

disjoint blocks, we may assume that, w.l.o.g.,

N1 x {Z},

ZEA',

are blocks of B A. (The case A' = 0 is not excluded.) Let B~ denote the set of the remaining (m + UA)2 - UA blocks of B A , and put (3.1.e)

B:=

U B~. AeA

Finally, we use (3.l.a) to construct further blocks. For each j E N~ there is a TD[k; h j ], say T j , with

:=N1

Pj x (SnHj ), {i} x (SnHj ),iE N

point set point classes

1,

and block set

Cj

•

Put (3.1.f)

A* :=BUC1 U ... UC/.

Note that the case S n Hj = 0, Cj = 0 is not excluded. We claim that (V*, A *, G*) is the desired TD[k; mg + s].

(b) The verification First we show that two points of a point class Gi are not joined by a block. Let x, y E G i and m, m' E M. The points (x, m) and (y, m') are not in a Pj , and hence not in a block of C := C 1 U ... U C/. If x =I y they are not both in a set PA , hence not in a block of B. If x = y the points (x, m), (x, m') are in a point class of each TA with x E Ao n G i , and hence in no block of a B A . Now consider two points (i, p), (i, q) with p, q E S. If p and q are in distinct point classes of T, they are joined by a unique A, and they are in a point class of T A.

699

§3. A construction of Wilson

Next we shall discuss some special cases of Theorem 3.1. We begin with the casesl = 1 and 1 = 2. In the case 1 = 1 we have UA E to, I}, and hence (3.1.b) reduces to m, m + 1 E TD(k). Thus we obtain 3.3 Theorem. If 0::: u::: g, then for mEN (3.3.a)

N(mg

+ u) ~ min{N(m), N(m + 1), N(g) -

1, N(u)}.

Proof. Put k := 2 + rnin{N(m), N(m + 1), N(g) - 1, N(u)}. Then let T be a TD[k + 1; g] and note that both (3.1.a) and (3.1.b) hold, where S is any usubset of HI' By Theorem 3.1, the assertion follows. (Recall the convention N(O) = N(1) = 00). • 3.4 Lemma. Let g ~ 3. In every TD[k; g] with g blocks.

~

k there are three disjoint

Proof. We first show the existence of two disjoint blocks. Let A be a block, say A = {al .... , ad. For i E there are g -1 blocks X # A with AnX = (ad. Hence the total number of blocks X with X n A # ¢ is 1 + keg - 1) ::: 1 + g2 - g < g2. Since the total number of blocks is g2, the assertion follows. Now let A and B be any two disjoint blocks. Then the total number of blocks Y with AnY # ¢or BnY # ¢isk(k-l)+2(g-k)+2:::g(g-3)+2g+2 < g2, implying the result. •

N1

3.5 Theorem. Ifu, (3.5.a)

N(mg

W E

N5, then

+ u + w) 2: min{N(m), N(m + 1), N(m + 2), N(g) -

2, N(u), N(w»).

Proof. Let k - 2 be the minimum on the right-hand side of (3.5.a) and start with T, a TD[k + 2; g]. For UA E to, 1, 2} there are TD[k; m + UA]'S. Since m + 1 ~ k, m + 2 > k, Lemma 3.4 implies that both (3. La) and (3.l.b) hold (with hi = U, h2 = w). Now Theorem 3.1 yields the assertion. • 3.6 Examples. (a) Let u::: g be an odd number and g (mod 18). Then, by Theorem 3.3, (3.6.a)

N(3g

+ u) ~2.

Thus it is easily seen that (3.6.b)

N(4n

+ 2) ~2

forn EN4'.

== 1,5,7,9,11,13,17

699

§3. A construction of Wilson

Next we shall discuss some special cases of Theorem 3.1. We begin with the casesl = 1 and 1 = 2. In the case 1 = 1 we have UA E to, I}, and hence (3.1.b) reduces to m, m + 1 E TD(k). Thus we obtain 3.3 Theorem. If 0::: u::: g, then for mEN (3.3.a)

N(mg

+ u) ~ min{N(m), N(m + 1), N(g) -

1, N(u)}.

Proof. Put k := 2 + rnin{N(m), N(m + 1), N(g) - 1, N(u)}. Then let T be a TD[k + 1; g] and note that both (3.1.a) and (3.1.b) hold, where S is any usubset of HI' By Theorem 3.1, the assertion follows. (Recall the convention N(O) = N(1) = 00). • 3.4 Lemma. Let g ~ 3. In every TD[k; g] with g blocks.

~

k there are three disjoint

Proof. We first show the existence of two disjoint blocks. Let A be a block, say A = {al .... , ad. For i E there are g -1 blocks X # A with AnX = (ad. Hence the total number of blocks X with X n A # ¢ is 1 + keg - 1) ::: 1 + g2 - g < g2. Since the total number of blocks is g2, the assertion follows. Now let A and B be any two disjoint blocks. Then the total number of blocks Y with AnY # ¢or BnY # ¢isk(k-l)+2(g-k)+2:::g(g-3)+2g+2 < g2, implying the result. •

N1

3.5 Theorem. Ifu, (3.5.a)

N(mg

W E

N5, then

+ u + w) 2: min{N(m), N(m + 1), N(m + 2), N(g) -

2, N(u), N(w»).

Proof. Let k - 2 be the minimum on the right-hand side of (3.5.a) and start with T, a TD[k + 2; g]. For UA E to, 1, 2} there are TD[k; m + UA]'S. Since m + 1 ~ k, m + 2 > k, Lemma 3.4 implies that both (3. La) and (3.l.b) hold (with hi = U, h2 = w). Now Theorem 3.1 yields the assertion. • 3.6 Examples. (a) Let u::: g be an odd number and g (mod 18). Then, by Theorem 3.3, (3.6.a)

N(3g

+ u) ~2.

Thus it is easily seen that (3.6.b)

N(4n

+ 2) ~2

forn EN4'.

== 1,5,7,9,11,13,17

701

§3. A construction of Wilson

assumption on U, V, W. To this end choose a first point Xo E Wand a v-subset V S;; H2 arbitrarily. Denote by X the (g - v)-subset of HI determined by the intersection points HI n B arising from the g - v blocks B joining Xo and the points of H2 \ V. Now choose any u-subset U C HI satisfying X S;; HI \ U, which is possible as u:S v; hence g - v:s g - u. Then the (g - u)(g - v) blocks joining some point in HI \ U to some point in H2 \ V determine at most (g - u)(g - v) - (g - v) + 1 distinct points of H3 (as Xo occurs g - v times in this way) and, as (g - u - 1)(g - v) < w we may include all these points in W. Thus any block A which misses both U and V will meet W, which proves our assumption. •

3.9 Corollary. (3.9.a)

N(90):::: 6.

Proof. Take m = 6, g = 11 and u = v = w

= 8 in Theorem 3.8. •

Wilson (1974a) proved N(n):::: 6 for all n:::: 91. It is remarkable that it was not realised that the critical result N (90) :::: 6 also followed from his Theorem 3.1 until some four years later. This shows that the task of finding suitable specialisations of a general result like Theorem 3.1 is not at all easy. Corollary 3.9 and (VIDA.30.c) allow us to prove N(n):::: 6 for n:::: 77, which will be done in the next section. A further interesting specialisation of Theorem 3.1 is the following result which is a little weaker than one obtained by Wojtas (1978).

3.10 Theorem. Assume 0 < s < g. Then (3.1O.a)

N(mg

+ s) 2:

min{N(m), N(m

+ 1), N*(m + s), N(g) -

s}.

Proof. Let k - 2 be the minimum occurring in (3.1O.a), take I = s in Theorem 3.1, and let S consist of s points of a block A. Then UA E {O, 1, s}, and (3.1.b) holds for UA = s, as there is a TD[k; m + s] with a parallel class, by Lemma VIDA.l3 . • 3.11 Corollary. (3.11.a)

N(91) 2: 7.

Proof. Take m = 8, g = 11, s = 3 in Theorem 3.10. •

701

§3. A construction of Wilson

assumption on U, V, W. To this end choose a first point Xo E Wand a v-subset V S;; H2 arbitrarily. Denote by X the (g - v)-subset of HI determined by the intersection points HI n B arising from the g - v blocks B joining Xo and the points of H2 \ V. Now choose any u-subset U C HI satisfying X S;; HI \ U, which is possible as u:S v; hence g - v:s g - u. Then the (g - u)(g - v) blocks joining some point in HI \ U to some point in H2 \ V determine at most (g - u)(g - v) - (g - v) + 1 distinct points of H3 (as Xo occurs g - v times in this way) and, as (g - u - 1)(g - v) < w we may include all these points in W. Thus any block A which misses both U and V will meet W, which proves our assumption. •

3.9 Corollary. (3.9.a)

N(90):::: 6.

Proof. Take m = 6, g = 11 and u = v = w

= 8 in Theorem 3.8. •

Wilson (1974a) proved N(n):::: 6 for all n:::: 91. It is remarkable that it was not realised that the critical result N (90) :::: 6 also followed from his Theorem 3.1 until some four years later. This shows that the task of finding suitable specialisations of a general result like Theorem 3.1 is not at all easy. Corollary 3.9 and (VIDA.30.c) allow us to prove N(n):::: 6 for n:::: 77, which will be done in the next section. A further interesting specialisation of Theorem 3.1 is the following result which is a little weaker than one obtained by Wojtas (1978).

3.10 Theorem. Assume 0 < s < g. Then (3.1O.a)

N(mg

+ s) 2:

min{N(m), N(m

+ 1), N*(m + s), N(g) -

s}.

Proof. Let k - 2 be the minimum occurring in (3.1O.a), take I = s in Theorem 3.1, and let S consist of s points of a block A. Then UA E {O, 1, s}, and (3.1.b) holds for UA = s, as there is a TD[k; m + s] with a parallel class, by Lemma VIDA.l3 . • 3.11 Corollary. (3.11.a)

N(91) 2: 7.

Proof. Take m = 8, g = 11, s = 3 in Theorem 3.10. •

§4. Six and more mutually orthogonal Latin squares

703

§4. Six and More Mutually Orthogonal Latin Squares The main result of this section is the following theorem on the existence of six MOLS which is essentially due to Wilson (1974a). Only the cases n = 76, 82 and 90 were settled later; they are due to Colboum, Yin and Zhu (1995), Mullin, Schellenberg et al. (1980a) and Wojtas (1978), respectively.

4.1 Theorem. One has (4. La)

N(n) 2: 6

for n 2: 76.

Proof. For the case n = 76, we refer to Colboum, Yin and Zhu (1995). The main part of the proof is an application of Theorem 3.5 with m = 7 which gives (4.l.b)

N (7 g

+ u + w) 2: 6

whenever 0 :s u, w and

:s g,

N (g) 2: 8

N(u), N(w) 2: 6.

Compare this result with the similar-looking formula (IX.4.3.c). Using (4.l.b) and (IX.4.3.d), we obtain (4.l.c)

N~~~g,7g+80) S; TD(8)

whenever N(g) 2: 8.

Using (4.l.c) for g = 79, 81, 83, ... and observing IX.4.3.(f), we obtain

(4. 1. d)

N560 S; TD(8).

To cover values n < 560 we will also require Theorem 3.7 for I E ~ (note that here g 2: 13 > (;)) and m = 7: (4.l.e)

N(7g

+ l) 2: 6

for I E N~ provided that N(g) 2: 12.

This allows an improvement in the lower bound in (4.l.c) as follows.

(4.1.f)

g g N;:{8 ,7 +80)

S; TD(8)

whenever N(g) 2: 12.

This now yields (4. 1. g)

by using (4.1.c) and (4.1.0 for g = 13, 16, 17, 19,23,25,27,29,32,37,41, 43,49,53,59,67 and 71 and by observing that for g = 13, 19,31 one even obtains N~P, Nl~~, N~g~ S; TD(8), which is seen from (4.1.b) and the table in

§4. Six and more mutually orthogonal Latin squares

703

§4. Six and More Mutually Orthogonal Latin Squares The main result of this section is the following theorem on the existence of six MOLS which is essentially due to Wilson (1974a). Only the cases n = 76, 82 and 90 were settled later; they are due to Colboum, Yin and Zhu (1995), Mullin, Schellenberg et al. (1980a) and Wojtas (1978), respectively.

4.1 Theorem. One has (4. La)

N(n) 2: 6

for n 2: 76.

Proof. For the case n = 76, we refer to Colboum, Yin and Zhu (1995). The main part of the proof is an application of Theorem 3.5 with m = 7 which gives (4.l.b)

N (7 g

+ u + w) 2: 6

whenever 0 :s u, w and

:s g,

N (g) 2: 8

N(u), N(w) 2: 6.

Compare this result with the similar-looking formula (IX.4.3.c). Using (4.l.b) and (IX.4.3.d), we obtain (4.l.c)

N~~~g,7g+80) S; TD(8)

whenever N(g) 2: 8.

Using (4.l.c) for g = 79, 81, 83, ... and observing IX.4.3.(f), we obtain

(4. 1. d)

N560 S; TD(8).

To cover values n < 560 we will also require Theorem 3.7 for I E ~ (note that here g 2: 13 > (;)) and m = 7: (4.l.e)

N(7g

+ l) 2: 6

for I E N~ provided that N(g) 2: 12.

This allows an improvement in the lower bound in (4.l.c) as follows.

(4.1.f)

g g N;:{8 ,7 +80)

S; TD(8)

whenever N(g) 2: 12.

This now yields (4. 1. g)

by using (4.1.c) and (4.1.0 for g = 13, 16, 17, 19,23,25,27,29,32,37,41, 43,49,53,59,67 and 71 and by observing that for g = 13, 19,31 one even obtains N~P, Nl~~, N~g~ S; TD(8), which is seen from (4.1.b) and the table in

§4. Six and more mutually orthogonal Latin squares

705

Put g := (n - u)/8; then N(g), N(u)?: 10 by MacNeish's theorem. If g?: u, i.e. n?: 9u, then (3.3.a) with n = 16 works. But n?: u ·239 = 2151 implies n?: 9u. Using Lemma IX.2.1, it is easily seen that n E B(P8') . •

4.4 Proposition. (4.4. a)

N~728 ~ TD(9).

Proof. Let n EN be even. We want to apply (3.3.a) with m uo, u 1, U as follows.

== 31 + n (mod 62),

(4.4.b)

Uo

(4.4.c)

u 1 = Uo

such that ul, n (4.4.d)

u=

ul

Ul

+ 62x,

x

E

= 31. Determine

0 < Uo < 62,

N~

¥= 0 (mod 3);

+ 186y,

Y ENci,

such that u, n - u ¥= 0 (mod 5, 7). Put g := n:;t. Then N(g), N(u)?: 10 and (3.3.a) works provided g ?: u, i.e. n ?: 32 . 929. •

4.5 Remark. A very important construction method not covered in this text is the use of "incomplete" transversal designs; loosely speaking, these are transversal designs with "holes", that is, sets of points which are not required to be joined by a line. Equivalently, one may talk about sets of mutually orthogonal Latin squares with holes. Such objects were first introduced by Dinitz and Stinson (1983); a nice discussion is contained in the survey by Heinrich (1991). For a recent paper with many relevant references, see Xu and Zhu (1995). Though MOLS with holes are interesting objects in their own right, their main use lies in constructing ordinary sets of M 0 LS; for a description of this method as well as the known construction methods for MOLS with holes, see Colboum and Dinitz (1996b). Another recent method which has resulted in many improved bounds on the number of M 0 LS of "small" orders is the uses of "thwarts" which constitutes a way to exploit Wilson's fundamental construction, see Colboum, Dinitz and Wojtas (1995) and Colboum, Dinitz and Stinson (1996). For recent discussions of construction techniques for MOLS, see Colbourn (1995a) and Colboum and Dinitz (1996b); the latter of these two papers explains how the extensive table of bounds (for orders:::: 10 000) given in Colboum and Dinitz (1996a) was prepared.

§4. Six and more mutually orthogonal Latin squares

705

Put g := (n - u)/8; then N(g), N(u)?: 10 by MacNeish's theorem. If g?: u, i.e. n?: 9u, then (3.3.a) with n = 16 works. But n?: u ·239 = 2151 implies n?: 9u. Using Lemma IX.2.1, it is easily seen that n E B(P8') . •

4.4 Proposition. (4.4. a)

N~728 ~ TD(9).

Proof. Let n EN be even. We want to apply (3.3.a) with m uo, u 1, U as follows.

== 31 + n (mod 62),

(4.4.b)

Uo

(4.4.c)

u 1 = Uo

such that ul, n (4.4.d)

u=

ul

Ul

+ 62x,

x

E

= 31. Determine

0 < Uo < 62,

N~

¥= 0 (mod 3);

+ 186y,

Y ENci,

such that u, n - u ¥= 0 (mod 5, 7). Put g := n:;t. Then N(g), N(u)?: 10 and (3.3.a) works provided g ?: u, i.e. n ?: 32 . 929. •

4.5 Remark. A very important construction method not covered in this text is the use of "incomplete" transversal designs; loosely speaking, these are transversal designs with "holes", that is, sets of points which are not required to be joined by a line. Equivalently, one may talk about sets of mutually orthogonal Latin squares with holes. Such objects were first introduced by Dinitz and Stinson (1983); a nice discussion is contained in the survey by Heinrich (1991). For a recent paper with many relevant references, see Xu and Zhu (1995). Though MOLS with holes are interesting objects in their own right, their main use lies in constructing ordinary sets of M 0 LS; for a description of this method as well as the known construction methods for MOLS with holes, see Colboum and Dinitz (1996b). Another recent method which has resulted in many improved bounds on the number of M 0 LS of "small" orders is the uses of "thwarts" which constitutes a way to exploit Wilson's fundamental construction, see Colboum, Dinitz and Wojtas (1995) and Colboum, Dinitz and Stinson (1996). For recent discussions of construction techniques for MOLS, see Colbourn (1995a) and Colboum and Dinitz (1996b); the latter of these two papers explains how the extensive table of bounds (for orders:::: 10 000) given in Colboum and Dinitz (1996a) was prepared.

§5. The theorem of Chowla, Erdos and Straus with 2h-

(5.3.b)

1

:::: x

+ 1 < 2h if n is odd, and h =

t;j=O,

707

0 if n is even, respectively, and

u;j=O(modq)

for all primes q :::: x.

Proof Since gcd (m, x!) = 1, the congruences (5.3.c)

t ;j= 0,

n - 2hmt ;j= 0 (mod q)

have a solution t for each prime q :::: x. By the Chinese Remainder Theorem, they have a simultaneous solution t. Now put

If t, u E Z satisfy (5.3.a) and (5.3.b), then t too. Hence we may assume

± Jrx

and u

+ 2hmJrx satisfy them,

(5.3.d) 5.4 Theorem. (SA.a)

lim N(n)

n-+co

= 00.

Proof. For given odd x EN, choose m as in Lemma 5.2. If the numbers u, t of Lemma 5.3 satisfy the inequality (5A.b)

then by MacNeish's theorem N(2 ht), N(u) > x, since x Condition (5A.b) certainly holds if (5.4. c)

+ 1 is not a prime.

n 2: m(m + 1)2h . Jrx =: no.

Now apply Theorem 3.3 with g = 2ht. Thus N(n) 2: x for all n 2: no . • As an application we have already introduced the numbers nr (r E N) in Remark 4.2. 5.5 Remark. Note that Theorem SA and the results quoted in 5.1 are of an asymptotic nature and do not provide us with an explicit function f such that

§5. The theorem of Chowla, Erdos and Straus with 2h-

(5.3.b)

1

:::: x

+ 1 < 2h if n is odd, and h =

t;j=O,

707

0 if n is even, respectively, and

u;j=O(modq)

for all primes q :::: x.

Proof Since gcd (m, x!) = 1, the congruences (5.3.c)

t ;j= 0,

n - 2hmt ;j= 0 (mod q)

have a solution t for each prime q :::: x. By the Chinese Remainder Theorem, they have a simultaneous solution t. Now put

If t, u E Z satisfy (5.3.a) and (5.3.b), then t too. Hence we may assume

± Jrx

and u

+ 2hmJrx satisfy them,

(5.3.d) 5.4 Theorem. (SA.a)

lim N(n)

n-+co

= 00.

Proof. For given odd x EN, choose m as in Lemma 5.2. If the numbers u, t of Lemma 5.3 satisfy the inequality (5A.b)

then by MacNeish's theorem N(2 ht), N(u) > x, since x Condition (5A.b) certainly holds if (5.4. c)

+ 1 is not a prime.

n 2: m(m + 1)2h . Jrx =: no.

Now apply Theorem 3.3 with g = 2ht. Thus N(n) 2: x for all n 2: no . • As an application we have already introduced the numbers nr (r E N) in Remark 4.2. 5.5 Remark. Note that Theorem SA and the results quoted in 5.1 are of an asymptotic nature and do not provide us with an explicit function f such that

§6. Transversal designs and orthogonal arrays

709

as in Theorem 11.8.8. Let us write ni for the number of points x =f:. p with Ax = i. Then we have (6.2.a) and rewriting equations (IT.8.8.b, c) yields (6.2.b)

:L>ni = LAx = r(sp, - 1), xip

(6.2.c)

L i (i -

l)ni

=L

We shall consider the function (6.2.d)

Ax(A x

-

1)

= r(r -

l)(fL - 1).

xip

i

f : Z -7 Z defined by

fey) = L(i - y)(i - 1 - y)ni' i

which is easily seen to be non-negative for all Y E Z. From (6.2.a, b, c, d) we obtain

fey)

= LiCi -1)ni i

- 2y Lini i

= p,[r(r - 1) - 2rsy + y(y

+ y(y + I) Ln; i

+ 1)s2)

- [r(r - 1) - 2ry + y(y + 1)]; and

f

(y) ?:: 0 implies

(6.2.e)

r(r - I) - 2ry + y(y + 1) --------=-..:......;--..,- r(r - 1) - 2rsy + y(y + l)s2

fL>

With (6.2.f)

C:= r -1- ys

one obtains (6.2.g)

D := r(r -1) -2rsy+ y(y+ l)s2 = res -1) - (C + 1)(s -1- C)

and then from (6.2.e) after some computation (6.2.h)

C))

S2 fL - 1 ( C(s --->r 1+---

s-1 -

D'

§6. Transversal designs and orthogonal arrays

709

as in Theorem 11.8.8. Let us write ni for the number of points x =f:. p with Ax = i. Then we have (6.2.a) and rewriting equations (IT.8.8.b, c) yields (6.2.b)

:L>ni = LAx = r(sp, - 1), xip

(6.2.c)

L i (i -

l)ni

=L

We shall consider the function (6.2.d)

Ax(A x

-

1)

= r(r -

l)(fL - 1).

xip

i

f : Z -7 Z defined by

fey) = L(i - y)(i - 1 - y)ni' i

which is easily seen to be non-negative for all Y E Z. From (6.2.a, b, c, d) we obtain

fey)

= LiCi -1)ni i

- 2y Lini i

= p,[r(r - 1) - 2rsy + y(y

+ y(y + I) Ln; i

+ 1)s2)

- [r(r - 1) - 2ry + y(y + 1)]; and

f

(y) ?:: 0 implies

(6.2.e)

r(r - I) - 2ry + y(y + 1) --------=-..:......;--..,- r(r - 1) - 2rsy + y(y + l)s2

fL>

With (6.2.f)

C:= r -1- ys

one obtains (6.2.g)

D := r(r -1) -2rsy+ y(y+ l)s2 = res -1) - (C + 1)(s -1- C)

and then from (6.2.e) after some computation (6.2.h)

C))

S2 fL - 1 ( C(s --->r 1+---

s-1 -

D'

711

§6. Transversal designs and orthogonal arrays

This is the result of Bose and Bush. As (b - p)(b + 1 - p)::: -~, we obtain 0= (b - p)(b

(6.4.d)

b

+ 1-

p) - s(b - 2p)::::

1

-4 -

s(b - 2p),

1

P -p.

This shows that Assumption 6.3 was reasonable. We have proved the following theorem of Bose and Bush (1952).

6.5 Theorem. a(s - 1)

(6.S.a)

If

a TDJL[r; s] with f.L > 0, s > 0 exists and 1, then

if

f.L - 1

+ b with a E No, 0 < b < s r<sf.L+f.L+a-p,

where p = -(s - b -

~) + JS(S

- 1 - b)

+~ .•

6.6 Remark. If ~ is small, then the order of magnitude of p may be seen from the Taylor expansion (6.6.a)

1 b (1 +b) 2 p >b+ - -s+Js(s -1- b) = - - -'----

2

2

8s

(1 +b)3

16s 2

Of course it is desirable to exhibit cases where (6.S.a) is best possible. To this end we introduce a construction using symmetric TD's. As these are simultaneously nets we may use the language of nets which is more convenient here. The basic idea goes back to Shrikhande (1964). It has been used by Hine and Mavron (1980) as well as by Jungnickel (1979) to construct TD's or nets with large values of r.

6.7 Lemma. Let:E be a symmetric (s, f.L)-net (i.e. an STDJL(s») with s I f.L, (cf I.7.17) and assume the existence of an (s, t; t;)-net. Then :E may be extended to an (s, Sf.L + t; f.L)-net by adding t new parallel classes of blocks. Proof. Let D be an (s, t; ;)-net and choose a bijection a mapping the Sf.L points of D onto the s f.L point classes of 'E. For each block B of D, write B" for

713

§7. Completion theorems for Bruck nets

6.10 Exercise (Bose and Bush 1952). Let p be a prime, i a positive, and j a non-negative integer. Define a, b by j = ai + b with 0 -:s. b < i. Show that

(6. 10. a) Compare this to (Vill.3.12.a).

6.11 Remarks. (a) Some infinite series of transversal designs which nearly meet the Bose-Bush bound (Theorem 6.5) were constructed by de Launey (1987b). (b) In view of Lemma VIDA A, the Bose-Bush bound may be viewed as a bound on orthogonal arrays of strength 2. In this context, we mention the following bound for orthogonal arrays of strength t which is due to Bierbrauer (1995a). For large values of t, this result improves the bounds of Rao (1947) which are also derived in Bierbrauer's paper. It also covers the cases s of the bound of Bush (1952) which concerns the special case J.. = 1.

t:::

6.12 Theorem. Assume the existence of an OA). (t, k, s). Then (6.I2.a)

b=

ASt

>Sk(I- -I)k) .• + -

(s s(t

1)

§7. Completion Theorems for Bruck Nets In this section we consider nets with f.t = 1, i.e. dual structures of transversal designs with A = 1. Bruck (1963) has studied possible embeddings of such nets into affine planes (by adjoining new parallel classes of lines). His results have been improved significantly by Metsch (199Ia). Together with the Bruck-Ryser theorem their results imply the non-existence of nets for certain parameters.

7.1 Definition. Let D be an (s, r; I)-net, that is an affine Sr(1, s; S2) or the dual structure of a TD[r, s]. Then the deficiency d of D is defined by (7. La)

d:= s

+ 1- r;

thus d measures how far r falls short from its conceivable maximum s + 1, cf. Corollary II.2.13. If D can be embedded to an affine plane, say E, by adjoining new parallel classes of lines, then E is called a completion of D, and D is said to be completable (or embeddable).

713

§7. Completion theorems for Bruck nets

6.10 Exercise (Bose and Bush 1952). Let p be a prime, i a positive, and j a non-negative integer. Define a, b by j = ai + b with 0 -:s. b < i. Show that

(6. 10. a) Compare this to (Vill.3.12.a).

6.11 Remarks. (a) Some infinite series of transversal designs which nearly meet the Bose-Bush bound (Theorem 6.5) were constructed by de Launey (1987b). (b) In view of Lemma VIDA A, the Bose-Bush bound may be viewed as a bound on orthogonal arrays of strength 2. In this context, we mention the following bound for orthogonal arrays of strength t which is due to Bierbrauer (1995a). For large values of t, this result improves the bounds of Rao (1947) which are also derived in Bierbrauer's paper. It also covers the cases s of the bound of Bush (1952) which concerns the special case J.. = 1.

t:::

6.12 Theorem. Assume the existence of an OA). (t, k, s). Then (6.I2.a)

b=

ASt

>Sk(I- -I)k) .• + -

(s s(t

1)

§7. Completion Theorems for Bruck Nets In this section we consider nets with f.t = 1, i.e. dual structures of transversal designs with A = 1. Bruck (1963) has studied possible embeddings of such nets into affine planes (by adjoining new parallel classes of lines). His results have been improved significantly by Metsch (199Ia). Together with the Bruck-Ryser theorem their results imply the non-existence of nets for certain parameters.

7.1 Definition. Let D be an (s, r; I)-net, that is an affine Sr(1, s; S2) or the dual structure of a TD[r, s]. Then the deficiency d of D is defined by (7. La)

d:= s

+ 1- r;

thus d measures how far r falls short from its conceivable maximum s + 1, cf. Corollary II.2.13. If D can be embedded to an affine plane, say E, by adjoining new parallel classes of lines, then E is called a completion of D, and D is said to be completable (or embeddable).

§7. Completion theorems for Bruck nets

7IS

7.5 Lemma. Let D be an (s, r; I)-net of deficiency d, and let p and q be distinct points, L a line, and T a partial transversal ofD. Then: (7.S.a)

ITI ::: s with equality if and only ifT meets each line.

(7.S.b)

p is joined to n! := res - 1) points and not joined to n2 := des - 1) points.

(7.S.c)

Ifp is not on L, then p is collinear with r - 1 points of L and not joined to the remaining s - r + 1 = d points of L.

(7.S.d)

If T is a transversal not containing p, then p is collinear with r points ofT and not joined to the remaining s - r = d - 1 points ofT.

(7.S.e)

If p and q are collinear; then there are precisely P~1 = s - 2 + (r - 1)(r - 2)

points x

p12 =

=I p, q which are collinear with both p and q;

d(r - 1)

points x collinear with p but not with q; and

P~2 = d(d - 1) points x collinear with neither p nor q. (7.S.£)

Ifp and q are not collinear, then there are precisely pi!

= r(r -

points x pi2

1)

=I p, q which are collinear with both p and q;

= red -

1)

points x collinear with p but not with q; and

P~2

= s - 2 + Cd -

I)(d - 2)

points x collinear with neither p nor q. • Remark: The notation P~k (i, j, k E (I, 2}) is chosen as follows. The upper index denotes whether or not p and q (the given two points) are collinear; the lower ones whether or not the points x to be considered are collinear with p or q. Here "being collinear" is denoted by 1, and "not being collinear" by 2.

7.6 Lemma. Let D be an (s, r; 1)-net of deficiency d > 0, and let T be a transversal and S a partial transversal of D such that S % T but IS n TI ::: 2. Then one has (7.6.a)

IS n TI ::: d - 1 and lSI::: Cd -

1)2;

§7. Completion theorems for Bruck nets

7IS

7.5 Lemma. Let D be an (s, r; I)-net of deficiency d, and let p and q be distinct points, L a line, and T a partial transversal ofD. Then: (7.S.a)

ITI ::: s with equality if and only ifT meets each line.

(7.S.b)

p is joined to n! := res - 1) points and not joined to n2 := des - 1) points.

(7.S.c)

Ifp is not on L, then p is collinear with r - 1 points of L and not joined to the remaining s - r + 1 = d points of L.

(7.S.d)

If T is a transversal not containing p, then p is collinear with r points ofT and not joined to the remaining s - r = d - 1 points ofT.

(7.S.e)

If p and q are collinear; then there are precisely P~1 = s - 2 + (r - 1)(r - 2)

points x

p12 =

=I p, q which are collinear with both p and q;

d(r - 1)

points x collinear with p but not with q; and

P~2 = d(d - 1) points x collinear with neither p nor q. (7.S.£)

Ifp and q are not collinear, then there are precisely pi!

= r(r -

points x pi2

1)

=I p, q which are collinear with both p and q;

= red -

1)

points x collinear with p but not with q; and

P~2

= s - 2 + Cd -

I)(d - 2)

points x collinear with neither p nor q. • Remark: The notation P~k (i, j, k E (I, 2}) is chosen as follows. The upper index denotes whether or not p and q (the given two points) are collinear; the lower ones whether or not the points x to be considered are collinear with p or q. Here "being collinear" is denoted by 1, and "not being collinear" by 2.

7.6 Lemma. Let D be an (s, r; 1)-net of deficiency d > 0, and let T be a transversal and S a partial transversal of D such that S % T but IS n TI ::: 2. Then one has (7.6.a)

IS n TI ::: d - 1 and lSI::: Cd -

1)2;

717

§7. Completion theorems for Bruck nets

Thus the new lines also satisfy the parallel axiom (I.2.4.b) and thus E is indeed an affine plane of order s. Assume now that F is any completion of D; then the lines of F \ D are transversals of D, and there are exactly sd such transversals. By (7.8.a), these are all the transversals of D, proving E = F . • Note that Proposition 7.8 provides a prooffor the first part of Theorem 7.2. We stress that one may well have t < sd even if (7.6.b) is satisfied; in §9 we will obtain examples of such nets which are even transversal-free, that is, they have t = O. In order to prove the second part of Theorem 7.2 as well as Theorem 7.3, we will associate two strongly regular graphs with each net. The following result is an immediate consequence of Lemma 7.5.

7.9 Lemma. Let D be an (s, r; I)-net, and define a graph G on the s2 points ofD asfollows:

(7.9.a)

{p, q} is an edge of G if and only

if p and q are collinear.

Then G is a strongly regular graph with parameters

(7.9.b)

n = S2, k = res - 1), J-L = r(r -1) . •

A = s - 2 + (r - l)(r - 2)

and

7.10 Definition. In view of Lemma 7.9, we call a strongly regular graph G with parameters (7.9.b), where 0 < r :::: s, apseudonet graph with parameters s and r. If G is actually the graph of an (s, r; 1)-net as in Lemma 7.9, it is called a net graph (or a geometric pseudonet graph). With these definitions, we obtain the following stronger version of Lemma 7.9; again, the proof is immediate from Lemma 7.5.

7.11 Proposition. LetD be an (s, r; I)-net, and define a graph G as in Lemma 7.9. Then G is a net graph with parameters sand r, and its complementary graph G is a pseudonet graph with parameters sand d := s + I - r. • In what follows, the second assertion of Theorem 7.2 and its improvement in Theorem 7.3 will be established by proving that any pseudonet graph with parameters sand d satisfying (7.2.b) or (7.3.a) is actually geometric; thus the graph G of Proposition 7.11 arises from an (s, d; I)-net in these cases, and adjoining the lines of this net to D yields the required completion of D. We will require two special types of subgraphs in this approach.

717

§7. Completion theorems for Bruck nets

Thus the new lines also satisfy the parallel axiom (I.2.4.b) and thus E is indeed an affine plane of order s. Assume now that F is any completion of D; then the lines of F \ D are transversals of D, and there are exactly sd such transversals. By (7.8.a), these are all the transversals of D, proving E = F . • Note that Proposition 7.8 provides a prooffor the first part of Theorem 7.2. We stress that one may well have t < sd even if (7.6.b) is satisfied; in §9 we will obtain examples of such nets which are even transversal-free, that is, they have t = O. In order to prove the second part of Theorem 7.2 as well as Theorem 7.3, we will associate two strongly regular graphs with each net. The following result is an immediate consequence of Lemma 7.5.

7.9 Lemma. Let D be an (s, r; I)-net, and define a graph G on the s2 points ofD asfollows:

(7.9.a)

{p, q} is an edge of G if and only

if p and q are collinear.

Then G is a strongly regular graph with parameters

(7.9.b)

n = S2, k = res - 1), J-L = r(r -1) . •

A = s - 2 + (r - l)(r - 2)

and

7.10 Definition. In view of Lemma 7.9, we call a strongly regular graph G with parameters (7.9.b), where 0 < r :::: s, apseudonet graph with parameters s and r. If G is actually the graph of an (s, r; 1)-net as in Lemma 7.9, it is called a net graph (or a geometric pseudonet graph). With these definitions, we obtain the following stronger version of Lemma 7.9; again, the proof is immediate from Lemma 7.5.

7.11 Proposition. LetD be an (s, r; I)-net, and define a graph G as in Lemma 7.9. Then G is a net graph with parameters sand r, and its complementary graph G is a pseudonet graph with parameters sand d := s + I - r. • In what follows, the second assertion of Theorem 7.2 and its improvement in Theorem 7.3 will be established by proving that any pseudonet graph with parameters sand d satisfying (7.2.b) or (7.3.a) is actually geometric; thus the graph G of Proposition 7.11 arises from an (s, d; I)-net in these cases, and adjoining the lines of this net to D yields the required completion of D. We will require two special types of subgraphs in this approach.

719

§7. Completion theorems for Bruck nets

since each ofthes(s-l) pairs (v, w) E C X C is adjacenttos-2+(d -l)(d -2) vertices, s - 2 of which belong to C. From (7.13.e, f, g) we obtain (7.13.h) Henceai =

ofor i i=d -1 and ad-l =

s2-s, proving (7. 13.b), sinced -1 < s.

It remains to consider two cliques C and C'. If C U C' is not a clique, then C U C' contains two non-adjacent vertices, and thus (7.9.b) implies (7.13.c). Similarly, if C n C' contains two vertices, then (7.9.b) implies (7.13.d) . • Note that the lines of a net graph yield cliques of order s, that is the maximum feasible value according to (7 .13.a). The following two results are due to Metsch (1991a), improving the work of Bruck (1963).

7.14 Lemma. Let G be a strongly regular graph with parameters n, k, J... and fh, and let a be a positive integer. Then G contains no claw of order a provided that

(7.14.a)

k < (a

+ 1)(J... + 1) -

1

l(a

+ 1)a(fh -

1).

Proof. Assume the existence of a claw (v, A) of order a {vo, ... , va}, and denote by

hypothesis, IVi I =

J...

+1

+ 1. Write A

=

Vi the set of common neighbours of v and Vi' By

for all i. Note

(7.14.b)

since every vertex that is counted x times on the left-hand side is counted 1 + 1x(x - 1) times on the right-hand side and since x :s 1 + 1x(x - 1) for all positive integers x. For distinct indices i and j, we have IVi n Vj I :s fh - 1, since every vertex in V; n Vj is a common neighbour of Vi and vj which is different from v. Since the set U~=O Vi consists of some of the k vertices adjacent to v which are not in A, it follows from (7.14.b) that

1

(a + 1)J... :::: k - (a + 1) + l(a + l)a(fh - 1),

contradicting (7. 14.a). •

719

§7. Completion theorems for Bruck nets

since each ofthes(s-l) pairs (v, w) E C X C is adjacenttos-2+(d -l)(d -2) vertices, s - 2 of which belong to C. From (7.13.e, f, g) we obtain (7.13.h) Henceai =

ofor i i=d -1 and ad-l =

s2-s, proving (7. 13.b), sinced -1 < s.

It remains to consider two cliques C and C'. If C U C' is not a clique, then C U C' contains two non-adjacent vertices, and thus (7.9.b) implies (7.13.c). Similarly, if C n C' contains two vertices, then (7.9.b) implies (7.13.d) . • Note that the lines of a net graph yield cliques of order s, that is the maximum feasible value according to (7 .13.a). The following two results are due to Metsch (1991a), improving the work of Bruck (1963).

7.14 Lemma. Let G be a strongly regular graph with parameters n, k, J... and fh, and let a be a positive integer. Then G contains no claw of order a provided that

(7.14.a)

k < (a

+ 1)(J... + 1) -

1

l(a

+ 1)a(fh -

1).

Proof. Assume the existence of a claw (v, A) of order a {vo, ... , va}, and denote by

hypothesis, IVi I =

J...

+1

+ 1. Write A

=

Vi the set of common neighbours of v and Vi' By

for all i. Note

(7.14.b)

since every vertex that is counted x times on the left-hand side is counted 1 + 1x(x - 1) times on the right-hand side and since x :s 1 + 1x(x - 1) for all positive integers x. For distinct indices i and j, we have IVi n Vj I :s fh - 1, since every vertex in V; n Vj is a common neighbour of Vi and vj which is different from v. Since the set U~=O Vi consists of some of the k vertices adjacent to v which are not in A, it follows from (7.14.b) that

1

(a + 1)J... :::: k - (a + 1) + l(a + l)a(fh - 1),

contradicting (7. 14.a). •

§7. Completion theorems for Bruck nets

721

Step 3. Let (v, A) be a claw with IAI = a v , and let W E A. Then there exists a unique grand clique C containing both v and w. The uniqueness of C is guaranteed by Step 1. Denote by C the set of vertices which are adjacent to both v and w, but to no vertex in A \ {w}. Since wand every other vertex of A have exactly fL - 1 common neighbours distinct from v, we obtain ICI :::).. - (a v - 1)(fL - 1). Any two distinct vertices u and u' of C are adjacent, since otherwise (v, (A \ {w}) U {u, u'}) would be a claw of size a v + 1. Hence C is a clique. As a v :::: a, it follows that C' := C U {v, w} is a clique with at least).. + 2 - (a - 1)(fL - 1) vertices. Therefore any maximal clique containing C' is a grand clique. Step 4. Any vertex v lies in exactly a v grand cliques. Let (v, A) be a claw with IAI = avo In particular, (v, A) is a maximal claw. Step 3 shows that every vertex of A occurs in a unique grand clique containing v, and Step 2 shows that every grand clique through v contains a vertex of A, proving the assertion. Step 5. Any two adjacent vertices v and w occur together in a unique grand clique. Let (v, A) be any maximal claw containing the claw (v, {w}). Then IAI :::: avo By Step 4, there exist exactly au grand cliques containing v, and by Step 2 each of these grand cliques contains a vertex of A. Since distinct grand cliques through v cannot contain any other common vertex by Step 1, it follows that IAI = av and that every vertex of A occurs in a grand clique containing v. Step 6. Let C be a grand clique, and let v and w be two distinct vertices contained in C. Then ICI :::).. + 2 - (a v - 1)(aw - 1). By Step 5, every vertex u which is adjacent to both v and w gives rises to grand cliques C u and Du with u, v E C u and u, WE Du. If u tf. C, then C u, Du i= C and, by Step 1, Cu n Du = {u}. Hence distinct common neighbours u and u' of v and w which are not in C determine different pairs of cliques (C u , Du) and (Cu" D u')' Therefore Step 4 implies that v and w have at most (a v -1)(a w -1) common neighbours which are not in C. Hence).. :::: IC I - 2 + (a v - 1) (a w - 1). Step 7. Every grand clique has order s, and a v = d for every vertex v. Let v be some vertex of G, putt:= a v , and let C 1 , ••• , C t be the grand cliques containing v. By Step 5, each neighbour of v lies in exactly one of these t cliques. Hence t

(7.15.e)

L(IC;I - 1)

= k = des -

1).

;=1

By (7. 13.a), we have ICd :::: s for all i; hence t::: d, with equality if and only if all cliques containing v have order s. Now assume t ::: d + 1 and, without

§7. Completion theorems for Bruck nets

721

Step 3. Let (v, A) be a claw with IAI = a v , and let W E A. Then there exists a unique grand clique C containing both v and w. The uniqueness of C is guaranteed by Step 1. Denote by C the set of vertices which are adjacent to both v and w, but to no vertex in A \ {w}. Since wand every other vertex of A have exactly fL - 1 common neighbours distinct from v, we obtain ICI :::).. - (a v - 1)(fL - 1). Any two distinct vertices u and u' of C are adjacent, since otherwise (v, (A \ {w}) U {u, u'}) would be a claw of size a v + 1. Hence C is a clique. As a v :::: a, it follows that C' := C U {v, w} is a clique with at least).. + 2 - (a - 1)(fL - 1) vertices. Therefore any maximal clique containing C' is a grand clique. Step 4. Any vertex v lies in exactly a v grand cliques. Let (v, A) be a claw with IAI = avo In particular, (v, A) is a maximal claw. Step 3 shows that every vertex of A occurs in a unique grand clique containing v, and Step 2 shows that every grand clique through v contains a vertex of A, proving the assertion. Step 5. Any two adjacent vertices v and w occur together in a unique grand clique. Let (v, A) be any maximal claw containing the claw (v, {w}). Then IAI :::: avo By Step 4, there exist exactly au grand cliques containing v, and by Step 2 each of these grand cliques contains a vertex of A. Since distinct grand cliques through v cannot contain any other common vertex by Step 1, it follows that IAI = av and that every vertex of A occurs in a grand clique containing v. Step 6. Let C be a grand clique, and let v and w be two distinct vertices contained in C. Then ICI :::).. + 2 - (a v - 1)(aw - 1). By Step 5, every vertex u which is adjacent to both v and w gives rises to grand cliques C u and Du with u, v E C u and u, WE Du. If u tf. C, then C u, Du i= C and, by Step 1, Cu n Du = {u}. Hence distinct common neighbours u and u' of v and w which are not in C determine different pairs of cliques (C u , Du) and (Cu" D u')' Therefore Step 4 implies that v and w have at most (a v -1)(a w -1) common neighbours which are not in C. Hence).. :::: IC I - 2 + (a v - 1) (a w - 1). Step 7. Every grand clique has order s, and a v = d for every vertex v. Let v be some vertex of G, putt:= a v , and let C 1 , ••• , C t be the grand cliques containing v. By Step 5, each neighbour of v lies in exactly one of these t cliques. Hence t

(7.15.e)

L(IC;I - 1)

= k = des -

1).

;=1

By (7. 13.a), we have ICd :::: s for all i; hence t::: d, with equality if and only if all cliques containing v have order s. Now assume t ::: d + 1 and, without

X. Transversal designs and nets

722

loss of generality, a w :::: a v = t for every vertex w i= v. Then Step 6 shows IC;! :::}.. + 2 - (t - 1)2 for all i, and thus (7.I5.e) yields (7.I5.f)

teA

Using A + 1 equivalent to

=

+ 1) -

t(t - 1)2 :::: des - I).

s - 1 + (d - I)(d - 2), inequality (7.15.f) turns out to be

-

t(t - 1)2 - t(d - l)(d - 2) t-d

= t(t

+d -

s < 1+ ----------

In view of d+ 1 :::: t = a v

2)

::::

+d +

d(d - 1) t_ d .

a, this contradicts (7.I5.c) and proves the assertion.

Step 8. G is the graph of a uniquely determined (s, d; I)-net.

By Step 7, all grand cliques have order s and every vertex lies in exactly d grand cliques. Thus the only feasible definition for the desired netD is to take all grand cliques of G as lines. Then each line has order s and any two lines intersect in at most one point. It remains to check the parallel axiom. By (7.13.b), a vertex v outside a given grand clique C has precisely d - 1 neighbours in C. Therefore Step 5 and Step 1 show that v lies on precisely d - 1 grand cliques meeting C in a unique vertex, and thus there is a unique grand clique through v which is disjoint to C. Hence D indeed is an (s, d; I)-net, and in view of Step 5 G is the graph of this net. • We can now prove the following results of Bruck (1963) and Metsch (1991a) on pseudonet graphs.

7.16 Theorem. Let G be a pseudonet graph with parameters s::: 2 and assume (7.2.b) and s::: d > O. Then G is the graph of a uniquely determined (s, d; I)-net. Proof. We apply Theorem 7.15 with a = d. Thus we need to check the hypotheses (7.I5.a) and (7.I5.b) of Theorem 7.15. Using (7.15.d), this requires showing 3

s > 2d - 4d

2

+ 2d

and

14

s > -d - d 2

3

+ d 2+1-d 2

1.

The second inequality is just (7 .2.b). If d i= 3, then the second inequality implies the first, proving the theorem for d i= 3. Thus let d = 3.Then the two inequalities require s > 24 and s > 23, respectively. This proves the theorem for d = 3 and

§7. Completion theorems for Bruck nets

723

s > 24. For the single remaining case d = 3 and s = 24, we refer to the original paper of Bruck or the proof given in the first edition of this book. (It would be quite difficult to explain why the present approach does not cover this particular case.) •

7.17 Theorem. Let G be a pseudonet graph with parameters s::: 2 and assume (7.3.a) and s ::: d > O. Then G is the graph of a uniquely determined (s, d; I)-net. Proof. For d :s: 4, the statement follows from Theorem 7.16. Thus let d ::: 5. We apply Theorem 7.15 with a := (4d - 2 - R)/3. By the definition of R in Theorem 7.3, we see that a is an integer with a:::d + 1. Using (7.15.d), the three hypotheses of Theorem 7.15 can be written as follows:

+ (2a - l)(d 2 a(a + l)(d 2 s > 12 := 1 + s>

fl

:= I

d - 1) - (d - I)(d - 2),

d - 1) - 2(a + l)(d - l)(d - 2) 2(a + 1 _ d) ,

s > f3 :=a(a +d - 2) +d2. Since a = (4d - 2 - R) /3, it is tedious but not difficult to see that condition (7.3.a) is equivalent to s > fl if R Eto, I} and to s > 12 if R = 2. Moreover,

6(a

+1-

d)(fl - h) = (d 2 - d - 1)(2 - R)(4d - R - 5) - 6(d - 1)2.

Because of d::: 5, it follows that fI ::: 12 if R Eto, I} and 12::: fl if R = 2. Hence s > II and s > 12 for R E to, 1, 2}. Since 5 :s: d < a :s: 2d, we have

II - /3 =

(a - d)(2d 2

-

4d - a)

+ 2d3 -

7d 2 + 4d > O.

Hence also s > f3. •

7.18 Remarks. (a) Theorem 7.16 and Proposition 7.11 at once yield a proof of the second assertion of Theorem 7.2, and Theorem 7.17 and Proposition 7.11 at once yield a proof of Theorem 7.3.ln §9 we will give examples showing that the bounds in (7.2.b) and (7.3.a) are fairly good. We note that the Metsch bound (7.3.a) is better than the Bruck bound (7.2.b) for all values d::: 5. (b) Metsch (I991a) obtained Theorem 7.17 from a more general result for strongly regular graphs; we have here only treated the specialisation to pseudonet graphs. Metsch's more general theorem allows further applications to partial

X. Transversal designs and nets

724

geometries and to the embedding problem for linear spaces into projective planes, cf. Theorems IX.I1.14 and IX. I 1.15. Theorems 7.2 and 7.3 can be restated as follows: 7.19 Corollary. Assume the existence of an (s, r; I)-net of deficiency d. Then either there exists an affine plane (and hence also a projective plane) of order s, or d satisfies neither of the inequalities (7.2.b) and (7.3.a). • 7.20 Example. A net of order s and deficiency d is completable in each of the following cases: d = 1; d = 2, s ¥= 4, a result of Shrikhande (1961); d =3, s::: 24; d =4, s::: 82; d = 5, s::: 179 etc. Using the Bruck-Ryser Theorem II.4.8 and Remark II.4.12, we obtain

(7.20.a)

N(s) :::: s - 4

for s = 6,10,14,21,22;

(7.20.b)

N(s) :::: s - 5

for s = 30,33,38,42,46,54,57,62,66,69, 70,77,78;

(7.20.c)

N(s) :::: s - 6

for s = 86, 93, 94, etc.

We finally mention another consequence of Theorem 7.16 which is due to Shrikhande (1959b). 7.21 Corollary. Let G be a strongly regular graph with parameters

(7.21.a)

n

= s2,

k

= 2(s -

1),

A=s - 2

and

f.-L

= 2.

If s ¥= 4, then G is uniquely determined. Proof. This is easily checked for s = 1,2,3. Note that the hypothesis means that G is a pseudonet graph with parameters sand d = 2. By Theorem 7.16, G is geometric for s > 4. Thus G is constructed from an (s, 2; I)-net as in Lemma 7.9. But clearly any two (s, 2; I)-nets are isomorphic . • 7.22 Remark. The graphs occurring in Corollary 7.21 are usually called the L2-graphs. Let us remark that there is a unique exceptional graph G in the case s = 4, see Shrikhande (1959b). G is the Cayley graph based on Z4 x Z4 with S = {±(1, 0), ±(O, 1), ±(1, I)}, where we use the notation introduced in Remark III. 10.2.

§8. Maximal nets with large deficiency

725

§8. Maximal Nets with Large Deficiency In this section we are again only concerned with nets with intersection parameter I-t = 1. The completion theorems of the preceding section guarantee that such nets satisfying certain parametric restrictions (i.e. with small deficiency d = s + 1 - r in the order of magnitude of 0) may always be completed regardless of their specific geometric structure. This immediately poses some interesting questions. First of all, one would like to know whether the bounds (7.2.b) and (7.3.a) are best possible or at least reasonably good. Then the more general question arises: for which parameter pairs (s, d) does a net of deficiency d exist which cannot be enlarged to a net of smaller deficiency or which perhaps has no transversals at all? 8.1 Definition. An (s, r; 1)-netD is called maximal if it cannot be embedded into an (s, r + 1; I)-net (which means that D has no set of s pairwise disjoint transversals). If D is not maximal, it is called extendible;J more specifically, if D may be embedded into an (s, r + t; I)-net, it is said to be step-t-extendible.

In this section we will construct maximal nets with few parallel classes, i.e. with large deficiency. The more difficult case of maximal nets with small deficiency (which is relevant to the completion theorems) will be considered in the next section. Maximal nets for I-t > 1 will be studied in Section 11. For I-t = 1, it is sometimes helpful to consider Latin squares instead of nets. 8.2 Definition. Let L = (lij )i,j=J, ... ,s be a Latin square of order s with entries from the set S. A transversal of L is a set T of s cells of L (i.e. T c {I, ... , S}2) such that the cells in T contain each element of S exactly once and intersect each row and each column of L exactly once. 8.3 Observations. Let L be a Latin square of order sand D the corresponding (s, 3; I)-net; cf. Lemma VIII.4.4. Then the transversals of L correspond bijectively to those ofD. More generally, ifD is an (s, r; I)-net and {L J , ••• , L r - 2 } a corresponding set of mutually orthogonal Latin squares, then the transversals of D correspond to those sets of cells T which are transversals for each Li (i = 1, ... , r - 2). 8.4 Example. Bruck's completion theorem is best possible for s = 4. To prove this, we have to find a maximal (4, 3; 1)-net. Consider the Latin square L of order

1

This is not to be confused with the notion of extendibility treated in Chapter II, §7.

726

X. Transversal designs and nets

4 which is the group table of Z4. It is easy to check that L has no transversal; hence the corresponding net D has no transversal and thus is maximal. In fact, D is the net we displayed in Remark 7.22. Example 8.4 may be generalised. If L is the group table of a group G of even order with a cyclic Sylow 2-subgroup, then L has no transversal; see Corollary 12.4 below. This provides examples of maximal (s, 3; I)-nets for all even values of s. We will now give a construction for such nets if s == 1 (mod 4). The following two results and the subsequent exercises are all due to Mann (1944).

8.5 Lemma. Let L be a Latin square of order s = 4m + 1 with a Latin subsquare of order 2m. Then there is no Latin square which is orthogonal to L; that is, L has no orthogonal mate. Proof. We may assume L to have the form

where A is a Latin subsquare of order 2m, say on the symbol set R. As L is a Latin square, no symbol in R can occur in either B or C; thus all entries from R occur in A or D, and D contains exactly (2m + 1)2 - 2m (2m + 1) = 2m + 1 entries not in R. Now let T be any transversal of L; then T contains the 2m symbols in R and thus T has at least 2m entries in A or D (and, possibly, more entries not in R in a cell of D). Now there exists some Latin square L * orthogonal to L iff L has a set of 4m + 1 pairwise disjoint transversals. Assume this to be the case. Then at most 2m + 1 of these transversals contain an entry not in R in a cell of D; so there is a transversal T which has no entry r ERin a cell of D. Hence its entries in the union U of A and D are exactly the 2m entries in R. Thus the corresponding symbol of L * occurs exactly 2m times in the cells of U. But if this symbol occurs precisely x times in the cells of A, then it occurs 2m - x times in the cells of B, and (2m + 1) - (2m - x) times in the cells of D, and thus 2m - 2x + 1 times in the cells of U, a contradiction. •

8.6 Theorem. There exists a maximal (s, 3; I)-net whenever s == 1 (mod 4). Proof. Let s = 4m + 1; by Lemma 8.5, it suffices to construct a Latin square L or order 4m + I with a subsquare of order 2m. We start with a Latin square A of order 2m + 1 with an orthogonal mate. We thus may choose 2m pairwise

§8. Maximal nets with large deficiency

727

disjoint transversals T I , ... , T2m of A. Let A be defined on the symbol set S = {2m + I, ... , 4m + I}; we shall define L on {I, ... , 4m + I}. Put

where D is any Latin square of order 2m defined on {I, ... , 2m}. Furthermore A is obtained from A by replacing all entries in the cells of Tj by j (j = 1, ... , 2m); finally, the entry in cell (i, j) of B (i = 1, ... , 2m + 1; j = 1, ... , m) is the entry of the unique cell ofTj in row i of A, and similarly for C (using the columns of A). It is easily checked that L is indeed a Latin square .

•

8.7 Exercise. (a) Let L be a Latin square of order s with a subsquare of order k. Prove k ::: ~. (b) Let s, k be positive integers with k ::: ~. Show the existence of a Latin square of order s with a subsquare of order k if s - k ;;j::. 2, 6 and discuss the exceptional cases s - k = 2 or 6. The result is then still true; cf. Denes and Keedwell (1974), p. 43.

8.8 Exercise. Show that a Latin square of order 4m + 2 with a subsquare of order 2m + 1 has no transversal. Construct such Latin squares. Hint: let L = (~ ~) be as in the proof of Lemma 8.5, and let x be the number of cells of a transversal T of L in A. Then count the number of cells in T containing symbols different from those in A to obtain a contradiction. We will next give another construction yielding maximal nets with large deficiency, this is due to Drake (1977). 8.9 Theorem. Let D' = (V', B') be an (s, s + 1; I)-net (i.e. an affine plane of order s) and D" = (V", B") a (t, s + 1; I)-net, where s does not divide t. Then D := D' x D" = (V, B) has no transversal. Proof. Forthe definitionofD' xD", see I.7.21. Let T be a transversal ofD, say T = {(PI, ql), ... , (PSI, qSI)} with Pi E V', qi E V"(i = 1, ... , st), and define f : Vi -+ N U {OJ by (8.9.a)

f(p) := I{i E {I, 2, ... , Sf} : Pi = p}l;

X. Transversal designs and nets

728

thus f (p) is the number of points of T with first coordinate p. Let L be any line of D'; then L gives rise to t parallel lines of D (of the form L x M, M E B"). As T is a transversal of D, it intersects each of these t lines exactly once. Hence (8.9.b)

L

f(L) :=

f(p) = t

for each L EB',

pEL

as the union of the t lines under consideration is precisely the set of all points

(p,q)EVwithPEL. Now assume that s does not divide t. By (8.9.b), the average value of f(p) is = tis (using a parallel class of lines of D'). As sft there is a point p with f(p) > ;; say f(p) = ~ + x, x > O. If L is any line containing p, then f(L \{p}) = t - ~ - x = (s - l)~ - x. SinceD' is an affine plane, any point q i= p is on exactly one line L through p. Hence we get

stls 2

st =

L

f(q)

= f(p) + L

qEV'

=

G+x) +

f(q)

qip

(s

+ l)((S -1)~

-x)

<st,

a contradiction. •

8.10 Corollary. Let s = ql ... qn be the prime power factorisation of the integer s, where ql < q2 < ... < qn. Then there exists a maximal (s, ql + 1; 1)net. • Note that Corollary 8.10 which is due to Bruck (1951) is a strengthened version of MacNeish's Corollary 1.7.8. Corollary 8.10 shows that the MacNeish bound cannot be improved by trying to enlarge a net constructed by MacNeish's method, where one affine plane has been used. It may be remarked that Drake's proof is much simpler than the original one by Bruck. Actually, Drake's results are even more general, since he considers a generalisation of the direct product construction, where several distinct nets of order t may be used as in Theorem 8.9. Also, Theorem 8.9 has further interesting consequences.

8.11 Corollary. (a) There exists a maximal (n, 3; I)-net whenever n =2 (mod 4).

(b) There exists a maximal (n, 4; I)-net whenever n == 3 or 6 (mod 9), n i= 6. (c) There exists a maximal (n, 5; l)-netwhenevern =4,8 or12 (mod 16) and n >40.

§8. Maximal nets with large deficiency

729

(d) Let s be a prime power and K := {t EN: sit and s2 t t}. Then there exists a maximal (t, s + 1; I)-net for all butfinitely many t E K. Proof. We shall prove part (c); the other parts are similar. We may use Theorem 8.9 with s = 4 and t = n / s; then the existence of D" is guaranted by Theorem 3.14 as t = n/s > 10. For part (d), use the Chowla-Erdos-Straus Theorem 5.4 .

•

We conclude this section with a few remarks. 8.12 Remark. In Theorem 8.9 we gave a construction method for nets without transversals and so the examples of Corollaries 8.10 and 8.11 are not only maximal, but indeed have no transversals. In Exercise 8.8, Latin squares (and thus nets with three parallel classes) without transversals were constructed, and in Section 12 we will show the existence of such a Latin square for every even order s. In contrast to this result, it has been shown that every Latin square has a large partial transversal, the best known bound being at present s - c(log s)2 for order s, see Shor (1982). No comparable result is known for nets in general. 8.13 Remark. By Theorem 8.6 and Corollary 12.5, we know the existence of a Latin square without an orthogonal mate for all orders s i= 3 (mod 4). Regarding the case s = 7, the complete classification of all Latin squares of order 7 is known; this is essentially due to Norton (1939) who had missed one species which was discovered by Sade (1951). It turns out that only six out of the 147 species of Latin squares of order 7 possess orthogonal mates, see van Rees (1990). It is still not known whether there exists any Latin square of order s == 3 (mod 4), where s?:: 11, which has no orthogonal mate. In this context, the following general result of van Rees (1990) is quite interesting; the proof uses ideas similar to those used in proving the classical results Lemma 8.5 and Theorem 8.6. In fact, van Rees also proves these older results and gives a nice general account of the topic of subsquares and transversals in Latin squares. We also note that van Rees conjectures that asymptotically almost no Latin square of order s has an orthogonal mate. 8.14 Theorem. For every positive integer m, there exists a Latin square L of order 4m + 3 for which no two orthogonal mates can be orthogonal to one another. In other words, L is contained in a maximal set of MOLS of size::: 2.

•

In some cases, a more precise result is possible, as the following theorem of Drake, van Rees and Wallis (1999) shows.

730

x. Transversal designs and nets

8.15 Theorem. Let v =1= 1,19 be a positive integer that satisfies one 0/ the congruences v '= 1 or 7 (mod 9), v'= 11 (mod 18). Then there exists a pair 0/ orthogonal Latin squares 0/ order v which do not have a common orthogonal mate. • Evans (1991) proved the following quite general theorem on the existence of maximal nets which complements the results by Bruck and Drake, respectively, given in Corollaries 8.1 0 and 8.11. The idea of proof of his result will be sketched in §12, see Remark 12.21.

8.16 Theorem. There exists a maximal (s, r)-net in each cases:

0/ the following

+ 1;

(8.16.a)

s is a power o/someprime p, andr = p

(8.I6.b)

s = mpG and r = p + 1, where p is a prime not dividing m provided that there exists an (m, r)-net. •

=

8.17 Corollary. Let s mpG and r = p + 1, where p is a prime not dividing m. Then there exists a maximal (s, r )-net whenever m is sufficiently large. • What "sufficiently large" means for some values of p can be seen from the table given in Remark 4.2. For instance, if p = 3, we may take m 2: 7; and for p = 5, we may take m 2: 23. Drake, van Rees and Wallis (1999) provided two further constructions for small maximal sets of MOLS. The first of these uses Singer cycles for the classical symmetric designs, whereas the second one makes use of vectors V(k, s) as discussed in Remark VIIIA.31.

8.18 Theorem. Let nand n + 1 be prime powers and d a positive integer. Suppose either that d is even or that n + I is prime. Then there exists a maximal set o/n MOLS %rder n d + ... + n + 1. • 8.19 Theorem. There exists a maximal set o/three MOLS %rder 8t every positive integer t =1= 3,5 such that 6t + 1 is a prime power. •

+ l/or

8.20 Remark. We finally give a table of the known results concerning the existence of maximal sets of mutually orthogonal Latin squares of small order, namely for orders s :::: 13. A larger table of this type may be found in Section IY.27 of Colboum and Dinitz (1996a); the results of Drake, van Rees and Wallis

1

§9. Translation nets and maximal nets

731

1999 and of Bedford and Whitaker (1998) allow some improvements in this table.

In the following table, non-existence always is a consequence of Bruck's completion theorem, see Example 7.20, with the following two exceptions. The non-existence of a pair of orthogonal Latin squares of order 6 will be proved in § 13, and the non-existence of a maximal set of three MOLS of order 7 was established by Drake (1977) using the complete classification of all Latin squares of order 7 already discussed in Remark 8.13. Except for the trivial case of complete sets of MOLS corresponding to affine planes, a reason for the existing cases is given in the column "references", where we refer to a result in the present book whenever possible. Order

Exists

Does not exist

Undecided

2 3 4 5 6 7 8 9

1 2 1,3 1,4 1 1,2,6 1,2,3,7 1,2,3,5,8

1 2 2,3 2,3,4,5 3,4,5 5,6 6, 7

4 4

10 11 12

1,2 2,3,4,10 1,2,3,4,5

7,8 8,9 9,10

3,4,5,6 1,5,6,7 6,7,8,11

l3

1,2,3,4,6,12

10,11

5,7,8,9

References

8.13, 12.23 12.5, 12.18 8.6,8.16, 12.18,9.28 8.8, VIII.4.9 12.23, Evans (1987) 12.5.8.11, Chang, Hsiang and Tai (1965), 12.18, Bedford and Whitaker (1998) 8.6,12.26, 8.18, 12.23

§9. Translation Nets and Maximal Nets with Small Deficiency The aim of this section is the construction of maximal nets (with J-i = 1) with smaIl deficiency for square orders q2. All the examples which we will present admit a nice type of point regular automorphism group and are constructed with the aid of projective geometry. This requires some effort; we begin with a basic study of "translation nets", which applies to the case J-i > 1 as well. 9.1 Definition. LetD = (V, B) be a net, and let G be an automorphism group of D. Then D is called a translation net with translation group G if G acts regularly on the point set V and fixes each parallel class of D.

732

X. Transversal designs and nets

9.2 Remark. In the special case of affine planes, Definition 9.1 agrees with the usual definition of translation planes; for detailed studies of these objects, see the books by Liineburg (1980) and Kallaher (1981) as well as the survey by Kallaher (1995) However, the general case exhibits features which are quite different from what holds for translation planes. For instance, Sprague (1982) gave examples showing that a Bruck net may be a translation net with respect to different, even non-isomorphic translation groups. Therefore a general translation net should always be considered as a pair (D, G). Indeed, the study of individual "translations" of a net poses certain problems; the reader is referred to Hachenberger and Jungnickel (1990) for a detailed discussion of this topic. We now give a group theoretic description of translation nets which generalises the representation of translation planes in terms of "congruence partitions" as given by Andre (1954) and Bruck and Bose (1964).

9.3 Definition. Let G be an additively written group of order S2p." and let U I , ... , Ur (withr::: 3) be subgroups ofordersp., ofG. ThenU:= {UI, ... , Ur } is called a partial congruence partition (in short, an (s, r; p.,)-PCP) in G if one has (9.3.a) Note that this condition is equivalent to (9.3.b)

Ui

+ Uj

= G

for i, j = 1, ... , r with i ;6 j.

The subgroups in U are called the components of U. If U cannot be extended to a larger PCP by adjoining a further component, U is called maximal. If U is any set of subgroups of some group G, we define an incidence structureD(U) as follows: (9.3.c)

D(U) := (G, {U

+ g: U E U, g E G}, E).

An incidence structure D is said to be group constructible if it is isomorphic to some incidence structure of the form D(U). 9.4 Lemma. The group constructible nets are precisely the translation nets. Moreover, D(U) is an (s, r; p.,)-translation net with translation group G if and only ifU is an (s, r; p.,)-PCP in G. Proof. Let D be a group constructible net, say D = D(U), where U is a set of subgroups of G. Then the points of D are just the elements of G, and the blocks

733

§9. Translation nets and maximal nets

of D through the point 0 are just the components U of U, by (9.3.c). For U E U, all cosets of U are likewise blocks of D(U). Since parallelism in a net is defined by disjointness, the parallel class of U has to be the set of cosets of U. Thus G fixes each parallel class, and D(U) is a translation net. Conversely, let D be a translation net with translation group G. As G acts regularly on the point set V of D, we may identify V with G after choosing a "base point" p; thus q E V is identified with g E G if and only if p8 q. Let B be any block of D through 0 (= p); as each g E G maps B onto a parallel block, we see that the trace of B is identified with the stabiliser G B of B in G, that is, with a subgroup of G. Thus the blocks of D through 0 define a set U of r subgroups of G; as G fixes parallel classes, the parallel class of U E U is {U + g: g E G}, and thereforeD ';::'; D(U).

=

Finally, D(U) will be an (s, r; tL)-net if and only if G has order s2 tL, U has r components of order s tL each, and any two distinct subgroups in U intersect in tL common points; that is, if and only if U is an (s, r; tL)-PCP in G . •

9.S Corollary. Let U be an (s, r; tL)-PCP in G. Then

(9.S.a)

S2tL - 1 r 2, we may choose three lines A, B, C ER" \ {L}. Since L ER(A, B, C) \ {R'}, the spreadS' is not regular. It is not difficult to see that the spread belonging to a translation net is essentially unique, cf. Liineburg (1980), Theorem 1.11. Hence A' ;t A, by Lemma 9.19 . • 9.21 Remarks. (a) Bruck and Bose (1964) also proved the converse of Lemma 9.19: if S is a regular spread of PG(3, q), then D(S) is isomorphic to AG(2, q2). (b) The planes constructed in the proof of Corollary 9.20 are precisely the socalled Hall planes; see Hall (1943) and Albert (1961) as well as Dembowski (1968), Hughes and Piper (1982), Liineburg (1980) or Kallaher (1981).

X. Transversal designs and nets

740

9.22 Exercise. Show that each regular spread S contains two reguli which have exactly one line in common. Hint: This follows from counting the reguli containing one or two lines of S. We now give the following general construction principle for maximal partial spreads over GF(qS) which stay maximal when considered as partial (2s + 1)spreads over the subfield GF(q). This result of Jungnickel (1993b) makes use of ideas going back to Bruen (1971) and Beutelspacher (1980). 9.23 Theorem. Let S be a regular spread in P = PG(3, qS), where qS :::: 4, and let Ro and RI be two reguli in S which intersect in exactly two lines, say L and M. Let U denote the partial spread of deficiency 2q s obtained by omitting the 2q s lines in Ro URI, and let W be a set of lines in P satisfying the following conditions: (9.23.a)

W is contained in the union R~ U R~ of the opposite reguli afRo and R I·

(9.23.b)

Each point in L U M is on at most one line of W.

(9.23.c)

W contains at least one line from each of R~ and R~ .

(9.23.d)

For each line G E (R~ U R~) \ W, at least one of the two points of intersection of G with L U M is on a line of W.

Then the set F (9.23.e)

= U U W is a maximal partial spread in P with deficiency

d = 2q s -IWI.

Moreover, any line not in F either lies entirely in P(R o) U P(RI), where P(X) denotes the set of points covered by the lines in a set of lines X, or contains at most four points not in P(F). Now denote by V = V(4, qS+I) the underlying vector space ofP, consider V as a (4s + 4)-dimensional vector space over the subfield GF(q) of GF(qs+I), and denote the corresponding projective space PG(4s + 3, q) by Q. (Thus the points and lines ofP are particular s -dimensional and (2s + I)-dimensional subspaces ofQ.) Then the maximal partial spread F constructed above remains maximal when considered as a partial (2s + 1)spread ofQ. Proof. Since the point set P (R) covered by the lines in a regulus R is the same as the point set covered by the opposite regulus R', conditions (9.23.a, b) imply that F is a partial spread of P. Now let K be any line not in F and consider the set K' := K \ P(F). As every point not in P(R o URI) is on a line in U, the set K' is contained in P(R o URI). By Remark 9.16, the point set peR) covered

§9. Translation nets and maximal nets

741

by the lines of a regulus R is a ruled quadric in P, and thus the only lines of P which intersect P(R o URI) = P(R o) U P(R I ) in more than four points are the lines which are entirely contained in P(Ro) or in P(RI)' that is, the lines in the reguli Ro, R 1, R~ and R~ . In particular, since q ::::: 4, any line X which is skew to every line of F has to belong to one of these four reguli. Because of (9.23.c), X cannot belong toRo or R I , and because of (9.23.d), X cannot be in R~ or R~ either, and thus F is indeed a maximal partial spread of P.

It remains to show that F remains maximal when considered over Q. Note first that the points of P form a partition T of Q into s-dimensional subspaces;2 thus each point v of Q is contained in a unique s-dimensional subspace Sv in T. Now assume the existence of a (2s + I)-dimensional subspace U of Q which is skew to every subspace in F. Then the point Sx of P cannot belong to P (F) for any x E U. Now choose a fixed point U E U, and let v be any point in U \ SUo The line Luv of P determined by Su and Sv is an (2s + I)-dimensional subspace of Q; since F is a maximal partial spread of P, we must have U =1= Luv. Therefore U is the union of the proper subspaces U n Luv, where v runs over the points in U \ Su. Obviously, this requires that each point u of U lies in at least three distinct lines Luv of P. On the other hand, the q + 1 points w of U on the line uv of Q (for any v E U \ Su) give rise to q + 1 ::::: 5 distinct points Sw of Luv none of which lies in P(F). Hence Luv is entirely contained in P(Ro U R j ) and therefore belongs to one of the reguli Ro, R I, R~ and R~. Since we may assume that Su lies in only one ofthe two ruled quadrics P(Ro) and P(Rl), say in P(Ro), we conclude that Luv has to be either the line in Ro or the line in R~ through Su, contradicting the fact that there are at least three distinct choices for Luv. This contradiction finishes the proof of the maximality of F as a partial (2s + I)-spread of Q . •

9.24 Construction. We now use Theorem 9.23 to construct the desired examples of maximal partial t-spreads. This requires the construction of suitable line sets W; we will achieve this by direct computation, using the representation of a regular spread given in Lemma 9.19. Thus let V = GF(q4s); then V is a two-dimensional vector space over its subfield K = GF(q2s), say with base {I, d}, which in tum is a two-dimensional vector space over its subfield F = GF(qS). If {I, e} is a basis for Kover F, then {l, e, d, ed} is a basis of the four-dimensional vector space V over F. As in Lemma 9.19, (9.24.a)

2

S = {aK : a E V \ {O}}

T is what is (among geometers) usually called a "geometric s-spread" of PG(4s

will not really require this tenninology and thus prefer to avoid it.

+ 3, q), but we

X Transversal designs and nets

742

is a regular spread of the projective space P = PG(3, qS). In the following examples, we use (in the notation of (9.19.a» the reguli Ro = R[,d and R[ = Rl,ed. Note that these two reguli intersect in the two lines L := lK and M := d K. It will be convenient for the subsequent explicit examples to list the lines in these reguli and their opposites explicitly. Because of (9.19 .a, b), we have:

(9.24.c)

= (A+d)K: AEF*}; R[ = (L} U (T). = (A + 8d)K: A E F*};

(9.24.d)

R~ = (L' = F

(9.24.e)

R~ = (L" =

(9.24.b)

Ro={L}U{S).

+ dF} U {S~ = (fL + 8)F + (fL + 8)dF: fL E F*}; F +sdF} U {T: = (fL+8)F + (fL+8)8dF: fL E F*}.

As (9.24.e) indicates, the computations depend upon the choice of the base element 8 of K: since the term 8 2 appears, the minimal polynomial of 8 is of importance in determining the intersections of lines in R~ and R~ .

9.25 Example. As a first application of Theorem 9.23 and Construction 9.24, we give a simple computational proof for the construction of maximal partial spreads with deficiency qS -1 in PG(3, qS) due to Bruen (1971). This requires finding two lines each in R~ and R~ which cover the same four points of L U M, provided that qS :::: 5 is odd, and turns out to be very easy if we choose the base element 8 such that v = S2 is contained in F. Thus let v be any non-square in F, and let s be a root of v in K. Then the lines L' and S~ of R~ contain the points F, sF, dF and sdF of L U M; and the lines L" and T~ of R~ contain the points F, sdF, sF and s2dF = dF of L U M. Thus the set

W:= {L', is a set of qS

S~} U

(R; \ {L", T~})

+ I lines satisfying the conditions in Theorem 9.23.

9.26 Example. We now show how one finds sets of qS, respectively of qS - 1, lines satisfying the conditions of Theorem 9.23, provided that qS :::: 5 is odd. Since the details of the computation are a little more involved, they will be omitted and we just indicate the respective definitions of W. One again begins with a suitable choice of the base element s which is adjoined to F in order to construct the quadratic extension K. Consider the equation (9.26.a)

x 2 - X = Y over F = GF(qS).

This equation has a unique solution for y = -1/4 and two solutions each for (qS -1)/2 values of y. Thus there are (qS -1)/2 choices of y for which (9.26.a) has no solution in F. We choose such an element y (so that the polynomial

743

§9. Translation nets and maximal nets

x 2 - x - y is irreducible over F) and also require y ¢ {I, 2}. In view of the preceding remarks, this is clearly possible for qS 2: 7; and for qS = 5, we may take y = 3. Thus we may assume our base element to satisfy

(9.26.b)

8

2

= 8 + y,

where

y

E

F \ {I, 2}.

We now define a set W of qS lines by putting (9.26.c)

W:=

{s~} U (R~

\ {T~, L"l).

It can be verified that W indeed satisfies conditions (9.23.a, b). Similarly, a suitable set W' of qS - 1 lines is obtained by putting

(9.26.d)

L'

= {s~, S;} U (R; \ {T~, L", T{, T;-d).

Applying Theorem 9.23 to Examples 9.25 and 9.26, we obtain the following series of maximal partial t-spreads due to Jungnickel (1993b).

9.27 Theorem. A maximal partial t-spread in PG(2t of exactly r components exists in the following cases: (9.27.a)

t

= 2s + 1, with

q 2: 5

odd,

r

= l+ I

-

+ 1, q) which consists

qS+ I

+0

OE{O, I,2} . •

We can now apply Theorem 9.27 together with Corollary 9.13 to obtain some series of maximal nets with small deficiency; this is due to Jungnickel (1993b) and considerably strengthens previous results of Bruen (1972, 1975). In fact, we shall include some further series of examples which may be derived by similar methods in the following result. These examples show that the bounds of §7, in particular the Metsch bound (7.3.a), for the completion of Bruck nets indeed are quite good.

9.28 Theorem. Let q be a power of the prime p. Then there exist transversalfree translation nets of order q2 and deficiency d in each of the following cases: (9.28.a)

d = q - I,

(9.28.b)

d=q,

P 2: 5;

p2:5;

(9.28.c)

d=q+l,

p2:5;

(9.28.d)

d = q - 1,

p= 2

(9.28.e)

d=q = 3.

and q = 2r for some prime r;

744

X. Transversal designs and nets

Proof. The results presented in this book suffice to prove (9.28.a, b) as follows. By Theorem 9.27, there exist maximal partial (2s + I)-spreads with deficiency ps+1 + 8 in PG(4s + 3, p), where 8 E {-I, 0, +1}. By Corollary 9.13, the corresponding translation nets are transversal-free if 8 =1= 1. Similarly, the case 8 = 1 leads to series (9.28.c) if one uses Corollary 9.30 below instead of Corollary 9.13. The example in (9.28.e) is obtained by applying Corollary 9.13 to a maximal partial spread with deficiency 3 in PG(3, 3); the reader may try and construct such a maximal partial spread as an exercise or consult Bruen (1971). Finally, series (9.28.d) was constructed by Jungnickel (I993b) using ad hoc arguments. • As mentioned above, series (9 .28.c) rests on the following analogues of Theorem 9.11 and Corollary 9.13; these results are again due to Jungnickel (1993b).

9.29 Theorem. Let D be a translation net of order s > 4 with translation group G, and assume that D has critical deficiency, say s = m 2 and d = m + 1. Then any net E extending D is either a translation net with the same translation group G, or E is a transversal-free net with deficiency m. • 9.30 Corollary. Any maximal partial t-spread of critical deficiency over a prime field GF(p) gives rise to a maximal (and hence transversal-free) net. • 9.31 Remarks. (a) Dow (1983) gave a construction of maximal nets of order q2 and deficiency q for each prime power q =1= 2, which are no longer translation nets but extend a translation net D of deficiency q + I; hence the second case in Theorem 9.29 can occur. In fact, D may be described as the translation net in common to the Desarguesian and the Hall planes of order q2; using the notation of the proof of Corollary 9.20, D is the net associated with the partial spreadS\R.

(b) Jungnickel (1984a) proved that any maximal partial spread U of deficiency q in PG(3, q) gives rise to a netD(U) which is not completable. In contrast to this result, he also constructed a maximal partial spread U of deficiency q - 1 in PG(3, q) such that D(U) is completable for each prime power q which is not a prime. On the other hand, if d = q - 1 and q == 2 (mod 3), one may also construct a maximal partial spread U for which D(U) cannot be completed to an affine plane; this is due to Johnson (1989). It would be very interesting to know if any of the non-embeddable examples actually give rise to extendible nets, since this would produce considerably larger maximal nets than we know up to now: the smallest deficiency known to occur is still q - 1.

§9. Translation nets and maximal nets

745

We conclude this section by discussing some related results. First we describe the present state of the existence theory for translation nets; as the results below will show, an extremely difficult problem such as the existence problem for nets can become much more tractable if one assumes the existence of a reasonably nice group. We first introduce some notation. Let G be a group of square order s2. Then we denote by T(G) the maximum cardinality of an (s, r; I)-PCP in G; we also write T (s) for the maximum value of T (G) as G runs over all groups of order S2. The following results due to lungnickel (198Ib, I989c) which are based on previous work of Frohardt (1987) determine the numbers T(s) and thus show the validity of the MacNeish conjecture for translation nets.

9.32 Proposition. Assume the existence ofan (s, r; I)-PCP in G, and let P bea Sylow p-subgroup o/G, say IPI = p2a. Then there also exists a (pa, r; I)-PCP in P. Hence (9.32.a)

T(G) ::: min {T(P): P E Syl(G)}.

IfG is nilpotent, one has equality in (9.32.a) . •

9.33 Corollary. Let s (9.33.a)

= ql ... qk be the prime power factorisation ofs. Then:

T(s) = min {qi

+ I : i = 1, ... , k} . •

The equality in (9.33.a) may be realised by taking G as the direct product of elementary abelian groups; however, note that these are in general not the only groups of order s2 satisfying T(G) = T(s). It is also known that one does not in general have equality in (9.32.a); the simplest counterexample is provided by taking G as the direct product of the alternating group of degree 5 with itself. Proposition 9.32 reduces the existence problem for translation nets with a nilpotent translation group to the same problem for p-groups. Bailey and lungnickel (1990) completely settled the abelian case by proving the following result.

9.34 Theorem. Let G be an abelian p-group containing an (s, 3; I)-PCP. Then G is isomorphic to a group of the form U ED U. If one has (9.34.a) where the qi are pairwise distinct powers ofp and where ai then (9.34.b)

T(U)

= pa + 1

with

a = min {ai: i

=1= 0 for i

= I, ... , n,

= 1, ... , n} . •

746

X. Transversal designs and nets

More generally, Bailey and Jungnickel (1990) considered splitting translation nets, that is, translation nets belonging to a partial congruence partition U in a group G which is the direct product of two components of U. The problem of determining T(G) for an arbitrary p-group G has not yet been settled and seems to be very difficult. Of course, one has T(EA(p2n» = pn + 1. Every other group G of order p2n has a considerably smaller value of T(G), as was first shown by Jungnickel (1981b). The best results known at present are due to Hachenberger (1992, 1993).

9.35 Theorem. Let G be a p-group of order p2n which is not elementary abelian, and assume the existence of a (pn, r; I)-PCP in G. Then: (9.35.a)

forn = 2;

(9.35.b)

forn = 3;

(9.35.c)

r ::: 4

(9.35.d)

r ::: pn-2

for p

+ pn-3 + ... + P + 1

= 2 and n =

3;

for n :::: 4 . •

The groups achieving equality in (9.35.a, b) have all been determined by Hachenberger (1992). As an example, we mention his results for the case n = 2.

9.36 Theorem. Let G be a p-group, and assume the existence of a (p2, 3; 1)PCP in G. Then one has one of the following cases:

+ 1.

(9.36.a)

G is elementary abelian and T (G) = p2

(9.36.b)

G

(9.36.c)

G ~ 71.,2 EB D4, where D4 denotes the dihedral group of order 8, and T(G) = 3.

(9.36.d)

p is odd and G is metacyc[ic, that is

~

Zp2 EB Zp2 andT(G)

= p + 1.

G = (x, y: Xp2 = yp2 = 1, y-1xy = x p+1 )

andT(G) = p (9.36.e)

+ 1.

p is odd and G is the direct product ofZp with the extraspecial group E(p3) of order p3 and exponent p, and T(G) = p + 1. Here E(p3) = (x, y, z: x p = yp = zp = [x, z] = [y, z] = 1, [x, y] = z) . •

9.37 Remark. The only 2-group achieving equality in (9.35.c) is the one already given in (VI.9.9.b); this result is due to Sprague (1982) and Gluck (1989). Regarding odd primes p, Hachenberger (1992) proved that, given p, there is

§9. Translation nets and maximal nets

747

precisely one group for which equality holds in (9 .35.b) which will be described in Theorem 9.38, a result generalising Theorem VI.9.9. Hachenberger (1993) obtained the even stronger result that, given p, there are precisely four groups of order p6 satisfying T(G) 2: P + 2. It is not known whether bound (9.35.d) is sharp for any pair (p, n). However, the results of Hachenberger (1991) show that this bound is not too bad. For instance, he constructs groups G of order p8 with T(G) 2: p2 + I and groups H of order pl2 with T(H) 2: p4 + 1 for all odd primes p.

9.38 Theorem. Assume the existence of a (pu, u; I)-PCP in some group G, where p is the smallest prime dividing IGI./f p = 2, then G is one ofthe groups described in Theorem VI.9.9./fp is odd, then G is a p-group; moreover, Gis either elementary abelian, or one ofthe groups oforder p4 described in Theorem 9.36, or the group of order p6 with exponent p and generators a, b, c, x, y, Z for which one has Z(G) = (x, y, z) and the relations [a, b] = x, [a, c) = y and [b, c) = z. In the last case, one actually has T(G) = p2 + 1. • 9.39 Remark. Recently, there has also been considerable interest in dual nets with a translation group, i.e. in translation transversal designs, which are defined in the obvious way. We refer the interested reader to Schulz (1984, 1985a, b, 1987a, b, 1988), Biliotti and Micelli (1985), Herzer and Schulz (1989) and Hachenberger (1994). As we have seen in this section, maximal partial spreads playa crucial role in the construction of maximal nets with small deficiency. For this reason, we mention a few results concerning these objects.

9.40 Theorem. Let F be a maximal partial spread in PG(3, q), and assume that F consists of r lines. Then one has the following: (9.40.a)

2q

:s r :s q2 + 1 - -li4.

for q > 2.

If q is not a square, then (9.40.b)

and even (9.40.c)

r

:s q2 + 1 -

(q

+ 3)/2

ifq is a prime.

i

X. Transversal designs and nets

748

If q is a square and if q = ph, where h > 2 is even and where p 2:: 5 is a prime, and if r 2:: q2 - q2/3 + 1, then the deficiency d = q2 + 1 - r ofF satisfies (9.40.d)

d

= s(.,jq + 1)

for some integer s 2:: 2;

similarly, if q = p2, where p is an odd prime, and ifr 2:: q2 + 1 - (q then d likewise satisfies (9.40.e)

+ 1)/2,

d = s(.,jq + 1) for some integer s 2:: 2.

Moreover, in either case the set of points of PG(3,q) not covered by F is the disjoint union of s Baer subgeometries. • The lower bound in (9.40.a) is due to Glynn (1982), and the upper bound to Blokhuis and Metsch (1993). For q = 5, we have a small improvement, namely r 2:: 11, given by Heden (1990). The improved upper bound (9.40.b) for non-squares q is due to Metsch (1991a); it is a consequence of Theorem 7.3. In particular, (9.40.a) shows that the case of deficiency d = .[ii + 1 can only occur for q = 4, disproving a conjecture stated by Hirschfeld (1985); the case q = 4 actually occurs, and all such maximal partial spreads have been classified by van Dam (1993). The bound (9.40.c) is due to Blokhuis (1994), and the strong restrictions for the case of square orders in (9.40.d, e) are recent results of Metsch and Storme (1999). All these results give some evidence to the long-standing conjecture that the smallest possible deficiency of a maximal partial spread in PG(3, q) is actually q - 1. We also mention the following very general construction for maximal partial spreads due to Heden (1993).

9.41 Theorem. A maximal partial spread in PG(3, q) consisting of exactly r lines exists for all odd q 2:: 7 and for all r in the range (9.41.a)

5q2

+ 4q 8

1 < r < q2 _ q

--

+ 2.

•

There are some similar results due to Heden (1995); a summary of the present state of knowledge regarding maximal partial spreads and maximal partial tspreads is given by Jungnickel (1996).

9.42 Remark. We finally mention the following interesting "q-analogue" of a t-design: A t-(n, k, A)-design over GF(q) is a collection B of k-dimensional subspaces of Yen, q) (called blocks) such that any t-dimensional subspace is

§10. Completion results for Ik > I

749

contained in exactly A blocks in B. In this terminology, a spread in Yen, q), n even, is the same as a I-(n, 2, I)-design over GF(q); a similar remark holds for (A-fold) t-spreads. A 2-(n, k, A)-design over GF(q) yields an ordinary 2-design S).(2, qk-l + ... + q + 1; qn-l + ... + q + 1) on the set of one-dimensional subspaces of V (n, q) by identifying every block with the set of one-dimensional subspaces which it contains. The first examples of t-designs over GF(q) with t:::: 2 were given by Thomas (1987) who constructed a 2-(n, 3, 7) design over GF(2) for all n with (n, 6) = 1. Ray-Chaudhuri and Singhi (1989) obtained an asymptotic existence result for t-designs over GF(q). Some other relevant references are Suzuki (1990a, b, 1992), Ray-Chaudhuri and Schram (1992), Abe and Yoshiara (1993), Ray-Chaudhuri and Schram (1994), Miyakawa, Munemasa and Yoshiara (1995) and !toh (1998).

§10. Completion Results for IL> 1 In this section we will consider the question of "completing" an (s, r; Ik)net with IL > 1 to an affine design A 2(1k); cf. Theorems II.8.7 and 11.8.8. As in Section 7, we will define a notion of "deficiency" and try to complete (s, r; Ik)nets of small deficiency. Unfortunately, this is much harder than in the case Ik = 1, and only the cases d : 1.

10.1 Definition. LetD be an (s, r; Ik)-net (i.e. an affine Sr(1, Slk; S21k» where s - 1 divides Ik - 1. Then the deficiency of D is the integer d defined by (10. 1. a) i.e. d measures how far r falls short of its conceivable maximum S:,~l which is reached precisely when D is an (affine) 2-design; cf. Theorem II.8.8. Assume thatD may be embedded in an affine design As (11) with parameters as described in Theorem II.8.7, say A; then A is called a completion of D, andD is said to be completable. The notions "extendible" and "step-t-extendible" are defined as in the case Ik = 1 (see Definition 8.1). We shall prove thatD is always completable for d = 1, and likewise for d = 2 provided s 1:4. It is helpful to introduce the notion of a transversal also for (s, r; J-L)-nets with J-L > 1. In view of (7.4.a) the

X. Transversal designs and nets

750

following definition is indeed a generalisation of the corresponding notion for fL=l.

10.2 Definition. Let D be an (s, r; fL)-net. Then a subset T of the point set of D is called a transversal of D if T intersects each block of D in precisely fL points. 10.3 Lemma. Any transversal of an (s, r; fL)-net D consists of SfL points. A further parallel class may be added to the blocks of D if and only if D has s pairwise disjoint transversals. • We will make essential use of Exercise n.8.20; therefore this result due to Jungnickel and Sane (1982) will be restated and proved.

lOA Theorem. LetD bean (s, SfL+t; fL)-netwhere t isanon-negative integer. Then (lO.4.a)

I(p, q)l::: t

moreover; I(p, q)1

for any two distinct points p, q of D;

= t implies that I(p, x)1 = I(q, x)lfor all points x =1= p, q.

Proof. Choose any two points p, q and assume I(p, q)1 = u. Removing the u parallel classes of D determined by the u blocks joining p and q leaves an (s, SfL + t - u; fL)-netD'. Now I(p, q)1 = 0 inD' and thus SfL + t - u :::: SfL by the dual of Theorem n.8.18; i.e. (lOA.a) holds. In the case t = u, the dual of n.8.18 also shows I(p, x) I = I(q, x) I in D' for all points x =1= p, q; but then clearly also I(p, x)1 = I(q, x)1 in D, as the blocks through p in D \D' arc precisely the t blocks joining p and q. • 10.5 Corollary. Let D be an (s, SfL + t; fL)-net where t is a non-negative integer. Then the point set ofD is partitioned into disjoint classes called cosets; we write p ~ q if p and q are in the same coset, where (l0.5.a)

p

~

q

~

I(p, q)1 = tor p = q

and

(lO.5.b)

I(p, q) I >

t

for p "" q .

Moreover; the joining number of two points in distinct cosets only depends on their cosets.

§1O. Completion results for J-t > 1

751

Proof. Exercise. • 10.6 Examples. (a) If D is a symmetric (s, J-t)-net (so t = 0), all cosets have size s. First examples were obtained in PropositionI.7.IS; also, each (s, sJ-t; J-t)difference matrix yields an example by Theorem VIII.3 .6, Lemma VIII.3.3 and Theorem II.S.21. Hence more examples are provided by Corollary VIII.3.12 and Theorem VIII.3.l4. (b) If D is a complete (s, J-t)-net (i.e. an affine design AI"(s», then any two points are joined by s:~ I = r - s J-t points and so there is just one coset of s2 J-t points. Removing one or two parallel classes from D leaves a net with coset sizes s J-t or J-t, respectively.

(c) Any symmetric (s, J-t)-netD is step-I-extendible. Take the SJ-t cosets (of size s each) and form a new parallel class Tby joining them to s blocks of J-t cosets each (in any way). This is dual to I.7.l6.(d); cf. the proof of Corollary VIII.3.S. By adjoining T to D we obtain an (s, SJ-t + 1; J-t)-net D' (still with cosets of size s). Now remove a parallel class distinct from T from D' to obtain again an (s, s J-t; J-t )-net D I ; then DI has only cosets of size l. We will now turn our attention to the completion problem again. For J-t > 1, there arises a further difficulty compared to the case J-t = l. If an (s, r; J-t)-net D of deficiency d is completable, then any completion E is an affine design AI" (s) and therefore any two points of E are joined by exactly A = ':_--/ blocks (see Theorem II.S.7). Thus a completion of D clearly is impossible ifthere exist points x, y ED with l(x,y)IA.

Now the first of these cases is impossible by Theorem lOA, but the second one was an open problem for a long time. For a given d, it is now settled for all but finitely many values of s by the following result due to Shrikhande and Singhi (I979a).

10.7 Theorem. Let D be an (s, r; J-t )-net of deficiency d with (1O.7.a)

s?:.. 2d(d - 1)

and strict inequality for d = 2. Then (1O.7.b)

A - d ~ I(x, y)1 ~ A for any two distinct points x, y ofD,

X. Transversal designs and nets

752 where

(1O.7.c)

S{L - 1 A=--. s-1

Proof. The lower bound in (10.7.b) is clearly valid if d 2: A. Thus let d < A and note that r = s:-~l d = S{L + A-d. Then I(x, y)l2: A - d by (lOA.a).3 It remains to prove the upper bound in (10.7.b). To this end, writebz = I(x, z)1 and Cz = I(y, z)1 for points z with z i= x and z i= y, respectively. Standard counting arguments yield the equations (using r = S {L + A - d)

-

(1O.7.d)

L

b z = (S{L

+A-

d)(s{L - 1) =

z#x

L

Cz

z#y

and (1O.7.e)

L bz(bz -

1) = (S{L

+A-

d)(s{L

+A-

d - 1)({L - 1)

z#x

We want to evaluate Lz#x,y bzcz and thus count pairs (B, C) with x, z E B and y, z E C. Abbreviating by = Cx by h, we obtain h(s{L - 2) such pairs with B = C; and h(h - 1)({L - 2) pairs where B i= C, X E C and y E B; and 2h(s{L + A - d - h)({L - 1) pairs where B i= C and either x E Cory E B; and finally (S{L + A - d - h)(s{L + A - d - h - 1){L pairs with B i= C, x rf. C and y rf. B. Altogether we have

L

(10.7.f)

bzcz = h[(s{L - 2)

+ (h -

1)({L - 2)]

z#x,y

+ (S{L + A -

d - h)[2h({L - 1) + (S{L

+ A-

d - h - 1){L].

We want to evaluate (henceforth, all summations are over all z i= x, y) CY

= 1;(bz + Cz - 2A)(bz + Cz + 1 - 2A)

= 1;b; + 1;c; + 21;bzcz + (1 + (s2 {L - 2)2A(2A - 1). 3

Here (IO.7.a) is not necessary.

4A)(1;bz + 1;cz)

§10. Completion results for f.L >

1

753

Using (1O.7.d) and (1O.7.e), note first (1O.7.g)

}:bz

= }:cz = (Sf.L + '). -

d)(sf.L - 1) - h

and (1O.7.h)

Also let us rewrite (l0.7.f) as (10.7.i)

}:bzcz

= -h(sf.L + 2').. -

2d)

+ f.L(SfJ, + '). -

d)(sf.L

+ '). -

d - 1).

From (1O.7.g, h, i) we obtain after some computation (J'

"2

= (sf.L+')..-d)[(sf.L - h 2 - h(1 - 2')..

= -h(h

+').. -d -1)(2f.L -1) + (sf.L -1)(2-4')..)]

+ SfJ, -

+ 1- 2').. +Sf.L -

2d)

+ (s2f.L -

2 2d) +A (2f.L

2)(2')..2 - ')..)

-1- 4sf.L + 2S2f.L)

+A[1 +Sf.L - 2d - f.L(s -1)(4d - 4sf.L -s +2)] + (Sf.L - d)(sf.L - 1)[(2f.L

+ 1) -

d(2f.L - I)].

Using A = s:~/ we rewrite this as (J'

- = -(h - A)(h - A + 1 +Sf.L - 2d) +2(sf.L -1)2f.L 2 +f.L(sf.L-1)(-4sf.L -s +2) 2

+ (d - d)(2f.L - 1)

= (d

2

- d)(2f.L -

+ (Sf.L -

1)(2sf.L

+ SfJ,)

I) - (h - A)(h - A + 1)

- (h - A)(Sf.L - 2d).

Since the bz , Cz are integers, we have (J' 2: 0 and therefore writing y = h - A, (l0.7.j)

y(y

+ 1) + Y(Sf.L -

2

2d).::s (d - d)(2f.L - I).

Note that the assertion is just y .::s 0; thus assume the contrary. Then (1O.7.j) implies

X. Transversal designs and nets

754 or, equivalently,

d2

-

3d + 2.::: JL(2d(d -1) - s),

which is clearly impossible because of (10.7.a). Thus indeed y .::: 0 and our assertion is proved. • The following result of Shrikhande and Bhagwandas (1969) is an easy consequence of Theorem 10.7.

10.8 Theorem. Any (s, r; JL)-net D of deficiency 1 is completable. Proof. By Theorem 10.7 with d = 1, we have (10.8.a)

A-I.::: I(x,

y)l.::: A

for any two distinct points x, y of D.

The cosets of D (cf. 10.5) are here defined by the joining number A. - 1 and in view of Example iO.6.(b) we have to show that each coset has SJL points. Thus choose a point x and let the number of points y with I(x, y) I = A-I (resp. A.) be denoted by ex (resp. f3). Then clearly ex + f3 = S 2 JL - 1 and (10.7.d) yields the further equation (A. - l)ex + Af3 = (s2JL + A - l)(sJL - 1). Substituting A = s:_~1 one obtains ex = s JL - 1 and thus the coset of x has indeed s f.h points. Then adjoining all cosets as new blocks yields an S)" (2, s JL; s2 JL) which clearly is resolvable; by Bose's Theorem 11.8.7 it is affine. Of course it is also possible to check that each coset is a transversal by another counting argument and then to use Lemma 10.3. • The case d = 2 is much more involved. Shrikhande and Bhagwandas (1969) proved thatD can be completed in this case if s =/= 4 and if no two points of Dare joined by A. + 1 blocks; but they could verify this condition only for s = 2 and 3. The problem remained open for 12 years till Shrikhande and Singhi (1979a) proved Theorem 10.7 and thus the condition needed for all s > 4. Let us state this as a lemma:

10.9 Lemma. Let D be an (s, r; JL)-net of deficiency 2, with s =/=4. Then (iO.9.a)

A. - 2 .::: I(x, y) I .::: A.

for any two distinct points x, y of D.

Proof. For s > 4 this is immediate from Theorem 10.7. The cases s .::: 3 follow from a more general bound for certain divisible designs due to Agraval (1964). We refer the interested reader to this paper as this result only covers the cases d = 1 and d = 2, s = 2 or 3. •

§lO. Completion resultsjor f-t > 1

755

We will prove the completability of D for d = 2, s =1= 4 using cosets, an approach due to Jungnickel and Sane (1982). The original proof of Shrikhande and Bhagwandas (1969) uses eigenvalue techniques and is rather involved. In view of Example 1O.6.(b), the cosets should have size f-t; let us begin by proving this.

10.10 Lemma. Let D be an (s, r; f-t)-net oj deficiency 2, with s =1= 4. Then all cosets oj D have size f-t.

Proof. Choose a point x of D and let a, (3, y denote the number of points y with I(x, y)1 = ).. - 2,).. - 1,).., respectively. Clearly (1O.1O.a)

a

+ (3 + Y =

S2f-t - 1;

and (10.7.d, e) yield (1O.1O.b)

().. - 2)a

+ ().. -

1)(3 +)..y = (Sf-t

+).. -

2)(sf-t - I)

and (1O.1O.c)

().. - 2)().. - 3)a = (Sf-t

+A -

+ ().. -

2)(sf-t

1)().. - 2)(3

+A -

+ )..(A -

1)y

3)(f-t -1).

Multiplying (lO.1O.a) by ).. and subtracting (lO.1O.b) from the result yields (lO.lO.d)

2a

+ (3 =

where we used sf-t equation (lO.1O.e)

2(sf-t - 1),

+A=

2(A - 2)a

S:-~l. Similarly we obtain from (lO.1O.b, c) the

+ (A -

1)(3 = 2(sf-t + A - 2)(f-t -1).

Then (lO.1O.d, e) yield a = f-t - I, i.e. the coset of x has size f-t . •

10.11 Theorem. Let D be an (s, r; f-t)-net oj deficiency 2 and assume s =1=4. Then D is completable.

Proof. In this case all cosets have size f-t by Lemma 10.10; we will denote cosets by capital letters P, Q, R. In view of Corollary 10.5 we may define (lO.l1.a)

[P,

Q] := I(x, y)l,

where x E P and Y E Q are arbitrary. We now define a graph G on the S2 cosets as vertices as follows: (P, Q) is an edge iff [P, Q] = ).. - 1. We write P ~ Q

X. Transversal designs and nets

756

iff (P, Q) is an edge of G. Our aim is to show that this yields a pseudonet graph with parameters sand 2 and then to apply Corollary 7.21. Now from (W.W.d) we see that any point x is joined to 2(s - 1)p, points by exactly A - 1 blocks and therefore each vertex of G has degree 2(s - 1). We next want to show that there are precisely s - 2 vertices joined to both P and Q, whenever P ~ Q. Put (lO.l1.b)

N(P):= {R: R ~ P, R,# Q},

N(Q):= {R: R ~ Q, R,# Pl.

Then clearly IN(P)I = IN(Q)I = 2(s - 1) - 1; we want to evaluate IN(P) n N(Q)I. Now choose x E P, Y E Q, let z ,#x, y be any point of D and define bz and Cz as in the proof of Theorem 10.7. Then we obtain from (lO.7.g, h, i) with d = 2 and h = A - I the following equation (1O.I1.c)

L (bz -

c z )2 = 2(sp,

+ 1..- 2)(2 -

A) + 2(1.. - 1)(sp, + 1..- 3)

z ;6x,y

= 2(sp, - 1).

Now if Z E P, then bz = 1..- 2 and Cz = A - 1; i.e. (b z - cz )2 = 1. A similar argument holds if Z E Q. Therefore (lO.I1.c) yields (10.I1.d)

L

(b z - c z )2 = 2(sp, -1) - 2(p, -1) = 2(s -1)p,.

z¥PUQ

For Z !if P U Q, we have bz, Cz E {A - 1, A} and thus (b z - cz? E {O, I}. As this value is independent of z as long as z E R for some fixed R and as each R contains p, choices for z, we obtain from (10.11.d) (1O.I1.e)

L

(b R

- CR)2

= 2(s -1),

R;6P,Q

where b R := [P, R) and CR := [Q, R). As noted, each term (b R -CR)2 is either Oar 1; and in fact (b R

- CR)2

= I{:=:::? [P, R) '# [Q, RJ {:=:::? R E (N(P)\N(Q)) U (N(Q)\N(P)).

Hence (1O.11.e) yields 2(s - 1) = IN(P)\N(Q)I

as IN(P)I

+ IN(Q)\N(P)I =

2IN(P)\N(Q)1

= IN(Q)I; but then clearly IN(P) n N(Q)I = s -

2, as asserted.

757

§10. Completion results for f-L > I

Next, let us show that there are precisely two vertices joined to both P and Q whenever P "" Q. With notation as before, this means (1O.1l.f)

IN(P)

n N(Q)I =

2

whenever p"" Q.

The proof of (1 0.11.f) is in complete analogy to that of the case P ~ Q; we will therefore just give the formulas corresponding to (10.II.c, d, e). These are: (1O.11.c')

L (bz - cz)2 = 4(sf-L - 2); L (bz - c z )2 = 4f-L(s - 2);

z ";'x,y

(10. I 1.d')

z\/,PUQ

(10.1 I.e')

L

(bR - CR)2

= 4(s -

2).

R";'P,Q

One then has IN(P)\N(Q)I = 2(s - 2) which yields (1O.1l.f). We have now verified that G satisfies the assumptions of Corollary 7.21; thus G is the graph of an (s, 2; I)-net. Hence we may label the S2 cosets as Pij (i, j = 1, ... , s) such that

.

(lO.n.g)

.,

OfJ=J.

We now define

T:= {

UPij: J=1

i = I, ... ,

s}

and

T':=

{U

Pi{ j= 1, ... ,

,=1

Adjoining T U T' to D gives a resolvable S)" (2, s f-L; II.8.7, this design is affine. •

S2 f-L).

s}.

By Bose's Theorem

10.12 Remarks. (a) The proof of Theorem 10.11 is basically that ofJungnickel and Sane (1982), but we have simplified the last part of the proof by using Bose's theorem; also, we have inserted a discussion of condition (10.11.f) omitted in their paper.

t=

(b) The use of Corollary 7.21 made it necessary to assume s 4 in Theorem 10.11. In fact we will see in the next section that the case s = 4 is truly exceptional; cf. (11.13.a).

(c) There are two further general completion results for (s, r; f-L)-nets with f-L> 1. Completion is possible for s = 2 and d :s 7 (Verheiden (1978), using the language of Hadamard matrices) and for d = 3, s :::: 104 (Shrlkhande and Singhi 1979b). The case d = 3 already requires the use of strongly regular multigraphs

758

X. Transversal designs and nets

and is much more involved than the cases d ::::: 2 or JL = 1. There is one partial result for d:::: 4: whenever s is large enough (e.g. s 2:: 2(d -1)2(2d 2 - 2d + 1», an (s, r; JL)-net D of deficiency d may be embedded into an 5)..(2, SJL; S2JL), say E. As it is not known whether E is resolvable or not, an application of Bose's theorem is not possible and the existence of E says nothing about the completion problem. Nevertheless, this result (due to Shrikhande and Singhi 1979b) supports the conjecture that D is completable if s is sufficiently large. Nothing better than what is true for JL = 1 can be proved, however, as we will see in the next section; thus it seems to be unimportant how large JL is.

§11. Extending Symmetric Nets In this section we will consider extensions of symmetric (s, JL)-nets with JL > 1. This will allow us to construct examples of maximal nets with small deficiency for JL > 1 as well (using the results of §9 with JL = 1). Our account will follow Jungnickel and Sane (1982). We will always assume JL > 1. We shall require a characterisation of the transversals of a symmetric net; we will derive this as a corollary to a more general result of Hine and Mavron (1983) characterising the subnets of a net as its sub-I-designs.

11.1 Lemma. Let e be a set of blocks of an (s, r; JL )-net D and assume that each point of D is on t blocks of e for some positive integer t. Then e consists of t parallel classes ofD. Proof. As e induces a I-design on the point of D, we have lei = st. Now let BEe and assume that C contains x blocks parallel to B. Counting pairs (p, X) where pI B, X and B =1= X E C, yields sJL(t - 1) = (st - x)JL; i.e. x = s . • 11.2 Corollary. Let T be a set of points of a symmetric (s, JL)-net D. Then T meets every block in t points if and only if T is a union of t point classes. In particular, T is a transversal ofD if and only if it is a union of JL point classes. Proof. As D is symmetric, it is simultaneously a TD. Hence the dual of Lemma 11.1 yields the assertion. • Corollary 11.2 implies the result of Exercise 1.7.16.(d) which we state here again for completeness.

11.3 Corollary. Any symmetric (s, JL)-net is step-I-extendible. •

759

§11. Extending symmetric nets

11.4 Theorem. Let D be a symmetric (s, J.1)-net and t 2:: 2 an integer. Then every step-t-extension E ofD induces an (s, t; fLls)-net N(EID) on the set of cosets of D, where the blocks of D' are the transversals of D used in the extension. Moreover, each (s, t; fLls)-net may be obtained in this way. Proof. Let E be any step-t-extension of D and let T be the set of blocks in

E \ D. Then T consists of transversals of D and thus each T E T is a union of J-L cosets of D, by Corollary 11.2. If x is a point of D, denote its coset by x. Define (l1.4.a)

X1T

{:=:}

x~

T

for any point x and any T

E

T.

The resulting incidence structure N (E ID) has s 2 m points with blocks of sm points each, where m = fL Is. Clearly each parallel class of E \ D induces a parallel class of N(EID). If T, T' are non-parallel blocks in T, we have IT n Til = fL inE and therefore IT n T'l = m in N(E ID). Thus N(E ID)is an (s, t; m)-net. The second assertion has already been proved in Lemma 6.7 . •

11.5 Corollary. Let D be a symmetric (s, fL)-net where s does not divide fL. Then the only extensions ofD are (s, sfL + 1; fL)-nets, and all such extensions are maximal. • 11.6 Corollary. A symmetric (s, fL )-net D is step-t -extendible for t > 2 if and only if s divides fL and if there exists an (s, t; fLls)-net. Hence D is step-7extendible if s I fL and s =j:. fL· Proof. The first assertion is immediate from Theorem 11.4; the second assertion follows from the existence of an (s, 7; m)-netfor all m 2:: 2 (the dual of Han ani's Theorem 2.1) . • In particular, we have the following result of Hine and Mavron (1980).

11.7 Corollary. A symmetric (s, fL)-net is completable if and only if there exists an (s, (SfL - 1)/(s -1); fLls)-net, i.e. an affine 2-design A/L/s(s) . • We will require the following strengthening of Theorem 11.4.

11.8 Theorem. Let E be a step-t-extension of a symmetric (s, fL)-net D with t:::: 2. Then E is step-n-extendible if and only if N(D IE) is step-n-extendible. In particular, E is maximal if and only if N (E ID) is maximal.

X. Transversal designs and nets

760 Proof. Exercise. •

11.9 Corollary. Let D be a symmetric (s, f.L)-net with s I f.L. Then D can be extended to a maximal (s, Sf.L + t; f.L)-net if and only if there exists a maximal (s, t; f.L/s)-net. • These results give us two general constructions for maximal nets with f.L > 1, due to Jungnicke1 and Sane (1982). 11.10 Theorem. Lets and c be positive integers, where c does not divide s, and assume the existence ofa symmetric (s, f.L )-net with f.L = cs n for all non-negative integers n. Then there also exists a maximal (s, r; f.L)-net with (1 LlO.a)

r

= cs n+ 1 + cs n + ... + cs + 1

for all n.

Proof. As that of Corollary 6.8, using Corollaries 11.5 and 11.9. • 11.11 Examples. In Theorem 11.10, one may choose e.g. (cf. Example 6.9): (ll.11.a)

s = pi, C = pj whenever p is a prime and i, j are positive integers with i > j (by Corollary Vrn.3.12);

(ll.11.b)

s an odd prime power, c

= 2 (by Theorem vrn.3.14).

The second construction is of even greater interest as here the corresponding nets might be completable (i.e. s - 1 divides f.L - 1). Thus we may speak: of the deficiency of the nets constructed in this case. 11.12 Theorem. Let s be a prime power and assume the existence ofa maximal (s, r; I)-net of deficiency d. Then there also exists a maximal net of order sand deficiency d for all f.L = sn with n E N. Proof. Symmetric (s, sn)-nets exist for all positive integers n by Proposition 1.7.18. Now use Corollary 11.9 and induction. • 11.13 Examples. Let n be any non-negative integer. Then there exists a maximal (s, r; sn)-net of deficiency d in each of the following cases: (l1.13.a)

d = 2, s = 4 (by Example 8.4);

(11.13.b)

d = 3, s = 5 or 9 (by Theorem 8.6 and Theorem 9.28);

§12. Complete mappings, difference matrices and maximal nets (11.13.c)

d = 4, s = 7, 8, or 25 (by Remark 8.20 and Theorem 9.28);

(11.13.d)

d = 5, s = 7, 8 or 25 (Remark 8.20 and Theorem 9.28).

761

More generally, we have the following series: (l1.l3.e)

s a prime power with s even or s == 1 (mod 4), d (Theorem 8.6 and Corollary 12.5 below);

(ll.l3.£)

s = q2, where q is a power of a prime p?::. 5, and dE {q - I, q, q + I}, by Theorem 9.28.

=s-

2

We remark that (11.13.a) shows that the condition s # 4 in Theorem 10.11 is necessary. Theorem 11.12. also shows that nothing better can be expected regarding the completion of (s, r; J.L)-nets with 11- > 1 than what is true for 11-=1.

§12. Complete Mappings, Difference Matrices and Maximal Nets In this section we will consider some more advanced results on difference matrices. We will deal with three interrelated topics, namely non-existence results for difference matrices and, in particular, generalised Hadamard matrices; connections to the notion of "complete mappings" which is one of the major themes in the study of Latin squares; and the use of difference matrices in the construction of maximal nets. Our first goal is the proof of a major non-existence result for difference matrices due to Drake (1979). For this purpose, we require the following auxiliary result from group theory which is a special case of Satz IV.2.8 in Huppert (1967); for the convenience of the reader, we will include an ad hoc proof.

12.1 Lemma. Let G be a finite group of even order, and let T be a Sylow 2subgroup of G. Assume that T is cyclic. Then G has a unique maximal normal subgroup O(G) of odd order; moreover, T is a complement of O(G), that is G = T· O(G). Proof. Since the product of any two normal subgroups of odd order is again a normal subgroup of odd order, it is clear that 0 (G) exists and is in fact a characteristic subgroup of G. Let a be a generator of T and note that G acts on itself by the right regular representation, see Exercise V.I.I. The permutation induced by a consists of 10 (G) I cycles of length IT I each and is therefore odd. Thus the subgroup of even permutations in G is a proper subgroup G I of index 2,

762

X. Transversal designs and nets

and O(G J ) = O(G), as each element of O(G) induces an even permutation. By induction, we may assume G 1 = O(G) . (T n G 1 ), as T n G 1 is either a Sylow 2-subgroup of GJ or of order 1. The assertion is now immediate. •

12.2 Theorem. Let G be a finite group of even order s, let T be a Sylow 2subgroup of G, and let A be an odd positive integer. Assume that T is cyclic. Then there is no (s, k; A)-difference matrix over G with k?: 3. Proof. It suffices to consider the case k = 3; thus assume the existence of an (s, 3; A)-difference matrix over G. By Lemma 12.1, T ;:::: G/O(G); thus Lemma VIII.3.l1 implies the existence of a (t, 3; Au)-difference matrix over T, where t := ITI and u := IO(G)I. As AU is odd, it suffices to prove the assertion for the special case T = G. Thus assume the existence of an (s, 3; A)difference matrix Dover Zs, where A is odd and s is a power of 2. We may assume that the first row of Dis (0, ... ,0). For i,j = 0, ... , s - 1, let f(i,j) denote the number of columns (O,i ,jl of D. As D is a difference matrix, we obtain the three conditions s-J

(12.2.a)

s-J

Lf(x, j)

s-J

= Lf(i,x) =

Lf(x,x +k)

= A,

where i, j, k are arbitrary elements of Zs and where all arguments are to be considered modulo s. Also note

(l2.2.b)

~

~n

=

s(s - 1)

2

n=O

s

=- (mods). 2

We now consider the f(a, f3) modulo s; then (12.2.a, b) imply

s

s-1

+ L(s -k) Lf(x,x +k) k=J

s-J

-

x=O s-J

Lf(i,j)(j-i+s-(j-i»=s. Lf(i,j)=O, i,j=O

i,j=O

a contradiction. • The special case A = I yields a famous result on Latin squares which is due to Hall and Paige (1955) and which yields an interesting series of transversal-free nets. We need a definition first.

§12. Complete mappings, difference matrices and maximal nets

763

12.3 Definition. Let G be a quasigroup, and let IX : G -+ G be a bijection. Then IX is called a complete mapping of G provided that f3 : x ~ xx a is a bijection as well. In other words, IX is a complete mapping if and only if the set of cells {(x, x a ) : x E G} is a transversal of the Latin square given by the multiplication table of G. Any permutation T) of G for which the mapping x ~ X-I x~ is a bijection is called an orthomorphism of G; in other words, T) is an orthomorphism if and only if x ~ x-I x~ is a complete mapping. 12.4 Corollary (The Hall-Paige Theorem). Let L be the Latin square given by the multiplication table of a group G of order s.lf s is even and if the Sylow 2-subgroup of G is cyclic, then L has no transversal. Proof. Assume otherwise; then G admits a complete mapping IX. Writing G additively, both IX and f3: x ~ x + x a are bijections of G. But then, with G = {gl, ... , gs},

g~) -g~

is an (s, 3; I)-difference matrix over G, contradicting Theorem 12.2. •

12.5 Corollary. Let s be an even positive integer. Then there exists a transversal-free (s, 3; I)-net. • The case s = 4 of Corollary 12.5 gives the net presented in Remark 7.22. For odd values of s, group tables cannot yield maximal nets as the following result of Sade (1963) shows.

12.6 Lemma. Let G be a group or a commutative quasigroup of odd order. Then G admits a complete mapping. Proof. It suffices to show thateach element of G has a (unique) square root, since then the mapping x ~ x 2 is a bijection which implies that the identity is a complete mapping for G. Quasigroups with this property are called diagonal. If G is a group and if g E G has odd order 2n -1, then g2n = g and hence gn is the square root of g. Now let G be a commutative quasigroup and note that each g E G occurs exactly IGI times in the multiplication table of G. Since there is an even number of ordered pairs (x, y) of elements of G satisfying x =1= y and xy = y x = g, there exists at least one element Z E G with Z2 = g. •

_I

\-...-

§12. Complete mappings, difference matrices and maximal nets

765

12.10 Theorem. Let s and A be odd integers, G an abelian group of order s, P a prime dividing s, and m some divisor ofthe square-free part of n := SA which is not a multiple ofp.lfm has even order modulo p, then there is no GH(s, Je) over G. Proof. Since G is abelian, there exists an epimorphism from G onto a (multiplicatively written) cyclic group C of order p. By Lemma VIII.3.II, the existence of some GH(s, A) over G implies that of a GH(p, UA) over C, where u := s/ p. Thus let H be a GH(p, UA) over C. Let m =PI ... Pt be the prime factorisation of m, and denote the order of m modulo Pi by ei, for i = I, ... , t. Then me == I (mod p), where e = el ... e t • Now assume that m has even order modulo p, that is, assume that ei is even for some i. Then me;/2 == - I (mod p), which will allow us to apply Lemma VI.3.10 to the element A := det H ofZC. As n is odd, the p-part of nn / p is a square, say q2. Hence Lemma 12.9 shows AA(-l) == 0 (mod pq2), and Lemma VI.3.10 implies A == 0 (mod pq). But then AA(-l) == 0 (mod p2q2), a contradiction. • 12.11 Examples. (a) Assume the existence of an abelian GH(s, 1) with s ::: 100. Then s is a prime power or s E {39, 55, 63}. This is a consequence of Corollary 12.8 and Theorem 12.10, as noted by de Launey (1986); the reader should check this assertion as an exercise. (b) There is no abelian GH(s, 1) withs == 15 (mod 18). To see this, lets = 18a+ 15 = 3(6a + 5) and choose p = 3. Clearly some prime m == 2 (mod 3) divides the square-free part of 6a + 5 (hence of s), and thus Theorem 12.10 implies the assertion. This application of Theorem 12.10 was noted by Jungnickel (1990a); the result in question was also obtained by Woodcock (1986) in the special case of cyclic groups. The following non-existence result is due to Brock (1988); its proof uses theorems on the hermitian congruence of matrices and is considerably more involved than that of Theorem 12.10. 12.12 Theorem. Assume the existence of some GH(s, A) over a (not necessarily abelian) group G, where n := SA is odd. Then the equation

has a non-trivial solution in integers x, y, zfor every t of a homomorphic image of G. •

# 1 which is the order

§12. Complete mappings, difference matrices and maximal nets

765

12.10 Theorem. Let s and A be odd integers, G an abelian group of order s, P a prime dividing s, and m some divisor ofthe square-free part of n := SA which is not a multiple ofp.lfm has even order modulo p, then there is no GH(s, Je) over G. Proof. Since G is abelian, there exists an epimorphism from G onto a (multiplicatively written) cyclic group C of order p. By Lemma VIII.3.II, the existence of some GH(s, A) over G implies that of a GH(p, UA) over C, where u := s/ p. Thus let H be a GH(p, UA) over C. Let m =PI ... Pt be the prime factorisation of m, and denote the order of m modulo Pi by ei, for i = I, ... , t. Then me == I (mod p), where e = el ... e t • Now assume that m has even order modulo p, that is, assume that ei is even for some i. Then me;/2 == - I (mod p), which will allow us to apply Lemma VI.3.10 to the element A := det H ofZC. As n is odd, the p-part of nn / p is a square, say q2. Hence Lemma 12.9 shows AA(-l) == 0 (mod pq2), and Lemma VI.3.10 implies A == 0 (mod pq). But then AA(-l) == 0 (mod p2q2), a contradiction. • 12.11 Examples. (a) Assume the existence of an abelian GH(s, 1) with s ::: 100. Then s is a prime power or s E {39, 55, 63}. This is a consequence of Corollary 12.8 and Theorem 12.10, as noted by de Launey (1986); the reader should check this assertion as an exercise. (b) There is no abelian GH(s, 1) withs == 15 (mod 18). To see this, lets = 18a+ 15 = 3(6a + 5) and choose p = 3. Clearly some prime m == 2 (mod 3) divides the square-free part of 6a + 5 (hence of s), and thus Theorem 12.10 implies the assertion. This application of Theorem 12.10 was noted by Jungnickel (1990a); the result in question was also obtained by Woodcock (1986) in the special case of cyclic groups. The following non-existence result is due to Brock (1988); its proof uses theorems on the hermitian congruence of matrices and is considerably more involved than that of Theorem 12.10. 12.12 Theorem. Assume the existence of some GH(s, A) over a (not necessarily abelian) group G, where n := SA is odd. Then the equation

has a non-trivial solution in integers x, y, zfor every t of a homomorphic image of G. •

# 1 which is the order

766

X. Transversal designs and nets

Let us once more consider Corollary 12.4. From its proof, it is clear that an (s,2; I)-difference matrix over G is "maximal" if and only if the corresponding (s, 3; I)-net (or Latin square) has no transversal. In order to generalise this observation, we give the following definition.

12.13 Definition. Let G be an additively written (but not necessarily abelian) group of order s and let D (dij)i=l, ... ,r;j=l, ... ,s). be a matrix with entries from G. One calls D a left (s, r; ).,,)-difference matrix over G, if the following condition is satisfied:

=

(12.13.a)

{-dij

+ dhj

:j

= I, ... , s} = )"'G

for any two distinct indices h, i

E

{I, ... , r }.

In other words, the "left" differences of corresponding entries of any two rows in D contain each element of G exactly)." times. D is called maximal if no further row may be adjoined to obtain a larger left difference matrix.

12.14 Remark. Similarly, one may define right difference matrices. Of course, it makes no essential difference which one of these two species of difference matrices is considered: D is a left difference matrix if and only if - D is a right difference matrix. In the abelian case, both conditions are trivially equivalent, anyway. Usually, the term "difference matrix" in the literature means "right difference matrix", as in Definition VIll.3.4. However, in the present context, where we want to consider the associated nets, the use ofleft difference matrices is more appropriate. We know that any right difference matrix D yields a TD[r + 1; s], by Corollary VIII.3.8, and thus an (s, r + 1; I)-net D; this net can be described explicitly by using the corresponding left difference matrix - D. The following result provides such a description and also shows thatD is transversal-free if and only if D is maximal.

12.15 Theorem. The existence ofa left (s, r; 1)-difference matrix D over some group Goforder s implies that ofan (s, r+ I)-net N(D). Moreover; D is maximal if and only ifN(D) is transversal-free. Proof. Let D be a left (s, r; I)-difference matrix over G. We construct N := N(D) as follows. As point set, we choose V={(i,x): i=I, ... ,S;xEG}.

§12. Complete mappings, difference matrices and maximal nets

767

We now define sets of lines

Po:={Lj:j=l, ... ,s}

and

Pi:={Li,x:XEG}

for i = I, ... , r, where

Lj:={(j,X):XEG}, Li,x := {(j, dij

+ x) : j

= 1, ... , s}.

It is obvious that all the Pi are parallel classes of lines. It remains to check that any two non-parallel lines intersect in a unique point. This is clear if one of the lines belongs to Po; thus consider any two lines L i •x and Lh,y where h i= i. Note that a point (j, z) is on both of these lines if and only if z = dij + x = d"j + y. This condition is equivalent to the equations

z

= dij + x

and

-dhj

+ dij

= Y - x;

the second of these equations has a unique solution for j by the defining property of a difference matrix, and then z is determined from the first equation. Thus N is indeed a net. Now consider any s-set T of points of N which intersects each line of the parallel class Po uniquely (and thus is a candidate for a transversal of N). Hence T has the form

T:= {(j,tj):j = 1, ... ,s}. By our definition of the lines Li,x, the intersection points of T with Li,x correspond to the solutions of the equation (l2.IS.a)

-tj

+ dij = -x.

Hence T is a transversal for N if and only if, for any given pair x and i, equation (12.lS.a) has a unique solution for j; but this just means that the row (t\, ... , ts ) extends D to a left (s, r + 1; I)-difference matrix. •

12.16 Remark. The equivalence between certain types of nets (or mutually orthogonal Latin squares) and difference matrices was first established by Jungnickel (1979, 1980b); see Hachenberger and Jungnickel (1990) for a considerably more detailed study. Sitnilar but weaker results also hold in the case of an arbitrary A, see Jungnickel (1979). Note that the net N constructed in the proof of Theorem 12.15 is a very special type of net (an observation similar

X. Transversal designs and nets

768

to that made for transversal designs in Theorem VIII.3.6): G acts on N as a group of "central translations" if we let the element g E G map the point (j, x) to (j, x + g) and use the induced action on the set of lines, namely, if we let g fix each line in Po and map the line Li,x to Li,x+g' Moreover, N is a so-called "Po-transitive" net. It can be shown that nets with such a "transitive direction" are equivalent to difference matrices. For a detailed study of the geometry of these nets, we refer the reader to Hachenberger and Jungnickel (1990). Given this geometric interpretation of a difference matrix as a net with a transitive direction, the maximality result in Theorem 12.15 can be recognised as a consequence of a theorem of Ostrom (1966) even though it was first explicitly stated in the first edition of the present book (with a minor mistake, because no attention was paid to the distinction between left and right difference matrices, and thus the proof given then only worked for the abelian case). 12.17 Exercises. (a) Show that Theorem 12.15 does not hold for (s, r; p,)difference matrices with p, > 1. Hint: consider Example 6.9.

=

(b) In the situation of Theorem 12.15, assume that N N(D) has small deficiency. Use Lemma 7.6 to prove that every extension of N corresponds to a left difference matrix extending D. This observation is due to Hachenberger and Jungnickel (1990). (c) Use Theorems 12.15 and IT.8.21 to prove the result already mentioned in Remark VIII.3.9: an (s, r; p,)-difference matrix D satisfies r = sp, if and only if - DT is also a difference matrix. 12.18 Examples. We will now use maximal difference matrices to construct some transversal-free nets of small orders. The examples we present are due to Jungni.ckel and Grams (1986) and were found by computer search, but their correctness can be checked directly, even though this is a bit tedious. Jungnickel and Grams (1986) in fact determined maximal (s, r )-difference matrices of all possible sizes r in all groups of order s :::: 10. This was extended to s :::: 15 by Bedford and Whitaker (1998). (a) The following matrix is a maximal (8, 4)-difference matrix over the elementary abelian group of order 8:

(~

000 000 000

000 000 000 100 010 110 101 100 001 111 011 100

000 000 001 101 010 111 110 001

000 011 110 101

111 ~) 011 . 010

§I2. Complete mappings, difference matrices and maximal nets

769

(b) The following matrix is a maximal (8, 3)-difference matrix over the quaternion group of order 8, written as G = {±1, ±i, ±j, ±k}:

G

1 -1

k

-i j

j -j

1 -j

-k

1 k -1

-!) -I

(c) The following matrix is a maximal (9, 6)-difference matrix over the elementary abelian group of order 9: 00 00 00 00 00 00

00 10 20 21 01 22

00 20 10 12 02 11

00 00 00 00 01 11 21 02 22 12 02 11 10 01 22 20 12 10 11 21 02 21 10 01

00 12 01 11 22 20

00 22 21 02 20 12

(d) The (12, 6; I)-difference matrix given in Lemma VllI.3.13 is maximal and hence yields a transversal-free (12, 7; I)-net. The following result shows that (s, r; f.L )-difference matrices with f.L > 1 can also be useful in the construction of maximal Bruck nets. This somewhat surprising fact was discovered by Evans (1991).

12.19 Theorem. Let D be an (s, r; I)-difference matrix over G, and assume that mD is a maximal (s, r; m)-difference matrix, where mD denotes the matrix obtainedfrom D by concatenating m copies ofD. If there exists some (m, r + 1)net, then there exists a maximal (sm, r + I)-net. • The proof of Theorem 12.19 is rather more involved than that of Theorem 12.15, though similar in spirit. As Evans noted, the maximality of D is trivially necessary but in general not sufficient for that of mD. Theorem 12.19 may be applied to give difference matrix proofs for some of the older results which were discussed in §8. However, its real interest comes from the fact that it allowed Evans to produce many previously unknown series of maximal nets. To this end, he first gave the following construction for maximal difference matrices of prime power order.

12.20 Theorem. Let s = pa be a power of some prime p. Then there exists an (s, p; I)-difference matrix Dover Zs for which mD is maximal whenever m is not a multiple of p. More precisely, every matrix of the form (12.20.a)

D

= (dij)i=I ..... p;j=I, ... ,s

with

dij

= CPU -

l)j

X. Transversal designs and nets

770

for some mapping : tl p -+ tl s satisfying

(12.20.b)

(i):=i (mods) for all i

has the desired properties. •

12.21 Remark. In particular, one may choose the desired matrix D in Theorem 12.20 to consist of the first p rows of the multiplication table of 7l s • Combining Theorems 12.19 and 12.20 gives a proof of Evans' Theorem 8.15. One wonders if it is possible to extend his work to prove the existence of a maximal (mpa, pb + I)-net, where b :::: a, provided that there exists some (m, pb + I)-net. This would follow if one could generalise Evans' work to solve the following problem which should have a positive answer at least when a is a mUltiple of b. 12.22 Research Problem. Construct a (pa , pb; I)-difference matrix D, where b :::: a, such that mD is maximal whenever p does not divide m, or show that no such matrix exists. We conclude this section with another important construction due to Evans (1992a) and some variations discussed by Pott (1993).

12.23 Theorem. Let p > 5 be a prime. Then there exists a maximal (p, r; 1)difference matrix, where

(12.23.a)

r = {(p -1)/2 (p

ijp:=3 (mod 4)

+ 1)/2 if p:= 1 (mod 4).

Hence there exist maximal (p, r)-nets with

(12.23.b)

(P+1)/2 r= { (p+3)/2

ijp:=3(mod4) ijp-=1(mod4) . •

12.24 Remark. Evans' proof of Theorem 12.23 is given in the language of orthomorphisms and uses permutation polynomials, in particular Hermite's criterion; see Lidl and Niederreiter (1983) for background. Pott gave a somewhat simpler and definitely more transparent proof in the language of difference matrices. We shall sketch his approach which is based on Redei's theorem; see Rectei (1973) and Lovasz and Schrijver (1981) for a simpler proof. The idea

§12. Complete mappings, difference matrices and maximal nets

771

is to use some rows of the trivial (p, p; I)-difference matrix D* given by the multiplication table of Zp. Thus we take G = (Zp, +) and D* := (mx)m,xEG'

We now want to extend a selected set of rows of D* by adding a row of the form (12.24.a)

/= (f(X»XEG,

where f is a suitable non-linear function on G. Let us consider the set of slopes determined by the graph of f in the affine plane AG(2, p), that is (12.24.b)

Sf= {

f(x) - fey) } :X,yEG,X¥Y· x-y

Then it is easily checked that f is compatible with the m-th row d m of D* if and only if m ¥ Sf; let D consist of all such rows together with row f Of course, there is no reason why D should be maximal in general, even though all further possible extensions g are quite restricted, as they have to yield the same set of slopes as f. By Redei' s theorem, Sf contains at least (p + 3) /2 elements, since f is non-linear, and so D has at most (p -1)/2 rows. Evans' result comes from choosing f as follows:

(12.24.c)

f(x) := X(p+1)/2 =

I~ -x

if x = 0 if x is a square if x is a non-square.

It turns out that the corresponding matrix D is indeed maximal if and only if p == 3 (mod 4). In the case p == 1 (mod 4), a maximal difference matrix is obtained by adjoining to D the further row (12.24.d)

-f = (- f

(x»x

E

G·

Pott's proof then makes use of Redei's theorem and of one ofits variations, i.e. the complete description of all functions giving a minimal set of slopes obtained by Lovasz and Schrijver (1981). Pott also considers possible extensions of Evans' result. Regarding a generalisation to prime powers, Pott explained why the construction cannot work for q = 9, a fact already noted by Evans; computer checks show, however, that it does work for q = 25, 27 and 49. Thus we have the following problem.

X. Transversal designs and nets

772

12.25 Research Problem. Does the analogue of Theorem 12.23 hold for prime powers q > 9? 12.26 Remark. Pott (1993) also considered possible extensions of Evans' results obtained from replacing the function f defined in (I2.24.c) by using higher power residues instead of squares. He suggested taking

(12.26.a)

ax f(x) = { bx

if x if x

E

Co

Ii Co '

where Co is the set of e-th powers if GF(q) for some divisor e of q - 1. In this case, the set of slopes of f has cardinality q - 1 - g, where ge = q - 1. With this approach, Pott used a computer search to show the existence of the maximal (p, r; I)-difference matrices given in the following table. Unfortunately, there is no theoretical explanation for his results and not even a reasonable conjecture comparable to the one suggested in Research Problem 12.25. p

g r

13 3 3

13 4 5

17 3 5

17 4 4

19 3 4

19 4 7

§13. Tarry's Theorem In this section, we prove the result of Tarry (1900) concerning the nonexistence of two MOLS of order six. Following Betten (1983) we show:

13.1 Theorem. A TD [3; 6] cannot have six mutually disjoint parallel classes.

• The proof will need some preparation.

13.2 Notation and Definition. We denote the cell of a Latin square in the i-th row and j-thcolumn by (~) and, if the entry in this cell is k = ioj we write k). Two Latin squares are called equivalent if their TD[3; 6]' s are isomorphic. Recall that the points of the TD of a Latin square are Xi (x E {I, 2, 3,4,5, 6}, i E {I, 2, 3}) and that the lines are the point sets {Xl, X2, (X 0 Yh}. Permutations of rows, columns, or symbols 1, ... , 6 yield equivalent Latin squares. The same holds for mappings Xi 1-+ X"i with J( E S3, e.g. replacing the quasigroup operation 0 by * which is defined by X 0 Y = z ~ y =X * z. Note that the transversals of a Latin square correspond to the parallel classes of its TD[3; 6].

e

773

§13. Tarry's theorem

13.3 Definition and Observations. The permutation n (i, j) of the rows i and j is the permutation n(i,j): iox

1-+

jox

of S6. As n has no fixed points, only the following permutation types can occur. b c

G G

c d

d e

b c

c d

(~

b c

~) (~

~),

e f

more briefly (a b c d e f),

type 6,

~) (; ~),

more briefly (a b c d)(e f),

type 4,4

~),

more briefly (a b c)(d e f),

type 3,4

more briefly (a b)(c d)(e f),

type 2,4

e f

(~ ~) (~ ~) (; ~),

The permutations of type 4 and 3 are even, the others odd. The main idea of Betten's proof is to establish and use the fact that each equivalence class of Latin squares of order 6 contains a Latin square with n(1, 2) of type 3.

13.4 Observation. Given two rows of permutation type 6, 4, or 2, there are three columns containing all six symbols. Given two rows of permutation type 4 or 2, any pair of columns with four distinct entries can be extended to three columns with six distinct entries. 13.5 Lemma. Any colouring of the 15 edges of the complete graph K6 with two colours, say red and blue, contains at least one monochromatic triangle. Proof. Any vertex x is joined to five other vertices, hence x is on three edges of equal colour, say red. Then either x is a vertex of a red triangle or the other vertices of the three red edges form a blue triangle. • This is the simplest case of Ramsey's theorem; cf. Ryser (1963). All the following reasoning is to be understood up to equivalence.

13.6 Corollary. The rows i E {I, 2, 3, 4,5, 6} and the row pairs ii, j} form the vertices and edges of the complete graph K6. Let the edge ii, j} be red

4

Note that this would usually be called type (4,2), etc.

X. Transversal designs and nets

774

if n(i,

j) is even, and blue otherwise. Then there is a red triangle, as a blue triangle is clearly impossible. •

13.7 Lemma. In each equivalence class there is a Latin square with n(1, 2) of type 3. Proof. Otherwise there are three rows, w.l.o.g. the first three ones with n(1, 2), n(2, 3), and n(l, 3) of type 4 (Corollary 13.6). Wl.o.g. the first two columns are

12) 21 . ( 34

By Observation 13.4, they can be extended to

(125---) 21- 6 -- . 346521

Then

203 E {3,4}, say 203 = 3 (w.l.o.g.). As n(1, 2), n(2, 3) and n(l, 3) are of type 4, the entries in the first three rows must be either

1 2 5 3 4 6) 2 1 3 6 5 4 (3 4 6 5 2 1

1 2 5 or

4 6 3) .

213 6 5 4 ( 346 521

In the first case the columns 3 and 4 are of permutation type 3. In the second case define * by x 0 y = z ¢:=} x * z = y. Then the Latin square belonging to * has the last two columns of type 3. • 13.8 Corollary. Each Latin square is equivalent to one ofthefollowing shapes, where the shaded cells are occupied by 4, 5, or 6 and the other cells by 1, 2, or 3.

Shape A

Shape B

Shape C

Proof. Exercise. • 13.9 Lemma. A Latin square of shape A has no transversal. Proof. A transversal must contain two entries from {I, 2, 3} in the upper left 3 x 3-square and one in the lower right one, or vice versa. Then three entries

§13. Tarry's theorem

775

from {4, 5, 6} must lie in one column of the lower left square and in one row of the upper left square, which is impossible. •

13.10 Corollary. A Latin square of shape B has at most four mutually disjoint transversals. Proof. Any transversal T avoiding the four central cells would also be a transversal of shape A. Thus it cannot exist. • Henceforth we only consider Latin squares of shape C.

13.11 Lemma. A transversal T of a Latin square of shape C has exactly one entry both in the upper left and the upper right 2 x 3-rectangle. Proof. Suppose T has two entries in the upper left 2 x 3-rectangle. Then it cannot have a third entry in the upper left 3 x 3-square, as the number of entries from {4, 5, 6} in the lower right 3 x 3-square cannot be two. Hence the third entry from {I, 2, 3} is in the third row on the right-hand side. Two of the entries from {4, 5, 6} are in (!), (;), (~), and the third one is in the lower left 3 x 3-square. But then one of the three rows 4, 5, 6 would contain two identical entries, a contradiction. Note that the entries in (!), (D, (~) must be distinct. The case that two entries of T are in the upper right rectangle is treated analogously. • 13.12 Lemma. If a transversal T of a Latin square of shape C has an entry in one of the cell sets {(~), (;), (~)} or {(!), (;), (~)}, then it has exactly one entry in each of these two sets. Proof. First assume (w.l.o.g.) that T has two entries in the set [(~), (;). m}. By Lemma 13.11, it has one entry in the upper left 2 x 3-rectangle, hence it must have exactly one entry (from [4.5, 6}) in {(!), n), (~)} and two in the upper right rectangle, which contradicts Lemma 13.11. Now assume that T has exactly one entry in {(i), (D, (~)}. By Lemma 13.11, T contains exactly two entries from {l, 2, 3} in the first three columns, and hence exactly two entries from {4. 5, 6} in the last three columns. Again by Lemma 13.11, one of these entries must be in {(:). (D, (~)}. •

13.13 Observation. Using the previous lemma, each Latin square (of shape C) with six mutually disjoint transversals contains one of the following two partial

X. Transversal designs and nets

776

squares (of course, up to equivalence), where three of the transversals are denoted by a, band c and where a letter a next to an entry means that the corresponding cell belongs to the transversal a, etc.

3

2 3 la

1 -

-

-

2b

-

I 2 3c

5 6 -

6 4

4 5

-

-

4c

-

-

-

5a

-

-

-

6b

5 4 -

or 2 3 la -

-

1 -

I 2

6 5

4 6

-

-

-

-

4c

-

2b

-

-

-

3c

-

3

5a -

-

6b

Note that this arrangement of the partial transversals a, b, c may be obtained by suitable pennutations of the rows, columns, and symbols. Moreover both of these partial squares remain unchanged under the simultaneous pennutation (4 5 6) of rows, (1 23) (45 6) of columns, (1 23) (4 5 6) of the symbols, and (a b c) of the transversals. Let r be the group generated by this simultaneous pennutation. Now we try to extend the partial transversals a, b, c. 13.14 Lemma and Observation. In the left-hand partial square of Observation 13.13, the partial transversal a cannot be completed. Proof. A complete transversal a must contain either (~ 3) and (~6) or (; 2) and 4), by Lemma 13.11. In both cases one cannot find an entry in row 6. •

(!

Henceforth only the right-hand partial square of 13.13 needs further attention. This partial square is also preserved under the pennutation

(J:

(45) (15)(24)(36) (15)(24)(36) (bc)

I

ofrows of columns of symbols of transversals.

§13. Tarry's theorem

777

13.15 Observations. In the last three rows the transversals a, b, and e can take the following places.

Ia

4

5b

I

5a

2

4a 3

6

4

2b

3 2a

6

5

6b

3

5a

2

6e 2

5

4

Ia

4e

4

2b

3e 4e

3b 5

2

6b

3

6

5

3e

Ie 6

Note that the group G generated by rand (J is isomorphic to the symmetric group S3 and permutes the above six 3 x 6-rectangles regularly. r permutes the upper three rectangles cyclically, and the three lower ones too, and (J interchanges each upper rectangle with a lower one. Now we combine the above rectangles. Up to equivalence there are only the following two possibilities.

Ia 6e

5b 2b 4a

4e

Ia 5a

6e

6b

3e

5b

4e

3e

2a

2b

5a 6b

These two rectangles can be uniquely extended to the following two partial squares (using the preceding lemmas).

2

3b

s=

Q=

4

3 Ie

I

6a

4b

5e

2a

5 Ib

6

4

2e

3a

4e 3 2

3

2

5 6

6

4

5

2b 4a

Ia 6e

5b 3e

5a

I

1

6b 5e 4a

2

3a

1

6

4b

3b

Ie

2

5

6

2e

6

6a 5b

Ib

Ia 6e

4e

3

3 2a

5a

3 2 1

I

6b

2b

4

3e

The four empty cells in Q have to be occupied by one of the Latin subsquares

(~~)

or

(~~).

778

X. Transversal designs and nets

13.16 Lemma. The partial squares Sand Q cannot be completed to Latin squares with six mutually disjoint transversals. Proof. The entries of a hypothetic fourth transversal T of S through (~ 5) in rows 3, 1,2 must be (; 6), 3) and 4). Now the extension to row 4 is impossible.

G

a

(!

G

a

A hypothetic fourth transversal T of Q through 2) must contain 1), 3), and (~ 6). The remaining two entries must form a transversal of a Latin subsquare (~~) or (~~), which is impossible. This contradiction proves Lemma 13.16 and hence also Theorem l3.1. • The method ofthis section, i.e. of Betten (1983), may also be used to determine all "species" of Latin squares of order 6; cf. Betten (1984) and Fisher and Yates (1934). Another simplified proof of Tarry's theorem is due to Stinson (1984) and a further proof, similar to that of Stinson, was given by Dougherty (1994); these proofs use the codes of a corresponding TD or net, respectively.

§14. Codes of Bruck Nets fu this final section, we report on some very interesting results due to Moorhouse (1991a, b, 1993) regarding the p-rank of Bruck nets; we remind the reader that we have discussed the p-rank of designs in §II.II. The proof of the following lemma is a simple exercise.

14.1 Lemma. Let A be an affine plane of order n, let P be the projective plane extending A, and let p be a prime dividing n. Then rankpA = rankpP - 1. • Combining Lemma 14.1 with Theorem II.I1.6, we have the following result:

14.2 Theorem. Let A be an affine plane oforder n, and let p be a prime dividing· the square-free part of n. Then rankpA = n(n + 1)/2. • Moorhouse (1991a) offered the following interesting conjecture regarding the p-rank of a Bruck net.

14.3 Coujecture. Let N be any (n, r; I)-net, and let p be a prime dividing the squarejree part of n. Moreover, let M be any (n, r - 1; I)-net contained in N. Then (14.3.a)

rankpN::;: (n - r + 1) +rankpM.

§14. Codes of Bruck nets

779

In particular;

(14.3.b)

rankpN;::rn - r(r - 1)/2.

This conjecture holds trivially for r = 1 and r = 2; also, Theorem 14.2 shows that (14.3.b) is true for r = n + 1. Moorhouse established that his conjecture is also true for r = 3; in fact, he proved the following considerably stronger result.

14.4 Theorem. Let N be any (n, 3; l)-net, and let p be a prime such that pe II n. Then

(14.4. a)

3n - 2 - e::::: rankpN::::: 3n - 2. •

14.5 Remark. More precisely, rankpN can be determined as follows by using some results from the theory of loops. For background and undefined terms, we refer the reader to Jungnickel (1990a) and Moorhouse (1991a). Coordinatise N by a loop G. Then (14.5.a)

rankpN

= 3n -

2 - s,

where pS = [G: K] for the unique minimal normal subloop of G for which G/K is an elementary abelian p-group. Theorem 14.4 immediately implies the validity of Conjecture 14.3 for r = 3. Moorhouse (1991 a) also obtained the following characterisations of nets based on a cyclic Latin square.

14.6 Theorem. Let N be any (n, 3; I)-net and assume that n is square-free or that N can be coordinatised by a nilpotent group G. Then rankpN = 3n - 3 for every prime p dividing n if and only if G is cyclic, that is, N belongs to the cyclic Latin square of order n given by the group table ofZn . • Moorhouse (1991a) also verified Conjecture 14.3 for many examples of nets of small orders. Moreover, he proved the validity of his conjecture for nets which are part of a Desarguesian affine plane AG(2, p), where p is any prime; his proof additionally provides an explicit basis for the GF(p )-code of AG(2, p). Note that one has to have equality in Conjecture 14.3 then, as is easily seen using Theorem 14.2. That Moorehouse's conjecture is extremely interesting but, presumably, also extremely difficult becomes clear from his following major result.

X. Transversal designs and nets

780

14.7 Theorem. Let A be an affine plane ofsquare-free order n, and assume the validity of Conjecture 14.3 for n. Then n is a prime and A is the Desarguesian plane AG(2, n) . • Moorhouse (1991b) proved the validity of Conjecture 14.3 for certain nets admitting translations; combining this with Theorem 14.4 we have the following results.

14.8 Theorem. Let N be an (n, r; I)-net. Conjecture 14.3 is valid in each of the following cases:

= 3;

(14.8.a)

r

(14.8.b)

n = p is a prime, r = 4, and N belongs to a (p, 3)-difference matrix over Zp;

(14.8.c)

N is a translation net with abelian translation group. •

Note that (14.8.c) contains a case previously mentioned, namely that N consists of some parallel classes of AG(2, p).

14.9 Remark. Moorhouse (1993) considers the p-ranks of direct products of nets, cf.1.7.21, and shows that Conjecture 14.3 holds in this situation if it holds for the direct factors involved. Codes of nets were also studied by Dougherty (1993) who gave a reformulation of Conjecture 14.3. A related problem is the determination of the p-ranks of net graphs; for this and for more general results on the rank and structure of graphs, we refer to Peeters (1995a, b).

XI Asymptotic Existence Theory Spat kommt Ihr, doch Ihr kommt! Der weite Weg entschuldigt Euer Saumen.

(Schiller)

§ 1. Preliminaries

1.1 Introduction. The aim of this chapter is the proof of Wilson's main theorem: the necessary conditions (IX.2.9.c, d) for the existence of an SA. (2, K; u) are sufficient eventually, i.e., for all but finitely many v E N; Wilson (1975). The proof uses the following previous results. (a) Dirichlet's theorem on primes in arithmetic progressions; see, for instance, Serre (1973) or Ireland and Rosen (1990). (b) The infinity of the closed sets B(k, A), cf. Theorem VII.6.6 and Corollary Vll.6.7. (c) The sufficiency of the necessary existence conditions for large values of A, cf. Theorem Vrn.7.1 and Corollary Vrn.7.15. (d) The theorem of Chowla, Erdos and Straus (1960) on the existence of arbitrarily many mutually orthogonal Latin squares of order g for almost all values of g, cf. Theorem X.5.4. First we shall prove some special cases of the main theorem: (i) the existence of infinitely many v E B(k) in any residue class u+k(k-l)N, if u == 1 (mod k - 1) and u(u - 1) == 0 (mod k(k - 1)), using Theorem Vll.6.6, Corollary Vlll.7.15, and linear algebra; (ii) the eventual periodicity (see below) of closed subsets K = B(K) of N; (iii) the main theorem for A = 1.

1.2 Definition. Let M be a subset of No = N U {O} and mEN. An m-fibre of

781

782

Xl. Asymptotic existence theory

M is any non-empty subset (1.2.a)

Ma,m := {x EM: x =.a (modm)} = Mn(a+mNo)

An m-fibre is called complete if it contains almost all x E a (a + mN) \ Ma,/n is finite.

with a E No.

+ mN, i.e. if

A non-empty subset M of N is called eventually periodic with period m if all m-fibres of M are complete. Then M is infinite. If M is eventually periodic with period m then it is eventually periodic with period nm for each n E N.

1.3 Lemma. If M is eventually periodic with periods m and n, then it is eventually periodic with period d = gcd(m, n). Proof. There are positive integers a, f3 such that d = am-f3n.Foreacha EM, there is an element b E M with b =. a (mod mn) such that b + mx, b + nx E M for all x E No. If hEN, thenb+hd = b+ham -hf3n. Letha = rn+y with 0:::: y < n. Then b + hd = b+ ym + (rm - hf3)n = b + ym + (h~ - y~)n. If h > Y;;', then b + hd E M; i.e. the d-fibre Ma,d is complete. • 1.4 Corollary. The periods of an eventually periodic subset M consist of the multiples of a primitive period d, i.e. the gcd of all periods. •

1.5 Lemma. Let K s:::: N with K Lemma IX.2.9. Then (l.S.a)

a(B(K» = a(K),

(l.S.b)

f3(B(K» = f3(K).

Proof. Obviously a(B(K»

i= rtJ, {I},

and define a(K), f3(K) as in

Ia(K) and f3(B(K» I f3(K). By Lemma IX.2.9

v-I =. 0 (mod a(K», v(v - 1) =. 0 (mod f3(K»

for all v E B(K), hence a(K) I a(B(K» and f3(K) I f3(B(K» . • 1.6 Observations. (a) Obviously, f3(K) is always even, and a(K) divides f3(K). If we define (1.6.a)

y(K) := f3(K)/a(K),

then a(K) and y(K) are relatively prime.

§2. Steiner systems with v in given residue classes

783

Proof. Let d be a common divisor of a(K) and y(K). Then d . a(K) divides k(k - 1) for each k E K. Now a(K) divides each k - 1, hence both a(K) and d are relatively prime to each k E K. This is only possible if d . a(K) divides k - 1. But then d . a(K) I a(K), i.e. d = ±1. (b) Let a, f3 be positive integers such that 2 and a divide f3, and gcd(a, ~) = 1. Then there is a set K S; N such that a = a(K), f3 = f3(K), namely K {v EN: a I (v - 1) and f31 v(v - I)}.

=

Proof. Exercise. (c) Let

(1.6.b)

y(K) =

n

pe(p)

pEP

be the prime decomposition of y(K), and v sary condition (1.6.c)

v(v - 1)

== 1 (mod a(K». Then the neces-

== 0 (mod f3(K»

holds iff (1.6.d)

v

== 0 or 1 (mod pe(p»)

for all pEP.

Proof. (1.6.d) is equivalent to v(v - 1)=0 (mod y(K», i.e. to pe(p)lv or pe(p) I (v - 1) for all pEP . • 1.7 Lemma. Let K be finite, say K (1.7.a)

kJk2 .... kn

= {kJ, ... , kn }. Then

== 0 (mod y(K».

Proof. Let pEP be as in (1.6.b). Then pe(p) divides each ki (k i - 1) but not eachki -1 (i = 1, '" , n). Thus there is an i E {I, ... , n} with pe(p) I k i (k i -1) and Pf(ki - 1), as ki and ki - 1 are relatively prime. Hence pe(p) I ki, which implies the assertion 0.7.a) . •

§2. The Existence of Steiner Systems with v in Given Residue Classes 2.1 Introduction. Following Wilson (1975) and Brouwer (1979c), we shall prove that every admissible residue class mod k(k -1) contains some v E B(k).

784

XI. Asymptotic existence theory

The idea is to use Theorem vn.6.6 and Corollary Vrn.7.1S and thus to reduce the existence question for A = 1 to the question for large values of A which was answered in Corollary vrn.7.1S.

2.2 Theorem. If there exists an 8J,.(2, k; u), where A = q is a prime power, and also an 8(2, k; qd), and if, furthermore, q 2: u + 2 and d 2: (;), then there is a Steiner system S(2, k; uqd) . • The proof will take several steps.

2.3 Notation and Construction. Let U = Ni = {1, ... , u}, and let (U,A) be an 8q (2, k; u). Let V = GF(q)d be a q-vector-space with d 2: (;) and with coordinates

of each vector x E V. Thus there are (;) coordinates with double index ij, i < j, and d' = d - (;) remaining coordinates with single index i. For each pair {i, j} E (~) there are exactly q blocks A E A containing i and j. Let

(2.3.a)

N ij

:

{A : i, j I A E A} -+ GF(q)

be a bijection of the set of these q blocks onto GF(q). Now assume IVI = v = qd E B(k), as in the hypotheses of Theorem 2.2. In order to construct an 8(2, k; uv) on the point set U x V, we first construct an S(2, k; v) on each "level" {i} x V. Let ({i} x V, B i ) be this S(2, k; v). Let W be a generator of (GF(q)*, .), and let H C V be the hyperplane consisting of all vectors x with

(2.3.b)

Xl2

+ ... + Xu-l,u + Xl + ... + Xd' = O.

For each i E U let T; : x ~ Y be the linear mapping defined by

(2.3.c)

Yjl

={

X'l J i XjlW

if i = j or i = l, otherwise,

and

(2.3.d)

Yj

= XjW i

for j = 1, ... , d'.

§2. Steiner systems with v in given residue classes

785

Finally, for each A E A, let fA : A -+ V be a mapping which will be specified later. Now we construct blocks B(x, y, A) for all (x, y, A) E V x H x A by (2.3.e)

B(x, y, A) := {(i,

z) : i

E A and

z=

x

+ 1i(Y) + fA(i)}.

We have to show that for an appropriate choice of the mappings fA these blocks, together with the blocks B E Bi (i = 1, ... , u), form an S(2, k; uqd). First note that we have got the right number of blocks. For given i, j with i < j there are q2d pairs {(i, z), (j, z')}. On the other hand, there are exactly q blocks A E A throughi andj. Thus the number of blocks B(x, y, A) defined by (2.3.e) is also q . qd . qd-l = q2d. We have to choose the mappings fA such that any pair {(i, z), (j, z')} with i < j is covered by at least (hence exactly) one block (2.3.e). This is true if and only if each pair {( i, 0), (j, z' - z)} is covered, where 0 is the zero vector.

2.4 Problem. For every A E A, find a mapping fA : A -+ V such that, for E (~), the expression

each {i, j} (2.4.a)

takes all values z

E

V, where (y, A) runs through H x A.

2.5 Observations. In order to solve Problem 2.4, we study the system of equations (2.5.a) for a given vector z coordinatewise. Assume i < j. For a given find y.

Case 1. Let I

E

{I, ... , d'}. Put

(2.5.b) Then (2.5.a) yields (2.5.c) hence the Yl are uniquely determined.

z we have to

Xl. Asymptotic existence theory

786 Case 2. Let{i', j'} (2.S.d)

E

(~), i' < j', and (i, j}

fA(i)i'j' := 0

ifi' < j' and i

n Ii', j'} = ¢. Put

'I Ii', n·

Then (2.S.a) reads

and the Yi' j' are uniquely determined. Case 3. i = i' < j' =1= j. Now (2.S.a) reads Tj(Y)ij' - T;(Y)ij' + fA(j)ij' - fA(i)ij' = Zij', j Yij'(W - 1) = ZiP - fA(j)ij' + fA (i)ij"

Hence the Yij' are uniquely determined, regardless of the choice of fA. The same holds for i =1= i' < j' = j and the coordinates Yi'j' and the similar cases i < j = i' < j' and i' < j' = i < j. Case 4. i = i' < j' = j. (2.5.a) reads Tj(Y)ij - T;(Y)ij

(2.S.e)

+ fA(j)ij -

fA(i)ij = Zij, i.e.

fA (j)ij - fA (i)ij = Zij'

Now, using (2.3.a), define for i < j (2.S.f) and (2.S.g)

fA (i)ij := O.

Then (2.5.e) has a solution with a unique A E A and arbitrary Yij' Hence Yij can be chosen such that Y E H; see (2.3.b). This proves Theorem 2.2. • 2.6 Theorem. Let m, u, kEN and k > 2. Moreover, let u - 1 == 0 (mod k - 1),

(2.6.a) (2.6.b)

u(u - 1) == 0 (mod k(k -

1».

Then there are infinitely many W E B(k) with

(2.6.c)

W

== u (mod mk(k -

1».

Proof. Let A. == 1 (mod k(k - 1». If A. is sufficiently large, then there is an S),,(2, k; u), by Corollary VIII.7.IS. In particular, A. can be taken from an infinite set of primes q == 1 (mod mk(k In addition choose d:::: (~) and

1».

§3. The main theorem/or Steiner systems S(2, k; v)

787

q :::: u + 2 such that qd E B(k), which is possible by Wilson's Theorem VII.6.6. Then w := uqd == u (mod mk(k Now the assertion follows from Theorem2.2. •

1».

2.7 Example. There are infinitely many S(2, 10; v)'s with v 90), even though no explicit example seems to be known. 2.8 Corollary. Define Rk := {x EN: (k - l)x Select mEN arbitrarily, and assume (2.8.a)

r(r - 1)

+I

== 46 or 55 (mod

E B(k)}, as in IX.3.7.

== 0 (mod k).

Then there are infinitely many x E Rk with (2.8.b)

x

== r (mod mk).

Proof. Put u := (k - 1)r + 1. Then (2.8.a) is equivalent to (2.6.a, b). There are infinitely many W E B(k) satisfying (2.6.c). Set -1 E Rk • k-l

W

. x:= - Then (2.8.b) holds. •

§3. The Main Theorem for Steiner Systems S(2, k; v) 3.1 Introduction. In this section we shall prove that r E Rk for almost all r satisfying (2.8.a). This is equivalent to the main theorem for the case K = {k} and J... = 1. Recall that Rk is closed, i.e. (3. 1.a)

also, from Lemma IX.3.13, (3.l.b)

if r, s

E

Rk and r E TD(k), then rs E Rk.

3.2 Lemma. There is an arbitrarily large ro E N with ro, ro ro == 0 (mod k).

+I

E Rk and

Proof. By the Chowla-Erdos-Straus TheoremX.5.4, there is a TD[k; u] for all sufficiently large u. As B(k) is infinite (cf. Corollary VII.6.7), we may choose

Xl. Asymptotic existence theory

788

u from B(k). Starting with a TD[k; u - 1] and a TD[k; u], we obtain k(u - 1)

+ 1 E GD(k, u -

1) + 1 S; B({k, u}) = B(k),

ku E GD(k, u) S; B(k, u) = B(k).

Now set ro = k~':}) .•

3.3 Lemma. If r, r + t E Rk.

+ 1, s, t

E Rk with s :::: t and s E TD(r

+ I),

then

rs

Proof. From a TD[r + 1; s] remove s - t points of one point class and thus get an S(2, {s, t, r, r

+ I}; rs + t). Hence rs + t

E B(Rk) = Rk . •

In the following we shall use this lemma without explicit reference. 8 > 0 be given. Then Rk contains an infinite sequence r[, r2, ... such that ri+1 > ri and ri+1/ri < 1 + 8 for all i E N.

3.4 Lemma. Let

Proof. Let ro be as in Lemma 3.2. Take any tI, t2 E Rk with tl < t2 and another mE Rk withm :::: t2 andm E TD(r+ I). Nowputu := rOm+tI, v := rOm+t2. If m is large, then 1
r(n) + nro + x with s', s' + 1 E Rk. Then

ros' + r(n)

+ iro + x

E

Rk

for i

and

ro(s' + 1) + r(n) Define r(n

+ nro + x

+ 1) := ros' + r(n) . •

E

Rk.

= 0,1, ... , n

790

Xl. Asymptotic existence theory

3.7 Lemma. Let ro be as in Lemma 3.2 and x any element of Rk. Then Rk contains almost all y == x (mod ro), that is, all such y with finitely many exceptions. Proof. By Lemma 3.5, there is an mEN and an infinite sequence Sl, S2, S3, ... of elements of Rk such that 0 < Si+l - Si < m for all i E No Put n := rom and apply Lemma 3.6. In detail, put ti := r(n)

+ iro + x

(i = 0, ... ,n)

and choose j EN such that s := Sj 2:: r(n) agE N such that (3.7.a)

ros

+ ti

= rOsj

+ nro + x. By Lemma 3.3, there is

+ r(n) + iro + x

E Rk

for i E {O, 1, ... , n} and for all j ::: g. As rO(sj+l - Sj) < n, these numbers (3.7.a) cover almost all integers y == x (mod ro) . • 3.8 Theorem. The necessary conditions (3.8.a) (3.8.b)

v(v -

°

== (mod k -1), 1) == 0 (mod k(k - 1»

v-I

for the existence of an S(2, k; v) are sufficient for almost all v E N.

Proof. We have to show that Rk contains all sufficiently large numbers r satisfying the condition r(r - 1) == 0 mod k, cf. (2.8.a). Let ro be as in Lemma 3.2. Select any r EN such that r(r - 1) == 0 mod k. We have to show that r E Rk if r is sufficiently large. By Corollary 2.8 with m = ro/ k, there is an x E Rk with x == r (mod ro). By Lemma 3.7, the set Rk contains almost all numbers in the residue class of x mod roo But this is the required result. • §4. The Eventual Periodicity of Closed Sets In what follows, K always denotes a closed subset of N which contains at least one element k > 1. Note that all the following results are trivial if 2 E K, since then K = No

4.1 Theorem. Let K =1= (/), {I} be a closed subset ofN. Then K is eventually periodic with period f3 (K). • The proof needs several steps.

791

§4. The eventual periodicity of closed sets

4.2 Corollary. Let K I- 0, {1) be a closed subset ofN, furthermore k, u E K and u == 1 (mod k(k - 1». Then K is eventually periodic with period u - L Proof. This follows immediately from Definition 1.2, assuming Theorem 4.1.

4.3 Lemma. Corollary 4.2 implies Theorem 4. L Proof. Choose k E K arbitrarily. By Theorem 3.8, there are u, v E B(k) with u =mk(k - 1) + 1, v =nk(k - 1) + u and gcd(m, n) = 1, hence gcd(u - 1, v - 1) = k(k - 1). By hypothesis, Corollary 4.2 holds, i.e. K is eventually periodic with periods u-1 and v- L By Lemma 1.3, K is eventually periodic withperiodk(k-1) and, by Corollary 1.4, the primitive period of K divides f3(K). Another application of Lemma 1.3 shows that it suffices to prove Corollary 4.2. • Henceforth we fix k, u

E

K with u

==

1 (mod k(k - 1).

4.4 Lemma. Letk, f, u E K andu == 1 (modk(k-1». Then there are infinitely many v E K with v == f (mod u - 1).

Proof. By Theorem 3.8, t(u - 1)

+1E

B(k) S; K

for sufficiently large tEN.

If t is large, then there is a TD[f; t(u - 1)

v := f(t(u - 1) Hence v

== f

+ 1]; i.e.

+ 1) E GD(f, t(u -

1)

+ 1) S;

B(K) = K.

(mod u - 1). •

4.5 Lemma. Let u E K and u == 1 (mod k(k - 1». Furthermore, let f E K and f - 1, f E TD(u). Then there are GD[K, {f - 1, u - I}; v] 's with both v = uf - 1 and v = uf - u which in case u I- f contain exactly one point class of size f - L Proof. In the case v = uf -1, construct aTD[u; fl which may be considered as an S(2, {u, f}; uf), say D. Delete one point x and consider as new point classes the point sets B \ {x} where B is any block of D containing x. If v = uf - u construct a TD[u; f - 1] and adjoin a point CXl to its point classes. From the

XI. Asymptotic existence theory

792

resulting S(2, {u, f}; uf - u + 1), say D', delete one point x and take as new point classes the point sets B \ {x} as before . • We finish the proof of Theorem 4.1 by establishing the following result which is equivalent to Corollary 4.2.

4.6 Proposition. Let k, u, f E K be given with u n(u -1) + f E Kfor almost all n EN.

== 1 (mod k(k - 1». Then

Proof. (a) By Lemma 4.4, we may assume that f - 1, f E TD(u). Hence the GDD's of Lemma 4.5 exist. (b) We will show that for almost all mEN with m allt E to, 1, ... ,m} (4.6.a)

==

1 (mod (u - 1)) and for

m(f - l)u

+ f(U -

1) E GD(K, {f - 1, u - I});

m(f - l)u

+ f(U -

1)

hence (4.6.b)

+1E

B(K U {f, u}) = K.

First note that m == 1 (mod k(k - 1)). Hence m E B(k) s;:; K for large m, by Theorem 3.8. Also (4.6.c)

m(f - l)u

+ t(u -

1) + 1 ==

f

(mod u - 1).

We will now find a GD[K, {f -1, u -I}; m(f -1)u+t(u -1)] for sufficiently largem EN. (c) In order to achieve this we use the recursive construction of Theorem IX.3 .2. But we describe it independently in the language of auxiliary matrices. LetD and E be the GDD's of Lemma 4.5, with incidence matrices

Mo No and .........

Here Mo and No have rows each.

f - 1 rows, and the other matrices Mi, Ni have u - 1

§5. The main theoremjor}.. = 1

793

Now let m == 1 (mod u - 1) be sufficiently large. Then there is a TD[f + 1; m J, say T. Delete m - t points from the second point class of T and obtain a GD[{f, f + I}, {t,m}; fm + tJ, sayG. Each block B ofGwith block length f + 1 or f determines a column in the incidence matrix of G. Replace the entries I in this column by Mo, M" ... , Mf if B has f + I points (one of them in the second point class of G), and by No, N" ... , N f-I otherwise. Replace the zeros by appropriate zero matrices. The result is the desired GD[K, {f - 1, u - I}; v] with v = m(f - l)u + t(u - I). (d) If m is sufficiently large we may assume that m(f - l)u

+ m(u -

1) 2: (m

+u -

1)(f - l)u.

Thus the numbers m(f - l)u + t(u - 1) cover the residue class f - 1 mod (u - 1) with finitely many exceptions, as required. •

§S. The Main Theorem for ,\ = 1 It suffices to prove the following lemma, with notation as before. 5.1 Lemma. Let fEN with (5. 1. a)

(5.tb)

f - 1 == 0 (mod a(K», f(f - 1)

Then there is a v

==

== 0 (mod f3(K».

f(mod f3(K» with v E B(K) . •

5.2 Main Theorem. The necessary conditions (5.1.a) and (5.l.b)for the existence of an S(2, K; f) are sufficient for almost all fEN. Proof. This follows from Lemma 5.1 and Theorem 4.1, as B(K) is closed. •

5.3 Proof of Lemma 5.1. (a) There is a finite subsetL of K witha(L) = a(K) andf3(L) = f3(K). HencewemayassumethatK is finite, say K = {k" ... , kn}. As B(K) is eventually periodic with period f3 (K), there are elements Xi E B(K) with Xi == k i (mod f3(K» for i = 1, ... , n, and Xi+' E TD(x, ... Xi)

fori = 1, ... , n - 1.

§5. The main theoremfor A. = 1

795

sinceki == 1 (moda(K)) fork = 1, ... , n which shows thata(K) divides k-l. Hence, by Observation 1.6.(a),

(5.3.1)

v == f(mod (J(K)).

Now Theorem 3.8 ensures v E B(k) s;: K for almost all values of v with the above congruence properties. This proves Lemma 5.1 and hence the Main Theorem for A = 1. • As a consequence, we obtain the following result due to Wilson (1972c), cf. Remark IX.7.3.

5.4 Corollary. For every K

~

B(L) = B(K). Hence every K

N there is a finite subset L

s;:

K such that

s;: N has a unique finite basis.

Proof. Let H s;: K be a finite subset with a(H) = a(K) and (J(H) = (J(K). Then B(H) contains almost all v with v -1 == 0 (mod a(K)) and v(v -1) == 0 (mod (J(K)). Hence K \ B(H) s;: B(K) \ B(H) is finite. Put L := H U (K \ B(H))

s;:

K.

Then B(K) ;2 B(L) ;2 B(H) U (K \ B(H)) ;2 K. As B(L) is closed, B(L) B(K) . •

=

# 0, {I} is eventually periodic with period if and only if d == 0 (mod a(K)) and d is odd.

5.5 Corollary. A closed set K d := {J(K)/2

Proof. (a) If K has period d, then there are x, x x - 1, x

+ dE

K, hence

+ d - 1 == 0 (mod a(K))

which implies d == 0 (mod a(K)). Furthermore (J(K) divides x 2 - x and (x + d)(x + d - 1) which impli~s d 2 + 2dx - d == 0 (mod (J(K)); this holds iff d is odd. (b) Let d == 0 (mod a(K)) and d == 1 (mod 2). By definition, x == 1 (mod a(K)) and x(x - 1) == 0 (mod a(K)) for all x E K. If y == x (mod (J(K)) is sufficiently large, then y E B(K), by Theorem 4.1. If y:=x +d (mod (J(K)) is sufficiently large, then y := 1 (mod a(K)) and y(y - 1):= x(x - 1)

and, since d

+ 2dx + d(d -

= (J(K)/2, by Theorem 5.2 also y

1) == x(x

E B(K) . •

1) (mod (J(K)),

§5. The main theorem for A. = 1

795

sinceki == 1 (moda(K» fork = 1, ... , n which shows thata(K) divides k-l. Hence, by Observation 1.6.(a), (5.3.1)

v == f(mod fJ(K)).

Now Theorem 3.8 ensures v E B(k) s;: K for almost all values of v with the above congruence properties. This proves Lemma 5.1 and hence the Main Theorem for A = 1. • As a consequence, we obtain the following result due to Wilson (1972c), cf. Remark IX.7.3. 5.4 Corollary. For every K

s;: N there is a finite subset L s;: s;: N has a unique finite basis.

K such that

B(L) = B(K). Hence every K

Proof. Let H s;: K be a finite subset with a(H) = a(K) and fJ(H) = fJ(K). Then B(H) contains almost all v with v -1 == 0 (mod a(K» and v(v -1) == 0 (mod fJ(K». Hence K \ B(H) s;: B(K) \ B(H) is finite. Put L := H U (K \ B(H»

s;:

K.

Then B(K) ;2 B(L) ;2 B(H) U (K \ B(H» ;2 K. As B(L) is closed, B(L) = B(K) . •

# 0, {I} is eventually periodic with period if and only if d == 0 (mod a(K)) and d is odd.

5.5 Corollary. A closed set K d := fJ(K)/2

Proof. (a) If K has period d, then there are x, x

x-I, x

+ dE

K, hence

+ d - 1 == 0 (mod a(K))

==

0 (mod a(K». Furthermore fJ(K) divides x 2 - x and (x + d)(x + d - 1) which implies d 2 + 2dx - d == 0 (mod fJ(K)); this holds iff d is odd. which implies d

(b) Let d == 0 (mod a(K)) and d == 1 (mod 2). By definition, x == 1 (mod a(K» and x(x - 1) == 0 (mod a(K» for all x E K. If y == x (mod fJ(K» is sufficiently large, then y E B(K), by Theorem 4.1. If y ==x +d (mod fJ(K)) is sufficiently large, then y == 1 (mod a(K)) and y(y - 1) == x(x - 1)

and, since d

+ 2dx + d(d -

= fJ(K)/2, by Theorem 5.2 also y

1) == x(x

E B(K) . •

1) (mod fJ(K)),

Xl. Asymptotic existence theory

796

5.6 Remark. An interesting application of Theorem 5.2 to partition lattices was given by Ganter and Gronau (1991). §6. The Main Theorem for .A> 1 In this section, we will finish the proof of the main result due to Wilson (1975).

6.1 Main Theorem (Wilson 1975). The necessary existence conditions for a B[K, A; v], i.e. (6. La)

(6.1.b)

A(V - 1) == 0 (mod a(K», AV(V - 1) == 0 (mod f3(K»

are sufficient for almost all v E N. • The proof requires some preparation.

6.2 Remark. If A satisfies (6. La, b), then for A' = mA+n.B(K) withm, n E Z obviously (6. La) and (6.1.b) with A' instead of A hold. If, in particular, A' is the greatest common divisor of Aand f3 (K), then the necessary existence conditions for A and A' are equivalent. Hence, in order to prove the main theorem it suffices to consider the special case that A divides f3(K). Henceforth we always make this assumption.

6.3 Lemma. Define M := B(k, A), where A divides k(k - 1), and (6.3.a)

k(k - l)/A .Bo = .Bo(k) := { 2k(k - 1)/A

if this number is even, otherwise.

Then (6.3.b)

.B(M) = .Bo.

Proof. As AV(V - 1)

== 0 (mod k(k -

1» for v

V(V - 1)

== 0 (mod k(k -

l)/A).

As {3(M) is even, f30 I f3(M), say (6.3.c)

f3(M) = m.Bo.

E

M,

§6. The main theoremfor J... > 1

797

By VII.S.13, all sufficiently large prime powers q with (6.3.d)

q

== 1(mod k(k -

1)/J...)

are in M. In particular, let q E M be a large prime of the form

q = fJo(fJo

+ l)x -

fJo

+1

(x EN).

The existence of q is assured by Dirichlet's theorem. Then q(q - 1)

== 0 (mod fJ(M».

If q > fJ(M), thenq and fJ(M) are relatively prime, hence fJ(M) divides q -1 = fJo(fJo + l)x - fJo and m divides (fJo + l)x - 1. Therefore, (6.3.e)

gcd(fJo + 1, m) = gcd(fJo

+ 1, fJ(M»

= 1.

Again, by Corollary VII.8.13 and Dirichlet's theorem, there are large primes p=fJ(M)t+fJo+lEM

withtEN,

hence pep - 1)

== 0 (mod fJ(M».

If p > fJ(M), then p is relatively prime to fJ(M). This implies p - 1 = fJ(M)t

+ fJo == 0 (mod fJ(M».

Hence fJ(M) divides fJo and, by (a), fJ(M) = fJo . •

6.4 Lemma. Let J... divide fJ(K), and let M := B(K, J...). Put (6.4.a)

fJ, :=

fJ(K)/J... { 2fJ(K)/)"

if this number is even, otherwise.

Then

(6.4.b)

fJ(M) = fJ,·

Proof. If v E M, then v(v - 1) divides fJ(M).

==

0 (mod fJ(K)/J...). As fJ(M) is even, fJ,

Xl. Asymptotic existence theory

798

For each k E K obviously B(k, A) s:; M = B(K, A). Hence (3(M) divides (3(B(k, A». Thus, using Lemma 6.3, f3(M) divides gcd{f3o(k) : k E K}. Let

e(k)

={

I

if k(k - l)/A is even,

2

otherwise.

Then gcd {f3o(k) : k E K} = gcd { e(k)

Case 1. e(k) = 1 for all k hence f3(M) divides f31'

E

k(k - 1) } A :k E K .

K. Then gcd {f3o(k) : k

E

K} = f3 2; see Theorem 2.10 below. We shall require some preliminary notions; but let us first observe that PGn - 1 (n, q) is not characterised by its parameters. As mentioned by Dembowski (1968), the following result is due to Bill Kantor. 2.2 Theorem. Let q be a prime power and n > 2 an integer. Then there exist . symmetric 2-designs with the same parameters as PGn - 1 (n, q) which are not isomorphic to PGn- 1 (n, q).

809

§2. Characterisations ofprojective spaces

= (V, B, E) be the symmetric design PGn-1 (n, q), and U a fixed block. Put A := {H n U : H E B \ {Un. Then (U,A, E) ';:!; PGn-z(n - 1, q). Letrr be a permutation ofA. We define a new incidence structureE = (V, C, E) by

Proof. LetD

(2.2.a)

e:= {U} U {(H \ U) U rr(U n H) : H

E B \ {Un.

Obviously ICI = IBI for any C E e, B E B. Now we show blocks of e intersect in the same number of points. If C E C = (H \ U) Urr(U n H), then IU n CI = Irr(U n H)I = C, DEB \ {U}, say C = (H \ U) U rr(U n H), D = (K \ U) with K, HE B \ {U}, then

that any two B \ {U}, say IU n HI. If U rr(U n K)

+ Irr(U n H) n rr(U n K)I UI + IU n H n KI = IK n HI.

Ie n DI = I(K n H) \ UI = I(K

n H) \

By Corollary II.3.3, E is a 2-design with the same parameters as D. Choose rr in such a way that the images of the qn-Z + ... + q + 1 2:: n blocks A E A containing a given point U E U have empty intersection. Let L be a line of D which intersects U in {u}, and choose two points a, bEL \ {u}. The blocks in E through a and b have exactly q points in common, namely L \ {u}, whereas in D the blocks through any two points have exactly q + I points in common. Hence D and E cannot be isomorphic. • Thus we will indeed require some new combinatorial concepts to characterise the designs PGn- 1(n, q), and these are natural generalisations of the concepts of line and plane. 2.3 Definitions. An incidence structure D = (V, B, l) is called cohesive if any two points are joined by at least one block. In this case, given two distinct points p, q, the line pq is defined as the intersection of all blocks containing p and q; formally (2.3.a)

pq:=

n

(B) = «p, q».

p,qEBEB

If p, q Est, then pq S; st. In general equality does not hold as easy examples show. Three points p, q, r are collinear if they are contained in a line. Otherwise {p, q, r} is called a triangle and (p, q, r) an ordered triangle. Triangles exist if there are blocks containing two points but not all the points.

810

XII. Characterisations of classical designs

q, r}. If A plane pqr is the intersection of all blocks containing a triangle {p, V. = pqr then r}, q, {p, ;2 there is no block B triangle A cohesive incidence structure with triangles is called smooth if every 0 is not = p that is contained in the same number p of common blocks. Note excluded. The set of lines will be denoted by L and the set of planes by P.

that 2.4 Lemma . Let D = (V, B, 1) be a cohesive incidence structure such propg followin the has D Then any two points are joined by exactly)... blocks. erties. (i) Any two points are contained in a unique line. (ii) Every line is contained in exactly)... blocks. (iii) For every line L ELand every block B I(B) nLI:::: 1.

E

B either L ~ (B) or

that Proof. Let p, q be any two distinct points of D, then p, q E pq. Assume blocks the)... of each in are they Then St. line a in p and q are also contained t s; pq, through sand t. As there are no other blocks through p and q we have s .• tely immedia follow (iii) and (ii) s assertion and hence st = pq. Now the

ng 2.5 Lemma . Let D = (V, B, 1) be a smooth incidence structure containi the Then blocks. )... precisely by joined are triangles such that any two points following assertions hold. (i) Any triangle is contained in a unique plane. (ii) Each plane is contained in exactly p blocks. line, or (iii) For each PEP and each B E B either P S; (B), or P n (B) is a Ipn(B) I::::l. containing (iv) For every line L and every point p ¢ L, there is a unique plane p and L. (v) p < ).... The p Proof. Let {s, t, u} be a triangle in the plane pqr. Then stu S; pqr. s, t, through blocks other no are there As u. t, s, blocks through p, q, r contain As (i). proves This stu. ~ pqr hence and and u, it follows that p, q, r E stu to left is s assertion other the of proof easy The there are triangles, (v) follows. the reader. •

SI1

§2. Characterisations of projective spaces

2.6 Lemma. Let D be a smooth 2-design SA (2, k; v). Then any line L of D has exactly m points, where (2.6.a)

A.k - pv m=---

A-P

Proof. Let L be any line and choose two points p, q E L. Counting all flags (x, B) with x i= p, q E B (and thus L s;; (B)) in two ways gives the equation

(ILl - 2)A + (v - ILI)p = A(k - 2) which implies the assertion. •

2.7 Lemma. Let D = (V, B, l) be a smooth 2-design with line set L, and let p E V be a point. Define an incidence structure D' (p) = (L p' B p' l) by putting Lp:= {L EL:p E L},

Bp:= {B EB:pIB},

and LI B iff L C (B). ThenD'(p) is a 2-design with parameters (2.7.a)

v-I v'=--, m-1

k-l k'=--, m -1

A'=p,

r' =A,

b' = r,

with m = ILl as in Lemma 2.6.

Proof. Exercise. • 2.8 Lemma. With notation as in Lemma 2.7, (2.S.a)

P::::

with equality

A(A - 1) r- I '

if and only if D' (p) is symmetric.

Proof. The assertion is trivial for A = 1. Now let A > 1. As there are triangles inD we have v > k, v' > k', hence Fisher's inequality r' ::: k' holds, cf. Corollary 1.8.6. Therefore the following mutually equivalent inequalities hold. k' (r' - A') :::: r' (r' - A'), r t2

e - rt2 + r'A' -

r' (r' (k' - 1))

k'A' ::: r'2k' - r'k',

+ r'A' -

k'A' ::: r'k' (r' - 1),

XII. Characterisations of classical designs

812

r'A' (v' - 1) + r'A' - k'A' ~ k'r' (r' - 1), A'k'(b' - 1) ~k'r'(r' - 1), , r'(r' - 1) A(A - 1) A=P< = . - b'-1 r-l This is inequality (2.8.a). Equality holds if and only if r' ll.3.3, this is true if and only if D' (p) is symmetric. •

= k'. By Corollary

2.9 Remarks. (a) Cameron (1974) calls a smooth 2-design "locally symmetric" if D' (p) is symmetric (for one and hence for each point p, by Lemma 2.8). We will consider such designs in §4 below. Note that the designs PGn-l(n, q) and AGn-l (n, q) are locally symmetric; in fact, here the designs D'(p) are isomorphic to PGn-z(n - 1, q). (b) Dembowski (1968) attributes Lemma 2.8 to Ryshpan. Using Lemma 2.8, Dembowski noted the equivalence of conditions 2.1 0 (i) and (v) below. The equivalence of the conditions (i) through (iv) is essentially due to Dembowski and Wagner (1960) who proved this result for the class of symmetric 2-designs. Later Kantor (1969d) remarked that the hypothesis could be somewhat weakened.

2.10 Theorem (The Dembowski-Wagner Theorem). Let D = (V, B, l) be a 2-design SA (2, k; v) with A > 1 and v > k + 1. Then the following conditions are equivalent. (i) D

~

PGn-l (n, q) for some prime power q and some integer n > 2.

(U) Every line meets every block.

(Ui) Every line has exactly ~=~ points. (iv) Every plane is contained in exactly A:~[ > 0 blocks.

(v) D is smooth and symmetric.

Proof. (a) We first note that PGn _ 1(n, q) satisfies conditions (ii) and (v). Next we shall show that conditions (ii), (iii), and (iv) are equivalent. Let L be any line of D. Then L is contained in exactly Ablocks and therefore meets precisely these and ILI(r - A) further blocks. Thus L meets every block iff ).,+ ILI(r -A) = b. This proves the equivalence of conditions (ii) and (iii). (b) We show that (iv) implies (iii). By (2.6.a) and (iv),

m=

Ak - pv Ak(v - 1) - (Ak - r)v -Ak + rv = =-c"--""'--:--A- P A(V - 1) - (Ak - r) r(k - 1) - (Ak - r)

bk - Ak k(r -).,)

=

b- A r - )., .

813

§2. Characterisations of projective spaces

(c) To prove the converse, we may also assume the validity of condition (ii) by the preceding argument. Let P = pqr be any plane of D and put L := qr. If P is contained in p blocks, then there are r - p blocks through p which do not contain L. Each of these blocks meets L in a unique point, by (ii). On the other hand, each point of L is on A - p such blocks, and therefore r - p = (A - p)ILI = (A - p)(b - A)/(r - A)

p=

= =

by (iii),

A(b - A) - r(r - A)

b-r A(b - A)(V - 1) - r(r - A)(V - 1)

--'..--'---,-::--~:---~----

(b - r)(v - 1)

(b - A)r(k - 1) - r(r - A)(V - 1)

(b - r)(v - 1) (b - r)(Ak - r)

Ak - r

(b-r)(v-l)

v-I

(d) Now assume that D satisfies the equivalent conditions (ii), (iii), and (iv). We will show that these conditions imply (i) by verifying the standard VeblenYoung axioms for projective spaces, i.e. (1.1.a, b, c, d), for the linear space (V, L). Thus (V, L) is isomorphic to PG 1(n, q) (n > 2 and q a prime power). Note first that any two points are on a unique line, by Lemma 2.4, which is the first axiom. The essential axiom is the following one: if four points p, q, r, s form a quadrangle and if the lines pq and r s intersect, then so do the lines pr and qs. Thus let t := pq n r s and consider the plane P := prt; then q, s E P. Now choose a block B with pr S; (B), but P % (B), which is possible as A > P by Lemma 2.5. Then Lemma 2.5 implies P n (B) = pro By (ii), the line qs intersects B in a point u. Thus u E P n (B) = pr, as qs S; P, verifying the second axiom. The third axiom requires that every line has at least three points. But otherwise every line would have exactly two points and thus b = 2r - A, by (iii). Then vr = k(2r -J,..), i.e. Ak = r(2k-v), andhencer(2k-v)(v-l) = Ak(v-l) = rk(k - 1), which implies (v - k? = v - k; i.e. either v = k or v = k + 1, contrary to hypothesis. Finally there are non-intersecting lines. Otherwise any four points would be contained in a plane. Let {p, q, r} be a triangle. Then x E pqr for all x E V. By (iv), pqr S; (B) for a block B E B; hence (8) = V, a contradiction. Thus

XII. Characterisations of classical designs

814

(V, L) is a projective space. Any block of D must be a linear subspace of (V, L). By (ii), it is a hyperplane of (V, L). As b 2::: v, all hyperplanes of (V, L) are blocks of D, and thus D is indeed isomorphic to PGn - 1 (n, q), as required.

It remains to show that (v) implies (iv). As D is symmetric, any two blocks intersect in A points. But then any two blocks of D' (p) intersect in ~=l points, where m is defined as in Lemma 2.6. Thus the dual structure of D' (p) is also a 2-design, and by Fisher's ineqUality D'(p) is symmetric. By Lemma 2.8 (case of equality), we obtain A(A - 1)

P=

r- 1

rCA - 1)

A(A - 1)

=

k- 1

=

v-I

Ak - r

= V-=!'

i.e. condition (iv) holds . •

2.11 Exercise. Show that the S7(2, 15; 31) = dev D, where D

= {1,2,4,S, 7,8,9,10,14,16,18, 19,20, 25, 28}

is the difference set of the non-zero squares in GF(31), has the same parameters as PG3 (4, 2) but is not isomorphic to PG3 (4, 2). Note that the three blocks D, D + 1, and D + 3 intersect in four points which is impossible for three hyperplanes of PG3 (2, 4). Both non-isomorphic 2-designs have a cyclic automorphism group. There are infinitely many examples for this phenomenon, see §V1.17. We next use Theorem 2.10 to give a characterisation of PGn -l(n, q) in terms of its "internal structures"; this is essentially due to Kantor (cf. Dembowski (1968, 2.2.14), when taking into account Mavron's Proposition II.8.14. In II.8.l3, 14 the block residue BD and the induced substructureD(B) of a symmetric 2-design D were defined by BD = (V \ (B), B \ {B}, I) and D(B) = «B), B \ {B}, l) for B E B.

2.12 Theorem. Let D = (V, B) be a symmetric 2-design with)", > 1 and v > k + 1. Then the following conditions are equivalent. (i) D :;:::; PGn - 1(n, q) for some prime power q and some integer n > 2. (U) D(B) is a multiple of a symmetric 2-designfor each block B E B. (iii) B D is an affine 2-design with parameter A' > 1 for each block B E B.

Proof. (ii) and (iii) are equivalent by Proposition II.8.14. Clearly (i) implies (ii): here D(B) :;:::; PGn - 2 (n - 1, q) for each block B E B. Finally, assume the validity of (ii) and (iii) and let B, C E B be any two blocks. Then a block X i= B

§2. Characterisations of projective spaces

815

contains B n C iff X \ B is parallel to X \ C in B D, as we have seen in the proof of Proposition II.8.14. Hence the line «B, C» of the dual design D* of D has preciselys+1elements,wheres = (v-k)j(k-J..) = (b-J..)j(r-J..)-1.Thus condition (iii) of Theorem 2.10 is satisfied for D*, and thus D* is isomorphic to some PGn - 1(n, q); but finite projective spaces are self-dual. • As a final consequence of Theorem 2.10 we present the second half of the original Dembowski-Wagner theorem characterising PGn - 1(n, q) by transitivity properties.

2.13 Theorem. Let D be a symmetric 2-design with J.. > 1 and v > k Then the following conditions are equivalent. (i) (ii) (iii) (iv)

+ 1.

D ;::: PGn-l (n, q)for some prime power q and some integer n > 2. AutD is transitive on ordered triangles. AutD is transitive annan-incident point-line pairs. AutD is transitive on unordered triangles.

Proof. (i) implies (ii), as ordered triangles correspond to ordered triples of linearly independent vectors in GF(q)n+l. Of course (ii) implies (iii) and (iv). On the other hand, if (iii) or (iv) holds, then obviously D is smooth, and, by Theorem 2.10, also (i) holds. •

2.14 Remarks. We have already seen that the existence of a regular automorphism group of a symmetric design is not enough to guarantee that this design is a projective space: difference sets provide many counterexamples. By Exercise 2.11, such counterexamples exist even with the parameters of a PGn - 1(n, q). Similarly, 2-transitivity on points is still not sufficient to force a symmetric design to be isomorphic to a projective space. As examples we mention the unique S2(2, 5; 11) (cf. 1.9.17 and IY.7.1O) and the complementary designs of the projective spaces. But here the following result of Kantor (1971a) shows that 2-transitivity together with the correct parameters forces a symmetric design to be classical.

2.15 Theorem. Let D = (V, B) be a 2-design with the parameters of some projective space PGn- 1(n, q). Then D is actually isomorphic to PGn- 1(n, q) if and only if Aut D is 2-transitive on points. • The proof requires some preliminary results.

816

XII. Characterisations of classical designs

2.16 Lemma. A line L of an SA (2, k; v) has at most ~=~ points. Proof. Any point x E L is on exactly r - A blocks not containing L. The number of all blocks not containing L is b - A, thus ILI(r - A) ::: b - A. Note that equality holds iff L meets every block. • The next two easy lemmas deal with permutation groups.

2.17 Lemma. Let G be a permutation group acting on the finite set X, and let P be a Sylow p-subgroup ofG.If pm divides IxGlfor some x E X and mEN, then pm divides IxPI.

= IGI and IxPI'IPxl Hence there is aCE N which is not divisible by p, such that

Proof. By Proposition Ill.3.lO, we have IxGI'IGxl

= IPI·

As IPxl divides IGxl, the assertion follows . •

2.18 Lemma. Let G be a permutation group acting transitively on two finite sets X and Y, where IXI and IYI are coprime. Then every stabiliser Gx(x E is still transitive on Y.

X)

Proof. By Proposition Ill.3.1O, both a := IXI and b := IYI divide IGI, say

IGI = abc with c E N. Then IGxl = bc and IGyl = acfor each y E Y. As G xy is a subgroup of both G x and G y' and as a and b are coprime, IG xy I divides c. Now the equation

which is derived from the natural action of G on X x Y, yields

I(x, y)GI ::: ab =

IX x Y!.

Therefore G is transitive on X x Y and G x is transitive on Y. •

2.19 Proof of Kantor's Theorem 2.15. By linear algebra, Aut PGn - 1 (n, q) is 2-transitive on points. Conversely, let Aut D be 2-transitive on points. In the case n = 2 the assertion follows by the theorem of Ostrom and Wagner (1959); see e.g. Hughes and Piper (1982) or Ltineburg (1980). Now assume n > 2.

§2. Characterisations 01 projective spaces Obviously all lines have the same size, say m Proposition 1.2.16 imply h ::: q.

817

= h + 1, and Lemma 2.16 and

We will show that actually h = q; then the assertion follows by Theorem 2.10. Choose a point x E V and two blocks B, C E B. There are (k - 1)1 h lines through x contained in Band (J.. - 1)1 h lines through x contained in B n C. Hence h divides k - J.. qn, and there is a prime p with q = pe, h pi for some e, 1 E N with 1 ::: e. We must show that actually e = I·

=

=

By Theorem IlIA.3, the group G is 2-transitive on blocks. Hence, for any block B, the stabiliser G B has two orbits on blocks. By Theorem IlIA. 1, G B has also two orbits on points. Thus G B is transitive on the k points x E B and also on the v - k points Y E V \ B. As k and v - k are coprime we can apply Lemma 2.18. Thus, for x E B, the stabiliser G B,x of both B and x is transitive on the v - k points Y E V \ B. Thus G is transitive on triples (B, x, y) with B E B, x E B, and y ~ B. Hence, given two distinct points x and y, it follows that G xy is transitive on the blocks B with x E Band y ~ B. Now there are k - J.. = qn-t such blocks. Hence qn-l divides IG xy I. Let X be the set of blocks B with x E B and y ~ B, and let P be a Sylow p-subgroup of G xy . By Lemma 2.17, qn-t divides IBPI for any B E X; i.e. P is transitive on X. Clearly P acts on the v - (h + 1) =q(qn-l + ... +q + 1) -hpointsz E V\xy. Now assume that e> I. Then ph divides q, and thus ph does not divide v - (h + 1). Hence there is a point z ~ xy with Iz PI = pi ::: h. Therefore, if B is a block with x E B, Y ~ B, then

IBP'I

= [Pz : Pz,B] = [P : Pz,B]/[P : Pz] = [P : PB][PB : pz.B]/[P :

Pzl

= IB P "ZPB I/lzPI 2:: IBPl/lzPI = qn-1lpi 2:: pqn-2. But as y ~ xz, there is a block B with fixes x and z, we have

x, z

E

Band y ~ B. As the group Pz

This contradiction proves the theorem. •

2.20 Remarks. (a) The 2-transitive collineation groups of finite projective spaces have been classified by Cameron and Kantor (1979) who also solved the harder problem of classifiying the antifiag-transitive collineation groups of these spaces. It is also interesting that all line-transitive collineation groups of

818

XII. Characterisations of classical designs

a finite projective space are already 2-transitive on points, with one exception (Kantor 1973). These results complement Theorem 2.15, but are much harder and will not be dealt with in this book. In this connection one should also mention Perin (1972) who has studied collineation groups of projective spaces acting transitively on the set of d-fiats for d > 1. (b) There are also characterisations of the designs PGn-l (n, q) in terms of special automorphisms, in particular of elations. Of course this needs a generalisation of the classical concept of elation. An elation is defined as an automorphism fixing some block pointwise and some point on this block blockwise. There are results saying that a symmetric design with many elations must be isomorphic to some PGn-l (n, q); see e.g. Kantor (1969d, 1970) and Liineburg (1961). More general symmetric designs still admitting comparatively many elations have been studied by Kelly (1979, 1984a, 1984b); his examples in fact all arise from the designs PGn-l (n, q) by some "twisting" and admit a nice characterisation. We finally mention Butler (1981) who considered symmetric designs with still fewer elations and the survey by Piper (1983). (c) There are some characterisations of other incidence structures related to the projective spaces PG(n, q). LeIevre-Percsy (1980) characterised the designs PGd (n, q) as those smooth 2-designs with the right parameters for which each line has size at least q + 1. Ray-Chaudhuri and Sprague (1976) characterised the incidence structures fO!1Tled by the d-fiats (as points) and the (d + I)-fiats (as blocks) in a projective space. (d) A Dembowski-Wagner type characterisation of the quasisymmetric design PG 2 (4, q) formed by the points and planes of PG(4, q) was given by Sane and

Shrikhande (1993). (e) Rahilly (1991) considered symmetric designs which possess (a spread of) lines of maximal size, that is, of size (v - A)/(k - A), cf. Lemma 2.16.

2.21 Exercise. Recall that a symmetric design is called skew if it has a (+1, -I)-incidence matrix A such that I + A is a skew-symmetric matrix. Clearly, any skew (Hadamard) difference set, cf. §VI.8, gives rise to a skew design; in particular, this holds for the classical Paley difference sets. Prove that the only smooth skew symmetric design is the projective plane of order 2. Hint: Use the Dembowski-Wagner Theorem 2.10; for an explicit proof, see Robinson (1995). The result discussed in Exercise 2.21 is contained in a more general result due to Cameron (1973b) which we now describe. A design D is called quasismooth, if

§2. Characterisations of projective spaces

819

any three points of D are in either Ihl or 1h2 common blocks (for some constants Ihl and 1h2). Then one has the following strengthening of Exercise 2.21; for a proof, see also Cameron and van Lint (1991).

2.22 Theorem. The only quasismooth skew symmetric designs are the projective plane of order 2 and the unique symmetric design S2(2, 5; 11) which belongs to the Paley difference set D = {I, 3, 4, 5, 9} in Z" . • We conclude this section with the following result due to Jungnickel (1984b) which shows that the classical symmetric designs are very exceptional structures indeed.

2.23 Theorem. The number of symmetric designs with the parameters of PGd-1 (d, q) grows at least exponentially. Froof. We first recall a method of constructing such designs which is due to Shrikhande (1951) and was already discussed in proving Proposition 11.8.15. Let A denote the classical affine design AGd-1 (d, q), and let T be any symmetric design with the parameters of PGd-2(d - 1, q). Note that the number of blocks of T equals the number of parallel classes of A. Given any bijection a from the set of blocks ofTto the set of parallel classes of A, define an incidence structure D = D(T, a) as follows. Enlarge the point set of A by adjoining the point set of T to it, and let these new points form a block Boo of D. For any block B of T, adjoin the points of B to all the blocks of A which belong to the parallel class B a and choose all these enlarged blocks as blocks of D. The reader is asked to check thatD is a symmetric design with the parameters of PGd_l(d, q); note that any symmetric design with these parameters which contains A as a residual design may be obtained in this way, cf. Proposition 11.8.15. In what follows, we call any such design a closure of A. Clearly, we may assume all closures of A to have the same point set V; in other words, all symmetric designs T used in the above construction are assumed to be defined on the same point set Boo. From the construction given above, (2.23.a)

D(T, a)

= D(U, fJ)

if and only if (T, a)

= (U, fJ),

since the blocks of T may be recovered from D(T, a) as the intersections of the blocks distinct from Boo with Boo. If we fix any specific T, we may construct from it distinct (though isomorphic) symmetric designs by applying arbitrary

XII. Characterisations of classical designs

820

permutations of Hoo. In this way, T gives rise to exactly (qd-l

+ ... + q + 1)! / IAutTI

distinct isomorphic copies. In view of (2.23 .a), we obtain the following auxiliary result. (2.23. b)

The number of distinct closures of A with point set V is precisely Lr«qd-l + ... + q + 1) !)2 /IAut where T runs over a complete system of representatives of symmetric designs with the parameters of POd-2(d - 1, q).

n

Now consider a fixed closure Do of A. We want to obtain an upper bound on the number of closures of A which are isomorphic to Do. Any such closure Di is the image of Do under a permutation of the point set V. Since Di has to contain A, the number of such closures is at most the cardinality of the set M of all those permutations y of V for which D~ contains A. But then AY-' is an affine design contained inDo, and hence IMI cannot exceed the number of affine designs contained in Do (which is clearly bounded by the number of blocks of Do) times the number of permutations y E M satisfyingAY =A. Since any such y induces an automorphism of A, there are at most IAutA I . (I Hoo J)! permutations Y EM satisfyingAY =A. Thus we have the following estimate (using Lemma III.S.9 and Proposition III.S.20): (2.23.c)

The number of distinct closures of A with point set V which are isomorphic to any given fixed closure Do is at most (qd-l + ... +q + I)! 'IArL(d, q)l. (qd + ... +q + 1) = (qd-l + ... + q + 1)! . IprL(d + 1, q)l.

Combining (2.23.b) and (2.23.c) gives the following estimate on the number Sed, q) of isomorphism classes of symmetric designs with the parameters of POd-lCd, q): (2.23.d)

Sd (qd-I+ ... + q +1)! ( , q) 2: IPrL(d, q)l· IPrL(d + 1, q)1 (qd-l

+ ... + q + 1)!

where q = pI, P prime. Using the power series representation for ek , one may show from (2.23.d) that the asymptotic rate of growth of Sed, q) is ek .1nk , where k = qd-l + ... + q + 1, which proves the assertion. The details are left to the • reader. •

§3. Characterisation of affine spaces

821

2.24 Remark. Kantor (1994) obtained considerably stronger results than Theorem 2.23 by proving that any given finite group G acts as the full automorphism group of a symmetric design with the parameters of PG(d, q) provided that q > 3 and that d is sufficiently large (meaning d > 50IGI 2 ); in fact, there are still exponentially many, namely more than [qO.8d]!, designs of this type. 2.25 Remark. As discussed in §VI.17, there is an infinite series of nonequivalent cyclic difference sets, namely the GMW-difference sets, which have the same parameters as the Singer difference sets; hence the associated symmetric designs have classical parameters. These GMW-designs were characterised in terms of their automorphism groups by Jackson and Wild (1997); the same authors also characterised a certain subclass of the GMW-designs (which actually consists of Hadamard designs) that can be obtained from quadrics in characteristic 2, generalising previous work of Maschietti (1992) and Jackson (1993) who had considered the planar case, where one uses hyperovals.

§3. Characterisation of Affine Spaces In this section we shall consider the designs obtained from affine planes and spaces. Recall that the affine planes are precisely the designs S(2, n; n 2 ). As in the projective case, the designs AG n - 1(n, q) are not characterised by their parameters alone; one obtains a characterisation similar to the DembowskiWagner Theorem 2.10. This is due to Dembowski (1964, 1967).

3.1 Definition. LetD = (V, B, l) be a resolvable 2-design and write B II C iff the blocks Band C are in the same parallel class of the given resolution. If L is any line and B any block of D, then we call L and B parallel (written L II B or BilL) iff there is a block C with L ~ (C) and B II c. Two lines Land L' are called parallel (again wiitten L II L') iff L is parallel to each block B with L' ~ (B). Note that resolvable incidence structures are simple, cf. 1.5.4; hence we identify B E B with (B). 3.2 Lemma. Any resolvable 2-design D has the following properties.

(i) Parallelism of lines is an equivalence relation. II L' implies L = L' or L L' = for all L, L' E L. (iii) Let p be any point and L any line of D. Then there is at most one line L' with pEL' II L. (iv) Two parallel lines are contained in at least one common plane. (ii) L

n

o

Proof. (i) Obviously parallelism of lines is reflexive. If L and L' are lines with L II L', then for each of the A blocks B containing L there is a block B'

822

XII. Characterisations of classical designs

containing L' and patallel to B. As parallelism of blocks is symmetric and transitive, the same holds for parallelism of lines. (ii) Assume L II L', L t L' and L n L' = {c}. There is a block B containing L but not L', say b E L'\B. There is a block B'II B containing h. As B and B' are disjoint, c 1 and let B be any block of D. By Lemma 5.11 and induction, we may assume thatD'(B) has rB blocks, i.e. fiats FeB, through any point. Now let p be a point contained in B. As each block C through p with B ::j C yields a fiat B n C through p, one sees that p is on exactly rp = prB + 1 blocks of D, by 5.10 (iii). Hence rp is independent of the choice of pin B, and rB is independent of the choice of B through p. As any two points of D are joined we conclude that r B does not depend on B and r r p does not depend on p. •

=

5.13 Proposition. Let D satisfy Assumption 5.7 with JL > 1. If D does not contain disjoint blocks, then D is isomorphic to some PGn - 1(n, q) with n 2: 3. Proof. By Lemma 5.12, the dual of D is a 2-design. The assertion now follows from the dual of the Dembowski-Wagner Theorem 2.10, since the designs PGn-l (n, q) are self-dual. (Note that the dual of (5.7.a) is the assertion that every line meets every block.) • Note that the dual of Proposition 5.13 is a strengthening of criterion (ii) in the Dembowski-Wagner Theorem, since the assumption that D is a design has been weakened. This can be strengthened further; cf. Theorem 5.17 below. We shall require some further auxiliary results first.

5.14 Lemma. Let D satisfy Assumption 5.7. Then there is at least one block incident with each ofrnin{/1 + 1, 3} given points. Moreover, all lines have the same size m, and any two points determine a unique line. Proof. Both assertions are trivial for /1 = 1, by Lemma 5.9. Thus assume /1 > 1 and let p and q be any two points ofD; then p and q are on a common block B of D. By Lemma 5.11, we may apply this argument toD'(B) to obtain a common fiat through p and q. Any further point r then is on a common block with p and q by (5.7.a). This proves the first part of the assertion. Next, let p, q, r be

838

XII. Characterisations of classical designs

any three distinct points and choose a block B containing these points and a flat in B containing p and q. Then the line pq contains the same points when regarded as a line of D or as a line of D' (B), and similarly for pr. By induction, we may assume Ipq I = Ipr I in D' (B), hence in D. Thus all lines of D have the same number m of points; so any two points of D determine a unique line. •

5.15 Lemma. Let D satisfy Assumption 5.7 with 11 > 1 and let p be a point of D. Define the incidence structure D' (p) as in Lemma 2.7. Then either all D' (p) are isomorphic projective spaces PGn - 1 (n, q)for some n ?: 3 or allD'(p) are projective planes of the same order. Proof. We verify the hypotheses of Proposition 5.13 for D'(p). As 11 > 1, any two blocks of D'(p) meet. The validity of (5.7.a) for D readily implies that for D'(p). By Lemma 5.14, the number of lines of D through p is v' = (v - l)/(m - 1), and the number oflines of D through p contained in a block through p is k' = (k - 1)/(m - 1). Similarly, the number of lines through p contained in a flat through p is 11' = (11- 1)/(m - 1). This proves(5.7.b) and part of (5.7.c) for D' (p); moreover, we see that the parameters of D' (p) do not depend on p. Clearly 11' < k', as 11 < k. Also v' ?: k' + 2; to see this, choose any block B of D, distinct points q, r ¢ B and a point s E B with s ¢ qr. This is possible, as v?: k+2 and IB nqrl ::: 1 < k. Then v' ?: k' +2 for D'(s) (and hence for every D'(p», since neither qs nor rs is in B. Thus D'(p) satisfies (5.7.c). Finally, (5.7.d) is immediate from the last assertion of Lemma 5.14. Thus D' (p) is isomorphic to some PGn - 1(n, q) by Proposition 5.13, provided that 11' > 1. But for 11' = 1, D'(p) is an S(2, k; v) by Lemma 5.9; as any two blocks of D' (p) meet, D' (p) is a projective plane in this case. • 5.16 Lemma. Let D satisfy Assumption 5.7 with 11 > 1. Then D' (B) is a 2design for each block B, and the parameters of D' (B) do not depend on the choice of B. Denote the number offlats in B by b'; then the number of blocks of D incident with a given point p ¢ B and disjoint from B is r b' and therefore independent of the choice of Band p. Proof. By Lemma 5.11, D'(B) has constant block size k' = 11. Now choose any two distinct points p and q of D' (B). The number of blocks of D joining p and q equals the number of blocks of D' (p) containing the point pq of D' (p); thus x does not depend on the choice of p and q by Lemma 5.15. Then p and q are joined by exactly I = (x - 1)/ p blocks of D(B) by 5.1O(iii). Hence D(B) is a 2-design SI(2, 11; k) for each B. The final assertion is now immediate using Lemmas 5.12 and 5.1 O(ii). •

§5. Good blocks

839

We now proceed to present the strongest version of the Dembowski-Wagner Theorem known at present, due to Kantor (1969d).

5.17 Theorem. LetDsatisfyAssumption 5.7 withM > landassumethatsome block ofD meets every other block. ThenD is isomorphic to some PGn- 1(n, q) with n :0:: 3. Proof. From Lemma 5.16, we see that any two blocks of D meet. Now the assertion follows from Proposition 5.13 . • We now prove the main result of this section, i.e. Kantor's common characterisation of affine and projective spaces of dimension at least 4.

5.18 Theorem. LetDsatisfyAssumption5.7withfL > Iand(fL-I)(v-k) =f. (k - fL)2; cf Exercise 5.8. ThenD is isomorphic to some PGn-1 (n, q) or some AGn_1(n, q) withn:o:: 4. Proof. The hypothesis (M - 1)(v - k) =f. (k - fL? implies fL(B) > 1, by Lemma 5.11. Let p be any point of D and B a block through p. Furthermore, let D'(p, B) be the design (D'(B»'(p), i.e. the design formed by the lines of D' (B) through p and the fiats of D through p contained in B. By Lemmas 5.11 and 5.15, D'(p, B) is a projective geometry of dimension :0::2. Hence it is clear that D' (p) is a projective geometry of dimension :0::3, again by Lemma 5.15. Moreover D has constant line size m by Lemma 5.14. If m > 2, then the assertion follows from the Doyen-Hubaut Theorem 4.8. Thus assume m = 2 and choose a fixed point p. Then the points of D' (p) may be identified with the points r =f. p of D, since pr = {p, r} for each r. Under this identification, D' (p) is isomorphic to D p; i.e., D is an extension of a design isomorphic to some PGn_l(n, q) with n ::: 3. By Cameron's Theorem 11.7.12, this is only possible for q = 2. By Remark 1.9.10, we then have D ~ AGn(n + 1,2) .

•

We conclude this section by considering an analogue of Theorem 5.17 for affine spaces; this is also due to Kantor (1969d).

5.19 Theorem. Let D satisfy Assumption 5.7 with fL > 1 and assume the existence of a non-incident point-block pair (p, B) such that there exists precisely one block C with pEe and B n C = 0. Then D ~ AGIl - 1(n, q) for some n :0:: 3 and some q.

840

XII. Characterisations of classical designs

1= (k - IL)2, then the assertion is immediate from Theorem 5.18, asD obviously is not aPGn-l (n, q). Otherwise we have IL(B) = 1 by Lemma 5.11; then each D'(B) is a 2-design with A = 1 by Lemma 5.9. Using the fact that any two points of D are joined and Lemma 5.10(iii), it is now easily seen thatD itself is a 2-design (with joining number p). By Theorem 5.17 and our hypothesis, D is an affine 2-design; thus D' (B) is an affine plane by Theorem 5.5. But the planes of D are just the blocks and thus the conditions of Proposition 4.7 are satisfied (with t = 1; cf. Lemma 4.6), provided that m 2: 3. Hence in this case D ~ AG2(3, q) with q 2: 3. Finally, for m = 2 one obtains D ~ AG2(3, 2) as in the last part of the proof of Theorem 5.18 . Proof. If (IL - 1) (v - k)

•

The following result is an obvious consequence of Theorems 5.19 and 5.5; it is essentially due to Kimberley (1971).

5.20 Theorem. Let D be an affine 2-design AI' (s) with IL> 1. Then the following conditions are equivalent. (i) D ~ AGn-l(n, q)forsome n 2: 3. (ii) Each block ofD is good. (iii) D'(B) is an affine 2-designfor each block B ofD. • McDonough and Mavron (1995) considered quasisymmetric designs with good blocks and obtained the following two interesting results which should be compared with Theorems 5.3, 5.5 and 5.18. The proofs use techniques which are similar to those already presented.

5.21 Theorem. Let D be a quasisymmetric design 8A (2, k; v) with intersection numbers 0 and y which has a good block. Then D is either a symmetric or affine design or a linear space or else has one of the following two parameter sets:

(5.21.a)

v = y4(l -

i + 2y -

1),

k=

i(i - y + 1),

A=l + y+ 1; (5.21.b)

v = y(y4 - l + y2 - Y + 1),

k = y(y2 - Y + 1),

A=i+l. In the last two cases, D(B) is a multiple of an 8(2, k; y)for any good block B .

•

§6. Concluding remarks

841

5.22 Theorem. LetDbeaquasisymmetricdesignS),.(2, k; v) with intersection numbers 0 and y, where A > 1. Then all blocks of D are good if and only if either (5.22.a)

D is isomorphic to AGn - 1(n, q) or PGn-1 (n, q) with n ~ 3

or (5.22.b)

D has parameters as in (5.21.a) or (5.2l.b), all lines ofD have size y and D p is a projective plane of order y3 + y or y2, respectively.

There is a unique example for (5.22.b) in the case y = 2, namely the Witt design S(3, 6, 22) . •

§6. Concluding Remarks In this final section of Chapter XII we report on some further characterisation results. Though we have concentrated in the preceding sections on results characterising the classical affine and projective spaces, there are many similar results for other interesting types of incidence structures. A common characterisation of finite projective spaces and affine planes was given by Beutelspacher and Delandtsheer (1981), see also Delandtsheer (1983b). Delandtsheer (1983a) gave a common characterisation of affine spaces, projective spaces, boolean lattices and the Witt design S(3, 6; 22), extending and correcting the work of Kantor (1974b). A common characterisation of affine and projective spaces and Hadamard 3-designs is due to Beutelspacher (1982b). We now report on some results involving transitivity conditions on linear spaces. The following fundamental result of Kantor (1985b) characterises the Steiner systems S(2, k; v) which admit an automorphism group acting doubly transitively on points. Its proof uses the classification of doubly transitive groups which was discussed in §Y.3.

6.1 Theorem. Let D be a Steiner system S(2, k; v) which admits an automorphism group acting doubly transitively on points. Then D is one of the following structures: (1) AG1 (d, q) or PG 1 (d, q)for some d ~ 2. (2) The affine plane over the near-field of order 9, see, for instance, Andre (1955).

XII. Characterisations of classical designs

842

(3) The affine plane of order 27 constructed by Hering (1969). (4) A hermitian unital as described in Remark VIII.5.31. (5) A non-classical unital associated with a Ree group. (The Ree groups are a family of simple groups due to Ree (1961); a short description of these groups may be found in Huppert and Blackburn (1982b). For the corresponding unitals, the reader is referred to Laneburg (1966) and Tits (1960).) (6) One of two designs S(2, 9; 729) constructed by Hering (1986). • The following result of Key and Shult (1984) and Hall (1985) is partially contained in Theorem 6.1; it gives more precise information for the special case of Steiner triple systems.

6.2 Theorem. Let D be a Steiner triple system admitting a doubly transitive automorphism group G. Then one of the following occurs: (1) D (2) D (3) D

PGI (d, 2) and PSL(d + 1,2) ::: G. PGI (3,2) and G ~ A 7 . ~ AG 1 (d, 3) and G contains the translation group of AG(d, 3) . •

~ ~

We also mention two results due to Delandtsheer (1984b, 1986b), respectively, which use Theorem 6.1.

6.3 Theorem. Let D be a non-trivial linear space and assume that Aut D is transitive on pairs of intersecting lines as well as on pairs ofdisjoint lines. Then D is either a Desarguesian affine plane AG(2, q) or a Desarguesian projective space PGl (d, q) . • 6.4 Theorem. Let D be a non-trivial linear space containing a proper subspace U which contains at least two lines. If Aut D is transitive on pairs of intersecting lines, then D is a Desarguesian affine or projective space. • For further similar results, we refer the reader to De1andtsheer (1984a, 1984b, 1986b, 1992a). The 2-homogeneous (but not 2-transitive) linear spaces have also been classified; this is due to Delandtsheer, Doyen et al. (1986).

6.5 Theorem. Let D be an S(2, k; v) admitting a group G which acts 2homogeneously but not 2-transitively on the points of D. Then v = pn for some prime p == 3 (mod 4), where n is odd. Moreover, one has one of the following two cases:

§6. Concluding remarks

843

(1) D is an affine space over a subfield of GF(pn); (2) D is a Netto triple system, k = 3 and p == 7 (mod 12). •

6.6 Remark. The Netto triple systems are, for instance, described on page 98 of Dembowski (1968) who seems to have coined this term (even though they are different from the examples due to Netto (1893), as noted by Robinson (1975». They are the derived triple systems of the quadruple systems of Liineburg (1965b) which were exhibited in Example Ill.6.17; see also Liineburg (1963). Theorems 6.1 and 6.5 completely classify the linear spaces admitting a 2homogeneous group. Thus one should now study weaker transitivity properties, of which the most natural are flag-transitivity and line-primitivity. Some general results on designs admitting a group of this type were already discussed in §Ill.4. Major progress was obtained by Buekenhout, Delandtsheer and Doyen (198586, 1988); one of their results is the following fundamental theorem showing that the flag-transitive linear spaces split into two large classes.

6.7 Theorem. Let G be ajlag-transitive automorphism group of an S(2, k; v) D, where 2 < k < v. Then one of the following two cases occurs: (6.7.a)

("Affine case") G has an elementary abelian minimal normal subgroup T acting regularly on the point set of D.

(6.7.b)

("Almost simple case") G has a non-abelian simple normal subgroup N such that N ::: G ::: Aut N . •

Based on this theorem, the classification of flag-transitive linear spaces is now reasonably complete, see Buekenhout, Delandtsheeretal. (1990). These authors have completely settled the almost simple case, with partial results being due to Delandtsheer (1986a) and Kleidman (1990).

6.8 Theorem. Let D be an S(2, k; v) admitting ajlag-transitive automorphism group G. In the almost simple case (6. 7.b), one of the following possibilities occurs (up to isomorphism): D = PG(d, q) with d ?: 2, and N = PSL(d + 1, q). D = PG(3, 2) and N = A7. D is a hermitian unital S(2, q + 1; q3 + 1) with e ?: 1, and N = PSU(3, q). D is a Ree unital S(2, q + 1; q3 + 1) with q = 32e +1for some e ?: 1, and N is the corresponding Ree group. (5) D = W(2 e ), cf Remark 6.9, with e ?: 3, and N = PSL(2, 2e ) • • (/) (2) (3) (4)

844

XII. Characterisations of classical designs

6.9 Remarks. (a) Regarding the examples in case (5) of Theorem 6.8, one way of describing them is as follows. Define W (2 e ) to be the Steiner system S(2, 2e - 1 ; 2e - 1 (2 e - 1» the points and blocks of which are the lines of P := PG(2, 2e ) disjoint from a regular hyperoval H (that is, a conic together with its nucleus) and the points of P outside of H, respectively; cf. Exercise VII1.5.17 and Remark VIII.5.34.(d). W(2e ) admits a group isomorphic to prL(2, 2e ) as its full automorphism group which is the stabiliser of H in Aut PG(2, 2e ); the observation that W(2 e ) is flag-transitive seems to be due to Kantor (1975a). This family of examples goes back to ideas of Bose, Shrikhande and Witt; we refer the reader to Buekenhout, Delandtsheer and Doyen (1988) for an extensive discussion of and references for the possible approaches to W(2 e ). Recently, the 2-rank of these designs was shown to be 3e - 2 e by Carpenter (1996), using results of Blokhuis and Moorhouse (1995). More generally, as noted in Exercise VIII.5.17, any hyperoval in a projective plane of even order q can be used to construct both an S(2, !f; (D), see Bose and Shrikhande (1973), and a Hadamard 2-design on q2 - I points, see Maschietti (1992). (b) The affine case of Theorem 6.7 presents considerably more difficulties, among them the classification of flag-transitive affine planes and the possible (if unlikely) existence of non-desarguesian sharply flag-transitive projective planes, cf. Theorem VI. 7 .24. Kantor and Williams (1996) produced huge numbers of flag-transitive planes of even order; in view of these results, a complete classification of such planes is impossible. (c) For some results on line-primitive linear spaces, see Delandtsheer (1988, 1989). Camina and Siemons (1989) studied block-transitive S(2, k; v)'s and characterised these in the case where k = 4 and the given block-transitive group G is solvable. Huiling (1995) proved that for k = 4 in the non-solvable case G necessarily is flag-transitive. Combining this with the work of Buekenhout, Delandtsheer et al. (1990) gives a classification of block-transitive S (2, 4, v),s. (d) For more on flag-transitive linear spaces, see also Zieschang (1992, 1993). The reader might also want to have a look at the survey by Doyen (1989) on designs and their groups and at the relevant parts of the survey by Delandtsheer (1995). In connection with the affine case of Theorem 6.8, we mention the following beautiful result of Hiramine (1990).

6.10 Theorem. Anyfinite affine plane with a point-primitive collineation group is a translation plane. •

§6. Concluding remarks

845

In a special case, one can say even more. The following result was independently obtained by Gluck (1990), Hiramine (1989) and R6nyai and Szonyi (1989).

6.11 Theorem. Afinite affine plane ofprime order p with a transitive collineation group is isomorphic to AG(2, p). • The next few results concern linear spaces embedded into projective planes. The hermitian unitals as described in Remark VIII.5.31 can be characterised in several ways.

6.12 Theorem. Let D be a unital S(2, m + 1; m 3 + 1) embedded in a finite projective plane P of order m 2, and assume that D is invariant under a group G of collineations of P. Then each of the following conditions implies that P is desarguesian and that D is hermitian: (6.12.a)

G is flag-transitive on D.

(6.12.b)

G is line-transitive on D.

(6.12.c)

G is point-primitive on D.

Moreover, PSU(3, m 2) :::: G .:::: prU(3, m 2 ) provided that m :::: 4. • Here the characterisation via (6. 12.a) is due to Kantor (1971 b); then (6.12. b) follows, since the line-transitivity of G implies that G is actually flag-transitive by Lemma III.4.21. Finally, (6.12.c) is due to Biliotti and Korchmaros (1989a, b), who actually study the more general case where G is point-transitive. As the occurrence of the Ree unitals in Theorem 6.1 shows, the condition thatD is embedded in some plane is essential. The following interesting characterisations of classical unitals which are assumed to be embedded in PG(2, q2) were obtained by Lerevre-Percsy (1982a), Faina and Korchmaros (1983) and Blokhuis, Brouwer and Wilbrink (1991).

6.13 Theorem. LetD bea unital S(2, q+ 1; q3+ 1) embeddedintoPG(2, q2). Then the following conditions are equivalent: (6.13.a)

D is classical.

(6.l3.b)

Each line ofD is a line of a Baer subplane of PG(2, q2).

(6.13.c)

D is contained in the GF(p)-code spanned by the lines of PG(2, q2), where q is a power of the prime p . •

846

XII. Characterisations of classical designs

Further characterisations of the classical unitals are due to Tallini-Scafati (1966), Kantor (1971b), O'Nan (1 972b), Wilbrink (1983), Hirschfeld, Storme and Thas (1991), Thas (1992) and Barwick (1994a). De1andtsheer and Doyen (1990) gave the following classification of the maximal arcs (which were discussed in §VIII.5) admitting a line transitive group; a similar question (regarding "semi-ovals") was dealt with by Delandtsheer (1991 b). 6.14 Theorem. Let P be a finite projective plane of order q, and let S be a maximal (k, n)-arc in P. If G is a collineation group of P fixing S which acts line-transitively on the linear space S (2, k; kq + k - q) D which is induced on S, then one of the following cases occurs: (1) q is a prime power, k = q, D is a line-transitive affine translation plane, and G contains the translation group ofD. (2) q = 2e for some e :::: 3, k = q /2, P = PG(2, 2e ), D = W(2 e ), and G contains PSL(2, 2e ), cf Remark 6.9. (3) q = 4, k = 2, S is a hyperoval in P = PG(2, 4), and G = A6 or S6.

Moreover, in each afthese cases, except for the sporadic case (3), G stabilises a line P disjoint from s. • 6.15 Remark. The Steiner systems W (2 e ) were already discussed in some detail in Remark 6.9.(a). It can be seen that W(2 e ) forms a maximal arc in the dual projective plane P* which is, of course, isomorphic to P. We also note that the maximal arcs of Denniston (1969) which we have constructed in Theorem VII1.5.20 were characterised by Abatangelo and Larato (1989) as those maximal arcs in PG(2, q), q even, which are invariant under a cyclic subgroup of order q + 1 of PGL(3, q). Linear spaces in which all planes are affine or projective were discussed in Remark 3.7. A more general result was obtained by Teirlinck (1977). We require a definition: An affino-projective plane is a linear space obtained by deleting a set of collinear points from a projective plane; one has the following result. 6.16 Theorem. Let D be a finite linear space whose planes are all affinoprojective and assume that at least one plane of D has order ::::4. Then D is isomO/phic to a projective geometlY from which part of a hyperplane has been deleted. •

§6. Concluding remarks

847

There is also some work on linear spaces having all their planes isomorphic to a given linear space rr; these are often called rr-spaces. Some papers the reader might like to consult are Buekenhout and Deherder (1971), Delandtsheer (1982, 1983c) and Leonard (1982). Laskar (1974) introduced certain threedim~nsional analogues of the (s, r; I)-nets which she called "3-nets". These are partial linear spaces whose planes are nets with {), = 1. Let us give the precise definition.

6.17 Definition. Let V, L, and P be sets, whose elements are called points, lines and planes, respectively, and let (V, L, II), (L, P, Iz) and (V, P, h) be incidence structures. Then the 6-tuple D := (V, L, P, h, h h) is called a 3net with parameters r, s, and b (where r, s, b are integers :::2) provided that the following axioms are satisfied.

(6.17.a)

phL and LIzP imply phP.

(6.17.b) (6.17.c)

If two lines intersect, then they are contained in a common plane. The points and lines of any plane form an (s, r; 1)-net.

(6. 17.d)

The planes are partitioned into b classes such that each class partitions V and such that any two planes from distinct classes intersect in the points of a line.

(6.17.e)

Each line is on at least one plane.

6.18 Exercise. Let P be a plane of a three-dimensional finite projective space P, put A := P \ P, and let Q be a subplane of P. Choose as points, lines and planes of D respectively the points of A, the lines of A meeting P in a point of Q, and the planes of A meeting P in a line of Q, with incidence induced from P. Show that this yields a 3-net. The following result of Sprague (1979) shows that Exercise 6.18 essentially provides all possible 3-nets.

6.19 Theorem. Every finite 3-net with r > 3 is isomorphic to a 3-net constructed as in Exercise 6.18. • 6.20 Remarks. (a) For generalisations of these results to dimensions d ::: 3, that is, for d-nets, we refer the reader to Laskar and Dunbar (1978) and Sprague (1981a); see also Sprague (1978, I981b) and Ray-Chaudhuri and Sprague (1979).

XlI. Characterisations of classical designs

848

(b) A rather general embedding theorem for planar spaces is due to Metsch (1989), and an interesting characterisation of the three-dimensional projective spaces among all planar spaces was given by Durante and Metsch (1996). (c) Extensive surveys on the classification of highly transitive "dimensional" linear spaces and on dimensional linear spaces in general are given by Delandtsheer (1994, 1995). Regarding extensions of AG(2, q), Thas (1990a, 1990b, 1994) settled a longstanding conjecture by proving the following result.

6.21 Theorem. The Mobius plane S(3, q + 1; q2 + 1) constructed in Theorem III.6.9 is the only extension of AG(2, q), provided that q is odd. • We now collect some characterisation results concerning designs with)" > 1. The 2-transitive symmetric designs were classified by Kantor (1985a).

6.22 Theorem. Let D be a 2-transitive symmetric design SJ.(2, k; v), where v > 2k. Then D is one of the following structures: (1) A classical design PGd-l (d, q). (2) The unique S2(2, 5; 11); (3) A unique symmetric design Sl4(2,50; 176) admitting the Higman-Sims

group, see Higman (1969) and Calderbank and Wales (1982). (4) A unique design with v = 22m, k = 2m - l (2 m -1) and).. = 2m - l (2m - l _l),

where m 0:: 2, see Kantor (1975b). •

6.23 Remarks. (a) In this connection we mention Lander (1988a) who among other results - classifies all biplanes (i.e., symmetric designs with ).. = 2) admitting an automorphism group G fixing some block B and acting 2-homogeneouslyon B. Biplanes admitting an automorphism group G isomorphic to PSL(2, q) such that G fixes some block B and acts on B in its usual permutation representation were considered by Kelly (1981) and Fanning (1991). (b) A symmetric design is said to have the symmetric difference property (and is, for short, called a symmetric SDP-design) if the symmetric difference of any three blocks is either a block or the complement of a block. Kantor (1975b) proved that every symmetric SDP-design has parameters of the form (6.23.a)

v

= 22m , m

).. = 2

-

k = 2m - l (21)> - 1) and l (2m - l - 1), where m 0:: 2.

§6. Concluding remarks

849

The designs occurring in case (4) of Theorem 6.22 are in fact SDP-designs. Kantor (1984) proved that the number of isomorphism classes of symmetric SDP-designs with parameters (6.23.a) grows exponentially. Both symmetric and quasisymmetric SDP-designs (which may be defined in a similar way) have found considerable interest, since they are connected to difference sets, "bent functions" and interesting classes of codes. We refer the reader to Rothaus (1976), Dillon and Schatz (1987), Wolfmann (1977), Assmus and Key (1992b), Iungnicke1 and Tonchev (1991a, 1992), Tonchev (1993), Parker, Spence and Tonchev (1994), McGuire (1997) and McGuire and Ward (1998). A summary of results on SDP-designs can be found in §V.1.9 of Colbourn and Dinitz (1996a). (c) The following analogue of Theorem 6.22 for affine designs is due to Pfaff (1991). His result shows that the property of admitting a 2-transitive group more or less characterises the classical affine designs, up to five sporadic exceptions; the situation in the symmetric case discussed in Theorem 6.22 is somewhat different, since there is a second infinite series in this case. Some interesting remarks on 2-transitive and flag-transitive designs may be found in Kantor (1993).

6.24 Theorem. Let D be an affine design admitting an automorphism group which acts 2-transitively on points. Then D is one of the following designs: (1) A classical designAGd_l(d, q).

(2) (3) (4) (5)

The unique Hadamard 3-design S2(3, 6; 12), see Theorem IV.7.12. The affine plane over the near-field of order 9. The affine plane of order 27 constructed by Hering (1969). One of two non-isomorphic affine designs with s = J1- = 3 which are not isomorphic to AG2 (3, 9), see Pfaff (1993a). •

Another class of geometries that should be mentioned here is the class of Hadamard designs. Kantor (1 969b ) characterised the Paley-Hadamard designs belonging to the Paley difference sets, see Theorem VI. 1. 12. His result is as follows.

6.25 Theorem. The Hadamard designs admitting automorphism groups which are transitive on incident point-block pairs but which are not 2-transitive on points are precisely the Paley-Hadamard designs. • We also mention a related result due to Norman (1968).

850

XII. Characterisations of classical designs

6.26 Theorem. LetD be a Hadamard design SA(3, 2A+2; 4),,+4), where).. is even.lfD admits a 3-transitivecollineationgroup G, then).. = 2 and G ;;: M ll .

•

6.27 Remark. Automorphism groups of Hadamard matrices were considered by Kantor (1969c). Norman (1973) considered automorphism-free Hadamard designs; and non-isomorphic Hadamard designs were studied by Norman (1976). Automorphism groups of Hadamard 3-designs were also studied by Kimberley (1973). The 2-transitive Hadamard matrices were classified by Moorhouse (1995). Next let us consider symmetric (s, f.L )-nets. The classical examples constructed in Proposition 1.7.18 may be characterised as follows, with appropriate modifications in the definitions of lines, triangles, planes, and smoothness; as there are points which are not joined, the definitions of §2 cannot be used exactly as stated there. The following result is due to Mavron (1981a) and Jungnickel (1981c).

6.28 Theorem. Let D be a symmetric (s, f.L)-net with s > 2 and f.L > 1. Then the following conditions are equivalent: (6.28.a)

D is isomorphic to a classical example as constructed in 1.7.18.

(6.28.b)

AutD is transitive on ordered triangles.

(6.28.c)

Aut D is transitive on point-line pairs (p, L), such that p is joined to every point of L.

(6.28.d)

D is smooth.

(6.28.e)

Every line of D has precisely s points. •

6.29 Remark. Jungnickel (1981 c) also exhibited symmetric (pi, pj )-nets with a group acting transitively on ordered pairs of joined points for each prime p and all positive integers i and j. The classical symmetric nets may be obtained from projective spaces by deleting all points on a hyperplane H and all hyperplanes through a point p for a flag (p, H). A corresponding characterisation of the divisible designs obtained from projective spaces by the same kind of deletion for a non-incident point-hyperplane pair (p, H) was given by Jungnickel and Vedder (l984b). 6.30 Remark. We finally mention two topics which are related to the material discussed up to now, even though they do not really belong to design theory (but rather to finite geometry). We begin with the polar spaces introduced by

§6. Concluding remarks

851

Veldkamp (1959, 1960). Two classical families of examples are obtained as follows. Let V be a vector space over a commutative field, f a non-degenerate quadratic form on V, and consider the point set S (f) of the quadric determined by f; then S(f) and its totally singular subspaces form a polar space. Let rr be a polarity of a projective geometry over the field K and assume that rr is not orthogonal if K has characteristic 2; then the point set S(rr) of all absolute points of rr together with all totally isotropic subspaces forms a polar space. Three important references on polar spaces are Tits (1974), Buekenhout and Shult (1974) and Cameron (1982). A nice treatment of projective and polar spaces is due to Cameron (1993). Polar geometry is only a small part of the broad area of the geometry of buildings and diagrams; this very active and important field which has close connections to the theory of finite simple groups has been initiated by the fundamental book of Tits (1974). Another fundamental reference is Buekenhout (1979a). The reader is also referred to Tits. (1981a, 1981b) and Buekenhout (l979b, 1981, 1982, 1983). A systematic exposition of diagram geometries was given by Buekenhout and Buset (1988). We also mention the monograph by Pasini (1994) and the following recent surVeys for diagram geometry, buildings, polar spaces, and related topics: Buekenhout (1995), Buekenhout and Pasini (1995), Scharlau (1995), Cohen (1995) and Rohlfs and Springer (1995).

XIII Applications of Designs

10

0 5

2 7 6

3 8 4

Du muSt verstehn! Aus Eins mach' Zehn, Und Zwei laB gehn, Und Drei mach' gleich, So bist du reich. Verlier' die Vier! Aus Fiinf und Sechs, So sagt die Hex', Mach' Sieben und Acht, So ist's vollbracht: Und Neun ist Bins, Und Zehn ist keins. Das ist das Hexen-Einmal-Eins! (Johann Wolfgang von Goethe)!

§1. Introduction From a historical point of view the title "Applications of Designs" is misleading. The development of the theory of combinatorial designs happens to be one of the few areas of mathematics which was initiated by questions arising in applications. When Leonard Euler was employed by the Prussian King Friedrich II, called "Der GroBe", he carried out a lot of applied science, ranging from the computation of layouts for draining channels for swamps to compiling handbooks for the management of sailing ships. Amongst these, the famous Euler problem, concerning the arrangement of officers of different ranks and troops in the form of a pair of orthogonal Latin squares of order six, was rather academic, but can still be considered as a problem in applied mathematics, which may have triggered most of the work described in this book. Even though magic squares have been used and discussed much earlier in history, this seems to be the first appearance of a type of problem

1

This solution of the "Hexen-Einmaleins" can be found in the Faust museum at Knittlingen, Gennany. For this most recent version of this quote and possible interpretations see Kracke (1992).

852

§1. Introduction

853

which developed into a systematic theory of designs of statistical experiments about two centuries later.

1.1. Design Theory Influenced by Applications While 19th century problems, like Kirkman's schoolgirl problem were an outgrowth of the so-called "Unterhaltungsmathematik" (recreational mathematics), regular combinatorial structures became important concerns in the geometric-group-theoretic schools of research towards the tum of the century, cf. Aigner (1990), page 97. Beginning with the 1930s, the area which is nowadays called design theory, underwent rapid development due to three influences: • the influence of synthetic finite geometry; • the beginning development of universal algebra in terms of which Latin squares are an example of algebraic systems with binary operations of some interest; • last, but not least, actual demands from an area of applications, namely the design of statistical experiments. The investigation of the connections of finite geometry and design theory with group theoretic means was initiated by Baer (1939, 1940, 1947) and his school, eventually providing as an inner mathematical application one of the tools of the classification of sporadic simple groups, cf. Gorenstein (1982). Methods and results from representation theory and algebraic number theory derived since Turyn (1965) and later Ott (1975, 1981), see also Pott (1995), have created the powerful link between the geometric algebra of designs and coding theory, providing one of the most successful applications of mathematics of this century. Fisher's and Yates' early statistical investigations established a systematic connection between this area of combinatorics with the then rapidly developing theory of statistical planning and inference. The successful application of the combinatorial methods to statistical experiments arising in such areas as agriculture, biology, medicine, industrial engineering, etc., created a considerable impact on the development of combinatorial theory during the last 50 years. This was especially influenced by the marriage of set-theoretic combinatorics with algebra. The applications of designs described in the subsequent sections use two important properties of designs: • balancedness and • orthogonality.

XlII. Applications of designs

854

The former refers to a purely combinatorial aspect described in the category of finite sets' of incidence structures and occurs in applications like packing, covering, partitioning and testing. The latter is concerned with the representation theoretic aspects of the modules generated by the incidence matrices. It is used mainly in coding theory, but also in instrumentation techniques such as measuring, weighing, correlations and projections. The success of design theory in many of the classical areas is well documented in Chapter V of the comprehensive handbook by Colbourn and Dinitz (1996a).

In the following ten sections we shall present many of the known, cf. Colbourn and van Oorschot (1989), and several new applications of designs and geometries. The applications range from communications engineering to physics, from computer engineering to cryptography and from statistical experiments to games. Thus we conclude this introduction by an example from gambling, in view of the 2000 year old motto: panem et circenses (Juvenal, Satire X.81).

1.2. Designs in Games and Gambling It has been known for a long time that designs are natural settings for the arrangements for matching and matches, Rev. T. P. Kirkman's schoolgirl problem (l850b) being one of the earliest examples of this type. Resolvability of designs has been a celebrated field for applications of designs. It has been used for arranging tournaments, for ball games (such as soccer, golf, etc.), or for card games (whist, bridge, etc.). The reader is referred to parts V.7.8 of the CRC Handbook, Colbourn and Dinitz (l996a), where an extensive listing of such applications is given. Even for gamblers, designs play a significant rOle in planning an "optimal" game strategy, cf. Colbourn (1996b). We give an example, cf. Oberschelp (1972), of this kind of application in the German national Lotto Block, where k = 6 randomly chosen numbers have to be predicted out of N =49. One way to predict the winning numbers is to choose all of the = (;i) = 13983816 possibilities or blocks. But in this strategy the total stake will be much higher than the winning prize.

G)

The Lotto company offers several predefined systematic betting patterns, which are solutions of the following problem. Choose a set of v different numbers (v 2: k) out of the numbers I, ... , N. Then the systematic patterns consist of b different k-subsets of the v chosen numbers. The b subsets have to satisfy the additional condition, that if one correctly predicts t :5 k numbers in the v-subset of

Table 1.1. YEW 22 of the German national Lotto Block fJ16

f3l7

x

x

x

X

X

fh fh fh f34 f3s f36 f37 f3s f39 f3lo f3ll f312 f313 f314 f3IS al a2 a3 a4 as a6 a7 as a9 alO all al2

x x x x x x

x x x

x x

x x

x x

x

x

x

x

x

x x x

x x x x

x x x x

x x

x x

x

x x x

x x

x x x

x

x x

x x x

x

x x

x

x

x x x x

x x x x x

x x

x x

X

X

x

x

x

x

x x X

X

X

x

x

x x

x x

x x

x x x

x x x x x

f3IS

f319

f320

x

x

x

X

X

x x x

x x x

x x

x

x

x

f322

x x x

x x

x

x

f321

x x x

x x x x x x

XIII. Applications of designs

856

chosen numbers, then At of the b subsets have to contain these t numbers. Thus one just needs at-design S,- (t, k; v).

It should be mentioned that in the German national Lotto Block a prize is given only in the cases 3 ::5 t ::5 6. One of the most popular patterns, the system YEW 22 ("verktirzte engere Wahl") is shown in Table 1. 1. Using this system a player chooses v = 12 numbers a1, ... , a12. The system YEW 22 defines b=22 combinations ({3i)i=l...22 of k = 6 of the chosen numbers aI, ... , a12. This means that if three of the six winning numbers are among the twelve numbers of the gambler's choice, then he hits them exactly twice if he tips all the 22 "blocks" of the YEW 22, each of them exactly once. Analogously for the following other predefined systematic patterns. These patterns facilitate tipping but do not improve the gambler's winning chance, such that in the case of three correct predicted numbers, two of the b subsets contain the correct predicted numbers. A list of predefined systematic patterns with corresponding parameters is listed in Table 1.2. The pattern YEW 132 has been generated using the Mathieu group Mh2, cf. ChapterIV. YEW 66 is a residual design (cf. II.S.13) of YEW 132, obtained by ignoring one element ai and all the blocks which contain ai. YEW 77 is based on the Mathieu group M zz , cf. Chapter IV. It should mentioned that the design corresponding to the pattern YEW 30 was found by the "Niedersachsische Lotto-Gesellschaft" and was not known in the mathematical literature previously; cf. Oberschelp (1972). Table 1.2. List of YEW systems of the German national Lotto Block, cj. Oberschelp (1972) Name

v

VEW22 VEW30 VEW66 VEW77 YEW 130 VEW 132

12 10 11 22 26 12

3 3

4 3 3 5

b=AO

Al

A2

A3

22 30 66 77 130 132

II 18 36 21 30 66

5 10 18 5 6 30

2 5

A4

8

3

1 1 12

4

§2. Design of Experiments In this section, we shall see that incidence structures provide a suitable method of reducing the number of statistical tests in experiments. This is based

§2. Design of experiments

857

on the underlying model of linear regression analysis. We illustrate this by a simple heuristic from agriculture: suppose there are v different artificial fertilisers (in general called treatments) whose effectiveness on b types of plants has to be examined. A complete experiment would require b plots each subdivided into v areas for the different applications of fertilisers. Even for relatively small values for v and b, the cost of the required area can exceed the available resources. For this reason, an incomplete experiment can be designed in which every type of plant is tested with k < v different fertilisers such that any two fertilisers are tested on ).. different types of plants. The reasoning behind balancing the occurrences of pairs of treatments on exactly).. of the b blocks of size k is that, heuristically speaking, the regular appearance of pairs of fertiliser treatments on the same plant provides a sufficiently strong coupling between the effects so that a complete correlation analysis of variance can be carried out. Note that this experimental design corresponds to a SA(2, k; v)-design - a balanced incomplete block design (BIBD). By using BIBD's, e.g. in agricultural tests, the underlying complexity measurearea (of plots) and time (for growth and harvest) - can be reduced significantly compared to a complete experiment. 2 Such incidence structures lead to efficient statistical tests, because the parameters to be determined can be estimated by variance analysis using an incomplete set of lists without a deterioration in quality. The undesired side-effects of experimental factors inherent in physical observations can be averaged out by taking the mean values of the occurring correlations of the first, second or third order, depending on the number of factors applied in the actual experiment. The mathematical setting for this model is given by the theory of linear models, and especially by the multiway classification models. In Section 2.2 ff. we give a formal derivation relying on the very detailed and informative book by Pukelsheim (1993).

2.1. Linear Models A statistical linear model is represented by an affine relationship, cf. Pfanzagl (1966),

(2. 1. a)

2

The above complexity measure was used long before a similar measure for the evaluation of VLSI circuitry in computer science had become fashionable.

XIII. Applications of designs

858

in which x T is a so-called regression vector, e E lI~k is the mean parameter vector to be estimated and the observed value Y E jR of an experiment is disturbed by an error e subject to a probability distribution p. With the usual moment assumptions on the mean parameter

and variance

a sequence Yi of observations (2.l.b)

xi

n

with different regression vectors in the i -th run gives a system of equations to determine the k + 1 parameter system consisting of the vector e and the variance parameter (J'2. In more compact notation we therefore write linear models as (2.1.c)

Y= X

·e+e,

with the response vector Y E jRn, the model matrix X e E jRn.

E jRnxk

and the error vector

In a general linear model with the usual assumptions the random vector Y is distributed according to (2.l.d)

xe

around the multidimensional mean vector and its dispersion matrix cr 2 y with the (n x k)-mode1 matrix X and the non-negative definite (n x n)-matrix y of known type. In general, if s linear forms of e given by Kt e with a (k x s )-coefficient matrix K are to be estimated from the linear model (2.l.c), a linear estimator LY for Kte is needed. Here L denotes an (s x n)-matrix fulfilling the unbiasedness condition (2. I.e)

LEU:= {U

E jRsxn

I UX = K}.

An optimal linear estimator [y for Kt e is obtained by minimizing the corresponding dispersion matrices cr 2 L YL t over the set U.

§2. Design of experiments

859

It is one of the more beautiful insights provided by mathematics that this analytical minimisation task can be characterised geometrically and combinatorially by the technique of LOwner ordering and can be solved algebraically by applying the Gauss-Markov Theorem.

2.2. Optimal Estimators As the dispersion matrices in question are non-negative definite matrices, we recall some facts about the geometry of these sets. For mEN the set NND(m) C Sym(m) of non-negative definite symmetric (m x m)-matrices is a pointed closed convex cone whose interior is the set PD(m) of positive definite matrices, with respect to the set Sym(m) of symmetric (m x m) matrices. Sym(m) is endowed with a partial order relation, the so-called L6wner ordering 2: by the definition (2.2.a)

A ::': B

{=:}

A- B

(2.2.b)

A > B

{=:}

A - B E PD(m).

E

NND(m)

The geometric properties of NND(m) are respected by the Lowner ordering through the necessary antisymmetry, reflexivity and transitivity to form a partial order. In addition we have (2.2.c)

VA. E jR+, A 2: 0:

(2.2.d)

VA, B 2: 0:

A+B::,:O

(2.2.e)

lim Al 2: 0:

for all sequences (Al)/ in NND(m).

/->0

A.A 2: 0

With this notation we are now prepared to formulate the Gauss-Markov Theorem, cf. Pukelsheim (1993), p. 20, which is fundamental to what follows. (2.2.f)

Gauss-Markov Theorem.

Let X be an (n x k)-matrix and V be a non-negative definite (n x n)-matrix. Suppose U is an (n x s )-matrix. A solution L of the equation LX = U t X attains the minimum of LV L t, relative to the £Owner ordering and over all solutions L of the equation LX = U t X,

LvLt=

min

LelR"":LX=U t X

if and only if LVRt =0,

LVLt,

860

XIII. Applications of designs

where R is the projector given by R = In - X G for some generalised inverse G of X, i.e. G E x-:= {G E jRkxll I XGX = X}. A minimising solution L exists; a particular choice is UtUn - V Rt H R), with any generalised inverse H of RV Rt. The minimum admits the representation

min

LVLt = Ut(V - VRt(RVRt)-RV)U

LelR"":LX=ul X

and does not depend on the choice of generalised inverses involved. •

With the notion of an optimal estimator being a minimum variance unbiased linear estimator in a classical linear model, cf. Equation (2.1.c), the GaussMarkov Theorem therefore implies, cf. Pukelshcim (1993), 1.23, that • the optimal estimator for the mean vector a 2 p and

xe is PY with dispersion matrix

(2.2.g) i.e. the orthogonal projector onto the range of X; • the optimal estimator for e is (2.2.h) with dispersion matrix a 2(Xt X)-l. While in the first case the optimal estimator P Y only depends on the range of the model matrix, i.e. the space (Xi liE [1, '" , n]) spanned by the regression vectors, in the latter case when estimating the parameter directly the choice of the regression vectors Xi influences the optimal estimator and its dispersion matrix.

e

The optimal estimator for e as given in (2.2.g) coincides with the least square estimator which minimises the sum of squares of deviations (2.2.i) With respect to each el this is defined by a solution (2.2.j)

VI E [1, ... , k]:

as ael

= 0

eof

§2. Design of experiments

861

given by (2.2.k)

2.3. Greenhouse Agriculture The following practical example, adapted from Chapter 7 of Mead (1988), illustrates the different concepts of optimal parameter estimates by least squares models allowing a trade-off of the number of tests against a controlled increase of the variance. To optimise the yield in greenhouse agriculture, an experiment is to be designed to investigate the effects of heating, lighting and carbon dioxide on the growth of green paprika peppers in a glasshouse. For technical reasons the number of possible treatments is six. Each observation on a treatment combination requires a greenhouse compartment. There are 10 greenhouse compartments available in two sets of five compartments. As compartments in different sets are likely to produce rather different yields, the problem is to design an experiment in two blocks, each of five units, to compare the six treatments. The primary idea behind blocking is that identifying blocks of homogeneous units allows a more precise comparison of treatments, by eliminating the large differences between units in different blocks from the comparison of treatments. It follows that information from blocked experiments is predominantly based on the comparisons that can be made between treatment observations in the same block. If two treatments do not occur in the same block, it is still possible to make a valid comparison between the two treatments if each treatment occurs in a block with a common third treatment. Essentially, a comparison between two treatments p, q E [1, ... ,6], within a block, has variance 20'2, where 0'2 is the variance of units within a block. If each of p and q occur in one block with treatment r then the p ___ q comparisons can be calculated via the intermediary by (p ___ r) and (r ___ q) comparisons, each of the two component comparisons being made within a block, and the variance of the implicit p ___ q comparison is 2(20'2). If the indirect comparison of p and q can be made through several different intermediaries, then the precision of the p ___ q comparison is improved. With two blocks of five units each available to compare six treatments, then, provided each treatment appears at least once, and no treatment appears twice in a block,

XIII. Applications of designs

862

a simple arrangement for this is shown in Figure 2.1. We now can set up the linear model, cf. Equation (2.1.c), for this problem.

Block

~ I:

r

s

u

r

s

u

Figure 2.1. Experimental plan for comparing six treatments in two blocks of five units (taken from Mead 1988). The yields Yij from a compartment in block i E {I, il} under treatment j E [1, ... , 6] are reasonably assumed to behave like Yij = i-L + b i + tj + eij with an expected mean i-L and deviations depending on the block bi or the treatment tj only with an error eij distributed with E(eij) = 0, Var(eij) = (12 and normalised constraints bI + bil = 0 and LjE[I, ... ,6] tj = O. The (10 x 9)model matrix X representing the experimental plan, cf. Figure 2.1, with respect to the vector

(2.3.a)

B=

E JJl9,

is given by 0 1

(2.3.b)

X=

0

0

0

0

0 0

0 0 0

1 0 0 0

0 0 0

0

1 0

0 0 0

0

0

0

1 0

0 0 0 0 0 0 0 1 0 0 0 0

0

0 0

1

0 0

0

0

0

0

0 0

0

0

1 0 0

0

0

1 0 0 0 0 0

0

1 0

863

§2. Design of experiments From this we compute 5

5 1

1

2

2

2

2

5

5

0 1 0

1

1

1

1

5 I

0 5 0 1 1 I 1 1 I 0 1 0 0 0 0 0

1

0

1 0

1 0 0 0 0

2

1

1 0

0 2

2

1

1 0 0 0

2

0 0

2

1

1 0

0

0

2

2

1

1 0

0 0 0 0 2

10

(2.3.c)

xtx=

0

0 0 0 0

giving the least square equations (2.2.i). These can be solved by Gaussian elimination giving e.g. the following estimates of the differences: (2.3.d)

A

A

tp - tq

=

1

4(4YIl -

YI3 - YI4 -

YIS -

YI6 - 4YII2

. + YII3 + Yn4 + Yus + YII6) (2.3.e)

A

A

tp - tr

1

= g(8 Yll

-

5YI3 - YI4 - YI5

- 3YII3 + Yn4 +

YIIS

YI6

+ Y1I6)

with (52

'iq) = 16 (2· 16 + 8· 1) =

(2.3.f)

Var(tp -

(2.3.g)

Var(tp - tr )

A

A

(52

5(52

2' 13(52

= 64 (64+25 +9 +6 ·1) = -8-'

We have given the details of this example from Mead (1988) here, as it shows a diagrammatic method of obtaining the linear combination of yields for complex least squares estimates, illustrated in Figure 2.2. The method is essentially just writing each total in (2.3.f) as the sum of the individual yields. The visualisation, however, provides a structure, reducing the scope for mistakes. The separate figures in each "unit" derive from the totals in (2.3.f); the circled figures represent the total coefficient for each unit in (2.3.g). 3

3

The hatched squares do not indicate broken windows of the greenhouse, but rather refer to the treatment not applied.

XIII. Applications of designs

864

Treatment 4 3

5

-1

-1

(]

(]

(]

@

6

-1

-1

Block +1

@

@

Figure 2.2. Diagrammatic representation of estimate of t p -

+1

+1

+1

@ tr

@

(taken from Mead 1988).

2.4. Linear Regression and Experimental Designs Following from this rudimentary example, we observe that in general the matrix X in fact fonns a list (table) of n regression vectors Xi. which we shall henceforth call an experimental design, even though it is in general not a t -design. The precision matrix of this design (2A.a) being the inverse of the dispersion matrix, therefore has to be maximised to give From Section 2.2 it is clear an optimal estimator for the parameter vector that for a given set S of regression vectors (which due to physical requirements can only be realised), the so-called regression range, each experimental design X is a sequence over S. For the computation of X only the multiplicity n(x) of the vector x in X is important, since

e.

xt

(2A.b)

~XiXr = ~n(x)xxT. XES

With the sample size n defined by (2A.c)

n = ~n(x) XES

of X we obtain the nonnalised moment matrix (2A.d)

M(X)

"n(x) T = 'L.t xx XES

n

of the design X. Thus the standardised precision grows directly proportionally to the sample size and decreases inversely proportionally to the model

865

§2. Design of experiments

variance 0- 2 . Even though the numbers ~ are rationals, by abuse of notation we define the convex hull (2A.e)

x=

conv(S)

as the set of all possible designs supported by S. Maximisation of M(X) is possible due to the following lemma. (2A.f)

Lemma (Pukelsheim 1993)

The set M (X) of moment matrices of all designs is the convex hull conv({xx T : XES})

of rank 1 matrices in the space Sym(k). If the regression range S is compact, then M(X) is a convex and compact subset of the cone NND(k) . •

e

If for the problem of estimating a parameter subsystem K t of the mean parameter vector a design x E X with moment matrix M = M (X) is used, the quality of the estimate (e.g. in the classical sense of least squares approximation) can be described by the generalised information matrix MK of the design X for Kte. MK is well defined by the mapping (2.4.g)

(2.4.h)

NND(k)

~

Sym(k)

due to the geometry and another application of the Lowner/Gauss-Markov argument. A moment matrix is called LOwner optimal for Kte in a set M C M(X) of competing moment matrices if (2A.i)

This, however, implies (cf. Pukelsheim 1993, Cor. 4.7) that no moment matrix is Lowner optimal for Kte in the set M(X) unless rank(K) = 1. While at first glance this result seems very pessimistic, it just means that in a class of designs conforming to practical realities and restricted to rational conditions, it may well be a daunting task to investigate special types of designs, for which the set M of competing moment matrices is more restricted. By means of geometric results of the Lowner/Gauss-Markov type, a bound for the number

866

XIII. Applications of designs

of points from S supporting a design can be derived, cf. Pukelsheim (1993), Theorem 8.1. This theorem states that every (k x k)-matrix M is achieved by + 1 different regression vectors. If the a design supported by at most (k x s )-coefficient matrix K has (full) rank s, the (s x s )-generalised information matrix can already be realised by a design supported by at most (s~l) + S . (rkM - s) vectors.

C!l)

Motivated by these results, somewhat resembling Fisher inequalities and rank formulae (cf. II.2.6), we now show how the class of block designs, forming the central topic of this book, is obtained from this general setting of operator theory and statistical estimation theory.

2.S. Two-Way Models and Block Designs We now describe the two-way classification model (with no interaction), which is the prototype of the type of design problems this book is concerned with. Suppose an experiment consisting of two factors gives responses

Yi,j,/

=

Cii

+ /3j + ei,j,/,

i E [1, ... , a], j E [1, ... , b], l E [1, ... , nij],

consisting of the mean effect Cii corning from so-called level i of the first factor A := [1, ... , a], i.e. the effect of treatment number i, plus the mean effect /3j stemming from the level j of the second factor B := [1, ... , b], i.e., the mean effect of the j-th block. Finally ei,j,/ denotes an error term coming from the l-th test replication and the replication number ni,j counts how often the "treatment" i is applied to "block" j. We encode these numbers in a rational matrix (2.S.a) which weighs the proportional contribution of observations that use a level i of factor A and level j of factor B. With the regression range given by (2.S.b)

S= ei

lei

liE A} x {fj

= (Oi/)/eA

E]Ra

Ij

and

E B} C ]Ra+b, /;

= (0 j/)/eB E ]Rb,

§2. Design of experiments

867

this leads to regression vectors of the form (2.S.c)

Xi,j

=(

i

e j) ' f

(i, j) E A x B,

with (2.S.d) defined as in Equation (2.4.d). With this regression set fixed, the design X can be uniquely represented by the (a x b)-weighing matrix W. With the following notation (2.S.e)

r := Wlb E JRa

(2.S.f)

s := WT la

(the vector of treatment replication numbers)

E JRb (the vector of block sizes)

the matrix W leads to a block design with the incidence matrix

N=n·W (having positive integer coefficients possibly exhibiting multiple blocks being multisets) corresponding to the experimental design X. The moment matrix M := M(X) can then be written in the form (2.S.g)

M

= L n(x)xx = L T

xeS

ieA

jeB

eie Wij

(

d

T

.;r

)

I

where for any mEN and x E JRm, Cl x denotes the m x m diagonal matrix formed by the vector x. In this model the parameter vector e has the form e = (p) where a = (al,"" aa)T and f3 = (f3I, ... , f3b)T. As MC~:) = 0, the optimal estimator (nM)-1 XTy for e cannot exist. We have to interpret this fact as saying that

868

XIII. Applications of designs

only certain subsystems mated.

Kt e of the mean parameter vector

e can be esti-

Indeed, the subvector, for instance a = (aI, ... , aal, can only be estimated when corrected by the mean effect (cf. Pukelsheim 1993, 3.20 and 3.25) (2.5.h)

a=- La; 1

a ;

giving the subsystem (al -

(2.5.i)

a, ... , aa -

Kaa = Kte with K

a)T of so-called centred contrasts,

= (~a)

defined by the centring matrix (2.5.j) and the averaging matrix (2.5.k) forming an orthogonal sum decomposition oflRa , which has special applications in measuring quantum systems (cf. Section 4.4). The generalised information matrix for the centred treatment contrast given by

(2.5.1)

!::"r MK= ( o

Kaa is

W!::";wt 0) 0

essentially being the Schur complement of !::"S in M, (2.5.m) here called the contrast information matrix of the block design

w.

We can now quote (cf. Pukelsheim 1993, Claim 4.8) the basic fact:

Let n E N+ and r E lRa be a given treatment replication with nr; E No, t! .r = 1. Then the product designs W = rsT are the only LOwner optimal

§2. Design of experiments

869

designs for the centred contrast Kj a in the set T (r) of block designs with blocksize vector r, having contrast iriformation matrix b. r - rrT . • With these results, we see that the incidence matrices of complete designs which test every unit against every other unit give optimal estimators. Here the following question arises: Amongst the many choices of possible matrices W in the class T of block designs of the form (2.5.a) with a . b tests, would it be possible to derive sufficiently many relations in order to determine the parameters or structure of the design? However, a close look at the dimensions involved shows that a block design with sample size n < a . b is sufficient even though it has incomplete support to the effect that some treatment-block combinations will not be observed.

2.6. Symmetry and BmD's For this we assume with some justification that the quality of a parameter estimation should be invariant with respect to the labelling of the treatments and blocks. More specifically, the concept of Lowner optimality has to be extended suitably, to remain invariant under permutations of the experimental domain; i.e. the sets A and B in the case of the two-way model. To simplify the problem, we consider a closed subgroup G < O(k) ofthe orthogonal matrices acting by left multiplication on the regression set, simultaneously acting by conjugation on the set M c NND(k) of moment matrices. Furthermore, this group affords a representation H < GL(s) on the group of invertible (s x s)-matrices acting by conjugation on the set MK C Symes) of corresponding information matrices for Kte. The design problem for Kte in M is called G-invariant, if (2.6.a)

\/g E G:

gMgt = M

and

g(imK) = imK.

The related information matrices are made H -invariant by defining the socalled Kiefer ordering (cf. Pukelsheim 1993, Chapter 14) as an extension of the Lowner ordering. Under this group theoretic-eombinatorial notion for optimality criteria, one sees that a moment matrix M is (Kiefer) optimal for Kte in M, if and only if Mis Lowner optimal for Kt() and fixed under the action ofG. With these preliminaries we can go back to the two-way classification problem. Applying the results about the centred contrasts (2.S.i) we consider now the

XIII. Applications of designs

870

maximal parameter system

(2.6.b)

where the dots in the index of the value x represent sums over the corresponding variables (e.g. Xi .. = Lj Lz xijz), with (2.6.c)

K

= (la 1b

Ka 0

0)

E ffi,(a+b) x (a+b+l) ,

Kb

with

-

1

1a:= -la. a

Within the moment matrix set M = M(T) of all block designs and the group G (2.6.d)

being the direct product of permutations of all treatments resp. all blocks represented by block diagonal permutation (a + b) x (a + b) matrices (2.6.e)

the identity

(2.6.f)

G~)K K(o1 0 0) =

0

0 0 a

p

holds as Ka and Kb are symmetric. Thus the group H afforded by G on MK is isomorphic to G. Now, we conclude that the uniform design (2.6.g)

W

= -1a ·l-bt

is the best block design for the two-way classification problem. Its realisation would imply that the sample size n must be a multiple of a . b, quite often an unfeasibly large number. A sample size n < a . b implies incompleteness, in so far as some combinations (i, j) E A x B are not observed. By requiring the other properties we arrive at the definition of a balanced incomplete block design, if W is

871

§2. Design of experiments

• a uniform (a x b)-weight matrix with

wij E

{O,

*}

• equiblock sized, WTl a = lb' • balanced in the sense that its contrast information matrix C (W) is completely symmetric and non-zero, i.e. of the form (2.6.h)

X/a

+yJa ·

A short calculation reveals that this is identical to the well-known formula for incidence structures of balanced incomplete block designs (2.6.i) with positive integers ~, b- and A = ~ . (1l~~;1. One can see from Equation (2.6.h) and the requirement that N and NT is to be a positive matrix, that this implies Fisher's inequality. We conclude this section by saying that by using functional analysis applied to linear estimation theory, the concept of a balanced incomplete block design can be derived in a purely functional analytic setting, producing the parameters of balanced incomplete designs as defined in Chapter 1.

2.7. Testing Speech Scramblers Similar questions about efficient experiments occur in many other areas of science. As a classical example we mention the problem of comparing the effects of different drugs in the vast area of medicine and pharmaceutics. Here we present another example concerning a rather unusual application of design theory to the design of cryptographic devices, namely quality tests for voice scramblers based on different permutations, cf. Beth, HeB and Wirl (1983), and Beker and Piper (1982). These scramblers are realised by a so-called time division multiplexing of the input voice data by dividing a segment of transmitted or recorded speech of length t = 600 ms into n blocks, which are shuffled by certain permutations, the scrambling keys. Such a scrambler can be viewed as a cipher, permuting the n blocks of a segment into a signal which sounds like noise rather than voice. The lack of objective criteria for the evaluation of the desired "non-understandability" of voice scrambling schemes, and the associated scrambling keys, necessitates subjective tests to validate a given metric used for the permutations.

872

XIII. Applications of designs

This technical goal has required that a series of psycholinguistic hearing tests be perfonned. During each test seven sample texts To, ... , T6 are used. With the exception of text To each text is scrambled by a fixed pennutation. Text To is scrambled by continuously varying pennutations. The permutations used for text TI-T6 are pairwise different. During the test, subjects listening to the scrambled texts have the task to "descramble" texts using their auditory system, i.e. by listening. To prevent errors caused by loss of concentration, each subject was asked to listen to three texts only. For planning the experiment a Steiner system 3(2, 3; 7) is used which leads to seven schemes (the blocks of the Steiner systems) each containing three texts. Each of the seven texts is contained in three schemes and occurs exactly once at the first, second and third position in a scheme (see Table 2.1). The outcome of these tests has been successfully applied to evaluate the quality and the design of voice scramblers, cf. Beth, HeB and Wirl (1983).

Table 2.1. Experimental scheme testing voice scramblers, which resembles the well-known incidence matrix of the Fano plane 3(2, 3; 7), cf Example 1.1.3 Subject

1st test

2nd test

3rd test

0 1 2

To TI Tz T3 T4 Ts T6

Tl Tz T3 T4 Ts T6 To

T3 T4 Ts T6 To Tl Tz

3 4

5 6

To solve this ranking problem by a regression analysis it is assumed that on a logarithmic scale the success Vtp of subject p in descrambling text t can be described by the linear model

where ILt denotes the log (understandability in % of text) and D!p describes the ability of person p on a logarithmic scale. The desired metric for nonunderstandability is then provided by the linear order of the number (- ILt), so that higher numbers represent higher crypt(ograph)ic quality. In analogy to

873

§2. Design of experiments

Example 2.3 the regression vector in this case is !J,l

(27) .. a

0 =!J,7

with model matrix

1 0 0

0 0

1 0 0 0 0

1 0 0

0 0

0

0 0 0

0

1 0 0 0

0

1 0 0 0 0 0 0

0

1 0 0 0 0 0 0 0 0

0

1

0 0 0 0 0 0 0 0 0

0

0

1 0 0 0 0

0 0

1 0 0 0 0

1 0

0 0

0

0

X=

1 0 0 0 0 0 0

1 0 0 0 0 0 0

0 0

(2.7.b)

0 0 0 0

0

0

0 0

1 0 0 0 0 0 1 0 0 0

0

1 0 0

1 0 0

1 0 0 0 0 0 0 0 0 0

1 0

0 0 0

1 0 0 0

0

0 0

1 0 0 0

0

1

0

0

0

0

0

0

0

0

0 0 0

1 0 0 0 0 0

0

0 0

0

0

0 0

0

0 0

0

1

0

0 0 0

0

0

1

1 0 0 0 0

0

0

1 0 0

0

1 0

0

0

0 0

0

1 0 0

0

0

0

1 0 0 0 0 0 0

0 0 0

0

0

1 0

0 0

0 0 0 0 0 0

0 0

0

0

0

0

0

0

1

1 0

1 0 0 0 0 0 0

1 0 0

1 0 0 0

0

0

1 0 0 0 0 0 0

1

0 0 0 0

1 0

0

0

1 0 0

0

0

0

0

0

1 0 0 0 0 0 0

0 0

1 0 0 0

0

1 0 0 0 0

XIII. Applications of designs

874 so that

300 0 000

101

000

o

I

3 0

0

0 0

0

0

I

0

003

000

0

0 0

I

0

I

000

3 0 0

0

000

I

0

000

I

0000300 000

0

0

000 000

100 0

I

0

3

0

o

3 0

o

I 0

0

0

0

0

100 3 0 0

0

0

000

000

0 1

000 000

0 0 0 000

o

I 0

1 103 000

000

o

0

000

3 0

(2.7.c)

o 0 o

0

300 0

100 000

0 3

0

0

0

1 000

0

003

0

o

000 0

3

1 0 0

has the form (2.7.d) where b. r = r . Iv, b. k = k . h, W is the incidence matrix of a proper 2-design Sl (2, k; v) with r = k = 3, v = b = 7, i.e. the Fano plane, as planned.

§3. Experiments with Latin Squares and Orthogonal Arrays

In Section 2, the theory of linear regression analysis has been described leading to the properties of balanced incomplete block designs for the so-called twoway classification models. However, a more general theory of multiway classification models could be envisaged. Here several factors of classification would have to be included leading to multidimensional tensorial incidence structures which are not dealt with in this context. However, the linear model of estimating three qualitative factors without interaction, cf. Witting and Nolle (1970), leads to a reduced complexity estimator for a linear theory of this type. First we give a simple example.

875

§3. Latin squares and orthogonal arrays

3.1. Three Factors on Four Tyres An example is the assessment of the wear of car tyres (see Mead 1988). Because there are performance differences between the four possible wheel positions of a car, which are also influenced by differences between cars, the experiment has to eliminate these influences. In the case of four different tyres, four cars can be used and the tyres are allocated to the cars so that one tyre of each brand is fitted to each car in the four different possible wheel positions distributed over the four tyres of the cars. This leads to the concept of Latin squares (cf. Table 3.1).

Table 3.1. Latin square for testing tyres.

Front left Front right Back left Back right

BMW

Mercedes

Porsche

Audi

Michelin Goodyear Conti Pirelli

Goodyear Conti Pirelli Michelin

Conti Pirelli Michelin Goodyear

Pirelli Michelin Goodyear Conti

We recall from Observation 4.5 in Chapter VIII that a Latin square of order n is an n x n array with entries 1, ... , n, such that each of these n elements appears exactly once in each row and each column. The corresponding 4 x 4 Latin square for our tyre experiment is given in Table 3.1. The formal method underlying this solution is taken from the theory of analysis of variance. It is described in Section 3.2.

3.2. Testing Three Qualitative Factors without Ihteraction Let Xijl be stochastically independent random variables with normal distribution D(Xijl)

= N(/-L

+ Vi +Kj + PI, ()2),

with (/-L,

V;, Kj, PI, ()2) E]R4 X ]R+,

i = 1, ... , m; j = 1, ... , m; l = lei, j), m 2: 3 where lei, j) describes a Latin square of order m. This property is obviously symmetric in the variables i, j and l so that summing over i for fixed j is equivalent to summing over the range of l.

876

XIII. Applications of designs

The mean vector a can be written in the form (3.2.a)

a -- fA' lie •..

+ '" L-t v·e· + '" i...J K·e J .J.. + "'Pie ..I , l

~

l..

I

j

with ei .. := Lj LI eijl, where eijl represents the unit vector of]Rn with a 1 at position (i, j, I) describing whether a treatment is or is not applied to a unit; e.j., e ..1 and e ... are defined in a similar way. This means that different treatments lead to additive effects during the experiment. In Equation (3.2.a) J1,e ... represents the effect of treatments which are performed for all experimental units, whereas Li Viei.. describes the effect based on differences in the treatments on a subset of units. The so-called averaged treatment effect J1,e ... is chosen so that the means V., K., p. of Vi, K j and PI are equal to O. Since the vector ei .. fixes one of the m dimensions, Li Viei .. spans an (m - I)-dimensional subspace of]Rn and because of the orthogonality of ei .. , e.j. and e ..I, the (k = 3m - 2)dimensional subspace of]Rn described by Equation (3.2.a) can be divided into three three-dimensional and one one-dimensional pairwise orthogonal subspaces (cf. Witting and Nolle 1970). A linear hypothesis H, which covers an h = 2m - 1 dimensional subspace of]Rn , and its alternative hypothesis K, which covers a k-dimensional subspace, can now be given by (3.2.b)

H : Vi = ... = Vm

= 0,

K:

I>l > o.

The estimator T*(x) is determined by a variance analysis which leads to

(3.2.c)

T* (x)

_l_m "(x. - X )2 = _---,;--_. . . . ._:.::m'--.:,.l--.:L..-=i~_I .._ _.._._ _ _ _ __ (m_l)1(m_2) Li Lj L/Xijl - Xi .. - X.j. - X .. I + 2x.. J2'

where dots in the index of the mean value x represent sums over the corresponding variables (e.g. Xi .. = L j LI Xijl), cf. Equation (2.6.b). The numerator can be interpreted as the projection of x onto the space of the hypothesis normalised by its dimension h = m - 1. Similarly the denominator can be viewed as the normalised projection of x onto the n - k = (m - I) (m - 2) dimensional space of the alternative hypothesis; cf. Equations (2.2.g) and (2.2.h). To prevent systematic bias, the Latin square is randomly chosen from the set of all possible Latin squares. The following two examples show the necessity of using layer families of Latin squares, so-called orthogonal arrays, for the design of experiments.

§3. Latin squares and orthogonal arrays

877

3.3. Four Factors on Four Tyres Suppose in addition to the situation given in Example 3.1 four differen t treads W, X, Y and Z (see Figure 3.1) are available for every brand, which could also influence the results of the experiment. So the treads have to be taken into consideration. On the one hand, a Latin square could be constructed ignoring the dependency on the car brand; on the other hand, the wheel position could be ignored. To overcome this situation two Latin squares have to be found which can be superimposed in such a way that all combinations of treads and tyre brands are considered. This leads naturally to the concept of orthogo nal Latin squares, as provided by the notion of orthogonal arrays in Chapter VIII.

w

~

W w

X

I

Y

Z

tt

~

~ yy

Figure 3.1. Correspondence between tread names and treads.

So, one Latin square corresponds to a parallel class in a column- row partition ed scheme. Every element is connected with another element not from the same row or column, a relationship which we could call a block decomposition, as described in the previous example of tyre positions. Another orthogon al Latin square would then allow for the putting in a different setting of an orthogon al factor, where no two units that have already been tested will be tested together on the same object. By contrast this, the total dimension of projectio ns carried out on the Hilbert space to estimate linear regressions, can be increase d as described by Pukelsheim (1993), see Chapter 1.22 f. Thus a sequenc e of mutually orthogonal Latin squares can be used to approximate the balanced incomplete block design experiment that comes from an affine plane, as describe d in Chapter Vill.4. Going back to Example 3.3, the Latin square in Table 3.1 is chosen in a Z4 circulant form. By Corollary X.l2.4, this Latin square has no transver sal and hence no orthogonal mate. Therefore, to solve the four-tyre problem for the orthogonal feature "tread" as well, different Latin squares (which are based on EA(4» have been chosen, corresponding to the parallel classes of lines in AG(2,2).

XlII. Applications of designs

878

It should be noted that the different behaviour of the constructions given in

Tables 3.1 and 3.2 is a consequence of the underlying algebraic structures. As this example shows, the choice of these algebraic structures can determine failure or success of an experimental design.

Table 3.2. Latin square for testing tyres with attention to different treads BMW Front left Front right Back left Back right

W Michelin Y Goodyear Z Conti X PireIIi

Mercedes Goodyear Michelin Y Pirelli W Conti

X Z

Audi

Porsche Y Conti W Pirelli X Michelin Z Goodyear

Pirelli Conti W Goodyear Y Michelin Z

X

In our example, obviously not all combinations between tread, brand of tyres, car type, and wheel position are tested. For this reason, we call the experiment incomplete. But each brand of tyres is tested with each tread exactly once, so that the experiment is called balanced. Special sets of MOLS, which are generated using abelian groups, correspond to translation nets (see Section X.9). Bailey and Jungnickel (1990) mentioned these for statistical applications which naturally lead to the use of group theory. In Chapters VIII and X, conditions for the existence and construction of sets of mutually orthogonal Latin squares are given. Most recently another application of design theory and algebra to the construction of higher factorial designs has been presented by Robbiano (1998). He describes the set of treatments of a full factorial design as a direct product of several factors, each factor being a finite discrete rational variety, of say integers. The defining ideal of this variety can be constructed as being generated by the canonical multinomial basis (almost analogously to Equation (6.2.d)). The problem of the design of experiments (DoE) to build up all the necessary inferences, cf. Yates (1937), from smaller fractions of the full design, cf. Street and Street (1987), such as sets of MOLS, is here reduced to solving an ideal theoretic problem by the use of Grobner bases, cf. Buchberger (1985). With Robbiano's approach another door to research in design theory may have been opened, as methods from algebra and geometry become more and more applicable through the use of advanced computer algebra packages, cf. Section 7.4.

880

XIII. Applications of designs

3.4. DNA Analysis A high-tech application of orthogonal Latin squares has been used recently in the design of so-called gene chips. By exposing a carrier plate (1 cm2 of specially coated silicon, "blank") to an on/off array of light patterns M while lying in a solution of one of the few nucleotides N E {adenine, guanine, thymine, cytosine} a 400 x 400 matrix of microplots of 25 /-tm x 25 /-tm with entries left blank or the nucleotide attached to the underlying surface can be formed. If subsequently alternating the patterns M and the choice of the nucleotide N, a 400 x 400 matrix of nucleotide chains (D ij )i,jE[I...400] E {A,G,T,C}* can be built up; cf. Figure 3.2. Each Dij is designed to pair with a special substring of a DNA sequence, due to the wellknown complementary docking reaction A ~ T and C ~ G in the genetic code. The position of an instance of a docking reaction is then read out by laser light of suitable wavelength giving fluorescent reactions at the special binding sites. With a suitably chosen orthogonal array both the production of the gene chips and the read-out pattern can be optimised.

§4. Application of Designs in Optics Amongst the many applications of design theory to the development of electronics communications, computer technology and optics which have been reported during the last three or four decades, one of the most surprising and successful examples is that of Hadamard transform optics used in the design and construction of modem imaging systems. In this chapter we dedicate a short section to this topic by presenting two examples. Firstly, even though the book Hadamard Transform Optics by Harwit and Sloane (1979) became famous amongst insiders after its publication about two decades ago, the results contained in this book seem not to have been adequately recognised in the areas of discrete mathematics, statistics and communications engineering. Even in optics it has remained a rather specialised topic, which is mostly used in the area of digital optics and advanced image reconstruction techniques in coherent optics, laser optics, speckle holography, etc. (Cf. Beth, Klappenecker et al. 1995.) As the intrinsic use of balanced incidence structures (i.e. designs) is a key tool in these applications, we give a brief summary of this application. The key to Hadamard transform techniques in optical processing and transform techniques lies in the fact that the signal-to-noise ratio of receivers and

§4. Application of designs in optics

881

measurement instruments can be enhanced considerably by applying special orthogonal transforms. As shown in Table 4.4 below, the principle idea behind these techniques for extracting information from quantities which are composed of several signal functions can be demonstrated easily by the example of weighing designs, which we define as follows.

4.1. Improving the Signal-to-Noise Ratio by Hadamard Matrices A number of physical quantities1{rl, ... , 1{rn, henceforth called objects (with unknown weights), are to be measured by a measuring device (4. 1. a)

P:1{rI-+(j"

which we call a weighing scale with output values or measurement (4.1.b)

disturbed by errors ej. We assume the stochastic process e = (ei)i, i E I, to be independent of the actual weight 1{rj and identically independently distributed with (4.1.c)

Vi

E

I:

E(ei)

=0

and

V(ei)

= 0- 2 •

If a weighing scale could measure the objects 1{r1, ... , 1{rn separately in a straightforward manner then, even though the mean error distribution of the actually measured values (j, is (4.1.d)

the mean square error in each scale equals (4. 1. e)

Thus the signal-to-noise ratio, defined by (4.l.f) is equal to the energy of the j -th object divided by the noise variance, that is

(4.l.g)

SNR(j) = 1{rJ2 . 0-

XIII. Applications of designs

882

This means that the signal quality is reduced by an average factor of degradation of 1 - ~ or, as in the language of communications engineering, going to a logarithmic scale the signal-to-noise distance is (in dB) (4.1.h) Shannon (1948) proved a celebrated result about the capacity of the communication channel, which in our case reads that the information transmission rate through a noisy channel can be increased as close as possible to channel capacity

(4.1.i)

1 C = -log2(1

2

bit ], + SNR) [measurement

with arbitrary small error by the use of random codes (cf. Chambers 1985 and McEliece 1977). This is, of course, related to the ideas of Yates (1935) who demonstrated that the mean square error can be reduced, by applying weighing strategies of several objects at once, i.e. by putting objects together in blocks. In this way the signal-to-noise ratio can be improved considerably, cf. Tables 4.3 and 4.4 below, as exemplified in the following. Suppose the same set of n physical quantities described above can be weighed in groups using a chemical balance having two scale pans, so that the right and the left pan action can be expressed by plus and minus signs. For the sake of simplicity we consider the case n = 4 to describe a weighing strategy. We call this a weighing design. Objects weighted with a plus sign have to be put into the left-hand side pan; objects weighted with a minus sign have to be put into the right-hand side pan. Additional standard weights 11i have to be added to the up going pan until balance is achieved. In this way we get 11 (111, ... , 114)T with

=

(4.1.j)

111 =

lh + 0/2 + 0/3 + 0/4 + e1

+ 0/3 - 0/4 + e2 + 0/2 -0/3 -0/4 + e3 0/1 -0/2 -0/3 + 0/4 + e4·

112 = 0/1 - 0/2 113 = 0/1 114 =

The weighing matrix W of this design is given by

(4.1.k)

W =

(~-~ 1

1 1 -1

1 -1

-~)

-1

-1

1

L

883

§4. Application of designs in optics

which is obviously a Hadamard matrix H4 of order four, hence W- I small calculation shows that for the vector (4.1.1)

=

~ W. A

I] = W1jr +e

the quantities of interest (4.1.m)

,

1 4

1jr = -WI]

are of the form (4.1.n)

, 1jr

1

== 1jr + -We. 4

For the mean value of the various {r j so derived, we obtain a mean square error (4.1.0)

which leads to an improved signal-to-noise ratio (4.l.p) Hence using a Hadamard matrix Hn improves the signal-to-noise ratio by a factor of ../ii, which in our example equals two, and this gives an improvement in dB by a factor of 10../ii = 20. Note that this improvement was obtained just by using a different weighing matrix W = Hn , instead of the trivial weighing matrix W = In. This shows that the use of designs is not only suitable for reducing the number of experiments while maintaining the same fidelity (as shown in the examples in Sections 2 and 3), but also will give a highly improved fidelity if the appropriate measurement strategy (i.e. weighing design) is chosen. The main idea behind this method of analysis of variance is based on blocking techniques as pointed out by Yates (1935). It is the nowadays well-understood idea of developing signal systems into suitable sets of orthogonal functions which has led to the theory of signal processing and fast signal transforms (cf. Section 5.4 of Blahut (1985) and Beth (1984». Special modulation techniques in communication engineering led to the widely known principles of soft decoding, etc. (cf. Clark and Cain 1982; Wozencraft and Jacobs 1965; Lazic and Senk 1992). These, after all, resemble an application of design theory which has been considered in the area of coding theory, see Section 5, where the codes generated by incidence matrices of designs have been studied; see Section 11.11 and Assmus and Key (1992a). In piuticular, the theory of modular codes and

XIII. Applications of designs

884

the associated modular representation theory plays an important role obtaining non-trivial combinatorial and algebraic results which can be applied in various settings such as, for instance, tomography; see Section 6.

4.2. Weighing Designs and Hadamard Transform Optics

Extending the weighing pattern of Example (4.1.j) to n = 8, (4.2.a)

T/I = VI + V2 + V3 + V4 + V5 + V6 + V7 + V8 + el ~=~-~+~-~+~-~+~-~+~

~=~+~-~-~+~+~-~-~+~ ~=~-~-~+~+~-~-~+~+~

~=~+~+~+~-~-~-~-~+~ ~=~-~+~-~-~+~-~+~+~ m=~+~-~-~-~-~+~+~+~ ~=~-~-~+~-~+~+~-~+q

the weighing matrix W becomes 1

-1

(4.2.b)

W=

1

-1

-1

-1

-1

-1

-1

1

-1

-1

-1

-1

-1

-1

1 -1

1

-1 -1

1

-1

-1

1

-1

-1

-1

-1

-1

-1

1 -1

1

-1

1

-1

1

1 -1

Even though the incidence matrices of designs over fields of characteristic zero are algebraically of no interest, since they have full rank, for decoding purposes they have been successfully used for the so-called Reed-Muller codes in early Mars missions, cf. Section 5.4. However, in signal processing square matrices of full rank, especially unitary matrices, play an important role for signal reconstruction as the above examples show. The following example does not use a (1, -1)-generalised incidence matrix of a Hadamard design, but rather a proper (O,1)-incidence matrix, which will apply to the same problem in a surprising manner.

885

§4. Application of designs in optics

If one uses a spring balance, which just weighs the total weights hung on it, the corresponding weighing matrix is obviously an (O,I)-matrix if we put those elements that have been hung on the spring at each weighing instance into common blocks. Thus they define an (O,I)-incidence matrix of a weighing design which in the case of n = 7 given below is the incidence matrix S7 of a PG(2, 2). This is equivalent to the matrix obtained from the Hadamard matrix Hg (see Example (4.2.b) above) by omitting the first row and the first column of H8, and changing the entries +1 into 0 and the entries -1 into I. After suitable column and row permutations, S7 becomes a circulant giving the measurements (4.2.c)

1J

= S71/r +e.

Note, that

-1 1 -1 (4.2.d)

•

-1

1/r=S7 '1J=

-1

-1

-1 -1

1 -1

1 -1 -1

-1

1 -1

1

1

-1

-1

1 '1J 1 -1

-1

-1

-1 1

-1

-1

-1

holds. Owing to the fact that the Hadamard matrix

H2k

1

-1 is obtained via the

AG(k,2) design related to the corresponding projective geometry with Singer

cycle, cf. Section 5.2, Sn and S;;' show a similar circulant structure, which can successfully be applied to optical technology. The properties of the optical design matrices are obtained by a theorem due to Harwit and Sloane (1979), which we summarise as follows. (4.2.e)

Theorem.

Hadamard matrices make the best chemical balance weighing designs, or the best masks when entries + 1, 0 and -1 can be used (Table 4.1). S-matrices make the best spring balance weighing designs, or the best masks when entries 0 and 1 can be used (Table 4.2).

If there is a number n of unknowns for which a Hadamard matrix of order n exists, then a weighing matrix ofHadamard type reduces the mean square error in each unknown by a factor of n. If the number of unknowns is nand an S-matrix of order n exists then the corresponding weighing S-matrix for

§4. Application of designs in optics

887

is, corresponding to (-1, 1) and (0, I)-matrices) are derived naturally. For instance, the example of (0, I)-matrices corresponds to a mask brought into the optical axis of an imaging system (cf. Figure 4.1), where an entry 0 describes a non-transmissive element at the corresponding matrix position, while an entry 1 means a fully transmissive hole in a so-called amplitude mask. While these masks are amplitude elements, cf. Section 9, the behaviour of phase elements - in which -I would correspond to a phase shift by Jr - is discussed e.g. by Aagedal, Beth, et al. (1996) for applications in diffractive optics or by Schroeder (1986) for acoustics.

. \. . . . . . . . . ~.,,:::::-....,".. . . ..

:.\

'· ·. · .Jti:;)\ I //'

1

·1

pinhole

...... _"::.

~.,::\.....

. . i/>4I ' lSI and as we shall see later the even stronger inequality H(M) > H(S) holds as well. Figure 10.1 schematically shows the essential features of what has been described thus far.

Figure 10.1. Schematic description of the source space S and the message space M (Figure 1 from Simmons 1985a). Any message in the shaded region of M would be rejected by the receiver, while any message in the set MJ would be accepted as authentic. Figure 10.1 also illustrates that it is possible for the opponent to fail to deceive the receiver, even though he succeeds in getting him to accept a message that was not sent by the transmitter. Assume that the state of the source is S2 and that the transmitter chooses to encode this information by sending message 11!2 to the receiver. If the opponent - not knowing the information shown in Figure 10.1 of course - intercepts the message 11!2 and replaces it with 11!3, the receiver would accept m 3 as being authentic since it is one of the messages that the transmitter might have sent, even though it was not the message actually sent in this case. However the receiver would interpret 11!3 to mean that the source state was s2 - as observed by the transmitter. The opponent would lose in this case, in spite of the fact that he succeeded in having the receiver accept a fraudulent message, since the receiver is not misinformed as to the state of the source. There is a well-known precept in cryptography, known as Kerckhoff's principle, that the opponent knows the system, i.e, the information contained in Figure 10.1. It is equally reasonable to assume the same for authentication. Consequently there would be no authentication possible for the receiver using the scheme shown in Figure 10.1 alone. What is done instead is to have many such encoding rules in an authentication system all of which are known to the opponent - with the choice of the particular encoding rule in use being known only to the transmitter and receiver, similar in many respects

959

§lO. No waves, no rules, but security

M

Figure 10.2. Suggestion of a general scheme (Figure 2 from Simmons 1985a).

to the "key" known only to the transmitter and receiver in a cryptosystem. Figure 10.2 suggests the general scheme. Each encoding rule, ei, determines a proper subset Mi of M, IMi I 2: 151, and a mapping - perhaps one to many - of 5 onto Mi. The inverse mapping D is a well-defined function, i.e., for any e E £ and m E Mi the function D(e, m) defines a unique state in 5 U 1/), where I/) is the null set. Even this very intuitive description of authentication should make clear the reason for describing authentication as a problem in "spreading" messages in M. If m] is an acceptable message only in set MI, then the opponent, knowing the system, would be able to conclude that el was the coding rule being used if he saw m 1 in the channel and would then be able to substitute another message with certainty of deceiving the receiver. To avoid this it is necessary that each message occur in sufficiently many authenticating sets to (ideally) leave the opponent no more able to "guess" at an acceptable message after he has observed what the transmitter sent than he could have before the observation. This ideal can be achieved in infinitely many perfect authentication systems.

We now transform this early description into the language of design theory as follows. The information to be transferred represents one of k states from a state set S. One uses a set M of v ~ k possible messages together with an encoding rule for mapping the states into messages. Each encoding rule e E E, of b = IE I possible ones, is a one-to-one mapping of the elements from some subset of S to those of some subset of M. Now an authentication code is defined by the set of states, the set of messages, the encoding rules, represented by the (b x k) encoding matrix over M. For authenticated information transfer two parties use an encoding rule, which is only known by these parties. The enemy may know the complete set of all possible encoding rules but not the e E E actually used. This will be illustrated by possibly the earliest example of an authentication code given by Gilbert, MacWilliams and Sloane (1974) after Simmons had

XIII. Applications of designs

960

• 2

Figure 10.3. Projective plane PG(2, 3). acquainted them with this problem. For this consider the projective plane PG(2, q), for instance with q = 3, given in Figure 10.3. Encode the state s E GF(q) U too} (considered as the line at infinity) under the encoding rule

(10.l.b)

e: s

1-+

(s, c) E (GF(q) U {oo})2,

where e is a point of PG(2, q) and (s, c) is the affine line of slope s and affine offset c of the unique block containing the point e in the derived design (PG(2, q»s of the point s, cf. Figure 5.10. For q = 3 Table 10.1 shows the principle for this authentication code without secrecy. There are two interesting probabilities which have to be considered. First the probability Pi that an enemy can insert a message without detection at the receiver's side and second the probability Pa that an enemy can alter a delivered message without detection at the receiver's side. In this situation (lO.l.c)

k v

Pi::: -

and

k-l Pa::: - - . v-I

For our example Pa = ~:::i is optimal. In general, designs are associated with authentication schemes almost in a one-to-one manner, as was shown by Stinson (1988, I992a). These results are summarised in the following theorem. First it should be noted that special classes of designs are connected to special types of authentication codes. For example Stinson (1988) shows the following result. Cartesian authentication systems (i.e. with H(S 1M) = 0, thus without secrecy) with N = kn and b = n 2 are uniquely determined as transversal de-

§10. No waves, no rules, but security

961

Table 10.1. Principle of authentication code for PG(2, 3) State

Encoding rule

Message

00

00

01

10

11

20 0 00 01 02 0 00 02 01 2 00 01 02 1

21 1 12 10

000 001 002 0000 20 21 22 200

2

0

11

00 11 10

12 0 10 11

12 2

02 12 22 2 21 22 20 1 22 21 20 00 20 21 22 00

10 11

12 100 00 01 02 000

signs TD[k; n] with (lO.I.d)

H(S) - H(M) = H(E

I M)

= log2(n)

and thus Pi = Pa = ~ are optimal. • Last but not least we mention a result which represents the culmination of the theory of authentication as conceived by Simmons (l985b). For this we introduce the notion of an authentication system with peifect secrecy, cf. Simmons (1992a), p. 15, which is defined by requiring that the probability Pd = max (Pi> Pa) of deception fulfils the equality (lO.I.e)

log Pd = H(M

I ES) + H(S)

- H(M).

Then the fundamental theorem, cf. Stinson (1992a) (Theorem 4.2), shows the importance and exclusive rOle of designs in authentication theory.

Suppose we have an authentication code which provides peifect secrecy in which Pi = ~ and Pa = ~::: Then b :::: ~~:::~, and equality occurs if and only

J.

if

962

XlII. Applications of designs

• v-I == 0 mod k(k - 1); • there exists a S(2, k; v); and • both the source states and encoding rules are equiprobable. •

10.2. Shared Control by Designs

Threshold schemes were independently discovered by Blakley (1979) and Shamir (1979) when they devised shared secret schemes for robust key management for cryptosystems. While originally devised for solving the problem of recovery from a loss of keys, this technique has become an important topic in modem system management and control. Not only can the access and authorisation structure in large organisations, e.g. multisignature systems used in banks, be implemented in this technology, there is also an almost unlimited use of such techniques with the recently growing demand for distributed computing and secure outsourcing of computational tasks in large networks, cf. Otten (1992). Last but not least we shall mention the immense demand for such shared control systems to be expected from legally conforming democratically controlled key escrow systems, which are presently discussed worldwide,1O cf. Beth (1995a). The reader is referred to Simmons (1992a), Chapter 9, where shared control systems are surveyed extensively. The main idea which relates the shared control systems via threshold schemes to design theory is the following. Given a cryptographic key or an access privilege, one wants to divide the key into several, say w, parts of information and (securely) distribute these pieces of information among a set of people in such a way that any t of these w pieces of information gain access but fewer cannot. That means that a reconstruction of the secret can be obtained by the knowledge of at least t parts of the secret key. Obviously the secret key even can be reconstructed if at most w - t partial secrets are lost. Such shared secret schemes are called (t, w)-threshold schemes. Before we describe design theoretic examples of such structures which can be provided by finite geometries or Latin squares, we refer to an algebraic solution which had been invented almost as early as that by Blakley (1979).

10 Beth, Th.: "Zur Sicherheit in der Infonnationstechnik", FinalReport at the Deutscher Bundestag,

Bonn 1990, see also Infonnatik-Spektrnm, 1990, no. 13, pp. 204-215. Beth, Th.: Report at public hearing "Zukunft der Medien in Wirtschaft und GesellschaftDeutschlands Weg in die Infonnationsgesellschaft", Deutscher Bundestag, Bonn 1997.

§lO. No waves, no rules, but security

963

Shamir's (1979) construction is based on interpolation of polynomials over GF(q). Using Lagrange interpolation and the fact that any polynomial of de-

gree t - 1 is uniquely determined by t sample values at different positions, one randomly chooses such a polynomial p(x) and considers the constant part p(O) as secret. Mter the evaluation of p(x) at I different positions the corresponding values are distributed among I people. Any t-subset of these I so-called shadows determines the original polynomial and, in particular, its constant term and therefore the secret. There is an obvious way to implement different levels of privileges, because in a hierarchy more important persons can have higher privileges than others, which are nicely represented by this application of classical mathematics. Note that this solution has the disadvantage of lacking perfect secrecy; it can however be used for a large range of applications, e.g. in outsourced anonymous computing, cf. Otten (1992). Here we describe, however, the appealing solution of the problem that is provided by applying design theory.

10.3. Blakley's and Simmons' geometric construction Originally Blakley described shared secret schemes in a geometrical setting in which he prepared a large technological step, invented soon after by Simmons. He used intersecting lines in a finite projective space PG(n, q) the points of which are the one-dimensional subspaces of GF(q)n+l. As there exist (10.3.a)

n7~Lr+2(qi - 1)

n;=, (qi -

1)

r-spaces in PG(n, q), it consists of ["i l ] = qn+~~l points and [,,;1] = (q(:~'-N(~"~/) lines, where the number of li~es thr~ugh a point in PG(n,;) is ~ q_1 ,see

Proposltlon .. 1216 .. .

[n

Blakley exemplifies this in the special case PG(3, q) in which k = = q+1 q points are lying on a line and v = [~]q = q3 + q2 + q + 1 points exist in all. Furthermore, in PG(3, q), each pair of different lines intersects in at most one point. On the other hand, each point lies on q2 + q + 1 lines. For this reason the secret, being encoded by the coordinates of a point p, can be divided into private information pieces, for instance, the different lines through p. Now, it is obvious that any two out of w :::: q2 + q + 1 pieces uniquely determine the point p. Hence we have a (2, q2 + q + I)-threshold scheme. As in the case of polynomials this secret sharing system is not perfectly secure, even if generalised to higher dimensions.

964

XIII. Applications of designs lineg

Figure 10.4. Geometric representation of a shared secret P (taken from Simmons 1992a).

In Figure 1004 a situation to overcome problems of this type is visualised, namely a (3, q + I)-threshold scheme. This was invented by Simmons (1990) after he pointed out that the uncertainty about the secret is reduced by every degree of freedom lost through coalition of partners. In the example of a so-called perfect secret sharing scheme described in Figure lOA, this disadvantage is overcome. Here all possible secrets lie on the given line g, the indicator variety. The concrete secret, the coordinates of P, has to be determined by three parties PI, P2 and P3 in the sense that the point P lies in the unique intersection from line g and the plane n spanned by PI, P2 and P3 in the domain variety. Obviously, all these three points are necessary to identify P. Note that the points have to be chosen such that no three lie on a line, so that we have q + 1 participants in this scheme if we use a conic in n, cf. Remark VIII.5 .34. With this geometrisation (and algebraisation) the problem is solved (and implemented) very efficiently. The main requirement of shared control is to divide access privilege among w people, such that any t of these w gain access but fewer cannot. Defining a set V of v shadows (partial privileges), a set !C of m keys and a set B of b disjoint w-subsets of V, a (t, w, v)-threshold scheme is defined by the pair (B, !C) together with a mapping f : B -+ !C such that for every t -subset T of V and for all blocks B E B which contain T, the equation f(B) = K holds for some fixed K E !C. With this definition the subset T determines a unique key. Now, this secret key can be shared among w people by selecting a block B with f (B) = K and distributing the shadows in B among w people. As a result of this construction, any group of t of these w people uniquely determine K by f. Obviously, Steiner systems S(t, k; v) directly lead to (t, k, v)-threshold schemes. Symmetric designs with index A. lead to (t, k, v)-threshold schemes with t = A. + 1. A (t, w, v)-threshold scheme is perfect if no s shadows, s < t, give any information about the block containing these shadows. The upper bound on the number m of keys in a perfect (t, w, v)-threshold scheme is due to Stinson and

§lO. No waves, no rules, but security

965

Vanstone (1998): (1O.3.b)

v - (t - 1) , - w - (t -1)

m
+++--->1 decoder

Figure 10.6. A sequential cipher system (taken from Beth 1983a).

The tempting idea to replace coin-tossing by so-called feedback-shift registers is easily implemented, cf. Gollmann (1994). For this and deeper reasons PRGs are still produced and sold worldwide, even though they are known to be totally insecure, cf. Herlestam (1982). Their replacement by so-called non-linear-feedback-shift registers has been a major topic in cryptographic research since the late 1970s, cf. Herlestam (1982), Beker and Piper (1982), when new non-linear PRGs have been proposed, cf. Jennings (1980), Beth and Piper (1984), Gollmann (1985), Chan and Games (1990), Wolfman (1992), Bradley and Pott (1995). They are usually composed from linear-feedback-shift registers (LFSRs), whose feedback polynomial f(x) E GF(2)[x] is primitive of degree n (Figure 10.7). Thus such an LFSR produces a maximal (0, 1)-sequence of period 2" - 1, which has linear complexity n, however; cf. Jungnickel (1993a). The art and science of stream cipher technology relies on the combination of these sequences to

Figure 10.7. A linear feedback-shift register with feedback polynomial j(x) L7=1 Ci xi (taken from Beth 1983a).

=I+

XIII. Applications of designs

968

I

J

initialisation S

---- r--

. . . . . .

I

-.

register j I 1x ,-d x ,-21 • 1 • 1 • 1 • 1 • 1 • 1 • I x 1 I xo 1

...

~\

. . . . . . . //II ··· - r--

~

g

fi x ,_" ... , x o)

II 1

Figure 10.S. A non-linear feedback-shift register (taken from Beth 1983a).

fonn a good non-linear pseudonoise generator (PN generator), the behaviour of which can be described wholly in tenns of number theory, polynomial algebra over finite fields, discrete Fourier transfonns, autocorrelation functions and automata theory, cf. Liineburg (1979), Blakley and Purdy (1981), Jansen (1989), Pott (1995), Vielhaber (1997); see Figure 10.8. To protect against the intrusion of an outsider who uses cryptanalysis to determine the predictability of the sequence, cf. Jansen (1989), it is usually assumed that the statistical properties of such sequences can be guaranteed. Quoting Beker and Piper (1982) p. 169, these can be described by the following termi-

nology: If (St) is a binary sequence then a run is a string of consecutive identical sequence elements which is neither preceded nor succeeded by the same symbol. Thus, for example, 011100 I begins with a run of one 0, contains a run of three I s and a run of two Os, and then ends with a run of one 1. A run of Os is called a gap while a run of Is is a block. Suppose that (sr) is a binary sequence of period p (i.e., sm+p = Sm for every m). For any fixed r, we compare the first p terms of (St) and its translate (SHT). If A is the number of positions in which these two sequences agree and D (= p - A) is the number of positions in which they disagree, then the autocorrelation function e(r) is defined by: (IO.4.a)

A-D

C("r)= - - .

p

These of course can be written as a convolution in the spectral domain, cf. Section 9.1, Equation (9. Le). We take some more notions from Beker and Piper (1982) as follows:

§10. No waves, no rules, but security

969

Clearly C (t + p) = C (t) for all t, so it suffices to consider only those t satisfying < p. When t = 0 we have in-phase autocorrelation. In this case, clearly, A = P and D = 0, so that C(O) = 1. For t =I 0 we have out-oJ-phase autocorrelation.

o :s t

The following three randomness postulates for binary sequence of period p were proposed by Golomb (1982). (IOA.b)

Rl If p is even then the cycle of length p shall contain an equal number of zeros and ones. If p is odd then the number of zeros shall be one more or less than the number of ones. R2 In the cycle of length p, half the runs have length 1, a quarter have length 2, an eighth have length 3 and, in general, for each i for which there are at least i+ 1 runs, of the runs have length i. Moreover, for each of these lengths, there are equally many gaps and blocks. R3 The out-of-phase autocorrelation is constant.

t.

Sequences fulfilling these "Golomb axioms" are called PN-sequences (pseudo noise sequences). While these axioms represent a most natural mathematisation of properties of a fair coin tossing sequence, i.e. a minimal set of properties, which cryptographically good sequences should bear, they imply the disappointing consequence for a cryptographer, in as far as from a design theoretic point of view they can be classified completely. This result is due to Piper and Walker (1984). For a binary sequence S So

(lO.4.c)

= (St) of period p the circulant (O-l)-matrix SI

Sp_1

A(s) =

SI

Sp_1

So

is the p x p-incidence matrix of a square I-design D(s) with regular cyclic automorphism group. The Golomb axiom R3 then implies that D(s) is a symmetric 2-design with a cyclic Singer group. Then from axiom Rl and R3 it especially follows that D(s) is a cyclic Hadamard design or the complement of a cyclic Hadamard design, having parameters SA (2, 2.:1. + 1; 4.:1. + 3) or SA (2, 2.:1.; 4.:1.-1) where .:I. is the weight of the intersection of any two row vectors of A (s). For instance, the PN-sequences of period 2n - I connected with the Hamming code Hn are generated via a primitive irreducible polynomial as described above. For detailed study of these cf. Jungnickel (1993a), and Chapter VI. A

XIII. Applications of designs

970

class of binary sequences of much higher linear complexity has been proposed by Gordon, Mills and Welch (1962) using the algebraic geometry of towers of extension fields, cf. §VI. 17. By completely different methods and criteria, other sequences of provably high complexity have been proposed, cf. Blum, Blum and Shub [1986]. The class of deterministic random sequences leaves room for many such constructions, cf. Beth and Dai (1989). Further developments can be found in Chapter 2 of Simmons (1992a). For recent results the reader is referred to the LNCS suite of Proceedings of the conferences CRYPTO, ASIACRYPT and EUROCRYPT, starting with Beth (1983a). Er hatte seine Kapazitat zu denken iiberschritten oder vielleicht konnte dort kein Mensch weiterdenken, wo er gewesen war. Oben, im Kopf, an seiner Schadeldecke, k1ickte etwas, es klickte beangstigend und harte nicht auf, einige Sekunden lang. Er meinte, irrsinnig geworden zu sein, und umkrallte sein Buch mit den Handen. Er lieB den Kopf vorniiber sinken und schloB die Augen, ohnmachtig bei vollem BewuBtsein. Er war am Ende. Ingeborg Bachmann, Das dreij3igste lahr ll

EPILOGUE Ach, es war noch allerlei; aber eigentlich haben Sie nichts versaumt. (Fontane)

II

Bachmann (1993).

Appendix. Tables Questo non piccioi libro e tutto pieno ...

(da Ponte)

In what follows we give a collection of tables of designs which may be used as a quick reference to the state of knowledge on certain parameter sets within a feasible range. In our experience such listings have proved helpful for both theoretical and practical purposes (e.g. in statistics). Naturally such a rather small collection always implies a selection; we think we have made a useful choice, and we hope that the reader will have access to the quoted main sources. In particular, a vast collection of tables is given in the recent CRC handbook of combinatorial designs edited by Colboum and Dinitz (l996a). We urge the reader to consult this handbook and its electronic update Colboum and Dinitz (1998) whenever he or she cannot find the desired information in the small set of tables presented here. No serious design theorist should be without this collection. The compilation we present would not have been possible without the support of several friends and colleagues, among whom we are indebted mainly to Andries Brouwer, Charles Colboum and Alexander Rosa as well as the late Haim Hanani.

§1. Block Designs We here present a table of block designs S)..(2, k; v) with k ~ vl2 and replication number r in the range from 3 to 17. This table is the most recent update of Hall's renowned Table 1 in his book Combinatorial theory originally published for r ~ 15 in 1967, with an extended range up to r ~ 20 in the 1986 edition, see Hall (1986). In our presentation, we rely heavily on the Tables ofparameters of BIBD's with r ~ 41 including existence, enumeration and resolvability results due to R. Mathon and A. Rosa which first appeared in 1985, see Mathon and Rosa (1985b, 1990); the most recent version can be found in Section 1.1.3 of Colboum and Dinitz (1996a). In Table A 1.1 below, we have listed all admissible parameter quintuples (v, b, r, k, A) with r ~ 17,

971

L

972

Appendix. Tables

i.e. those satisfying the necessary existence conditions (I.2.ll.a, b) and Fisher's inequality II.2.6. The parameter quintuples are ordered lexicographically with respect to r, k and J... (in this order); therefore, our enumeration coincides with that of Mathon and Rosa. We note that the tables of Hall go beyond the other tables mentioned by giving specific constructions, but warn the reader that the numberings do not coincide, since Hall does not include parameters of multiples of known designs. The column ND gives the number of non-isomorphic designs with the parameters in question or a lower bound for this number; similarly, the column NR contains information on the number of non-isomorphic resolutions of designs S),.(2, k; v). Note that this number is, in general, larger than the number of non-isomorphic resolvable designs with the given parameters. Here the entry "-" indicates that no resolution is possible, since k does not divide v. If the existence of a design (or a resolution) is - to our knowledge - still in doubt, we indicate this by writing a question mark "?" in column ND or NR, respectively. If the non-existence of a symmetric design is proved by Schutzenberger's theorem or by the Bruck-Chowla-Ryser theorem, we indicate this by "tI II.3.9" or by "tlIIA.6", respectively, in the column Comments and References. Similarly, if the existence of a quasiresidual design (cf. II.8.13) is excluded by Proposition 1.2.7 or by the Hall-Connor theorem II.lO.6, since the corresponding "master design" does not exist, we indicate this fact by "tlRes(#x)". Finally, if the design under consideration cannot be resolvable since it satisfies r = k +J... and would thus be affine by Bose's theorem whereas its parameters are not of the form (II.8.7 .a), we refer to this fact by the entry NR II.8.7. In a similar way, the entry Res(#x) or Ind(#x) indicates that the design in question may be obtained as a residual or an induced subdesign from the symmetric design No. x; cf. II.8.13, II.8.l4. By Ext(#x) we denote the (unique) extension of design No. x, which is a Hadamard design, cf. Theorems.1.9.9 and II.8.lO. An s-fold (quasi-)multiple of design No. x is denoted by s#x. We also indicate special cases of infinite series (like projective or affine geometries, Hadamard designs belonging to a Paley or a twin prime power difference set, etc.) by appropriate symbols (such as PG, AG, Paley, TPP, ... ).

In contrast to Hall's table, we usually refrain from giving exact constructions in general, there are many solutions. Nevertheless, we indicate at least one method or source of construction in the column Comments and References, preferably from the present book. If a design may be obtained as a multiple of a smaller one, this is indicated by the appropriate entry s#x; however, we will include an alternative construction for a simple design, if such a construction is known (to us) and reasonably easy. We also give references for further details in particular concerning the numbers ND and NR. Once again, we refer the reader to Section 1.1.3 of Colbourn and Dinitz (1996a) for corresponding information on designs in the range 18 :s r :s 41.

(

Table A1.I. Existence of designs with r::: 17 No.

v

b

r

k

A

1 2 3 4 5 6 7 8 9 10 11 12

7 9

7 12

13

13

14 15 16 17 18 19 20 21 22

6 16 21 11 13 7 10 25 31 16 15 8 15 36 43 22 15 9 25

10 20 21 11 26 14 15 30 31 16 35 14 21 42 43 22 15 24 50

3 4 4 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8

3 3 4 3 4 5 5 3 3 4 5 6 6 3 4 5 6 7 7 7 3 4

1 1 1 2 1 1 2 1 2 2 1 1 2 1 3 2 1 1 2 3 2 1

1 1 1 1 1 1 1 2 4 3 1 1 3 80 4 0 0 0 0 5 36 18

23 24 25

13 9 21

26 18 28

8 8 8

4 4 6

2 3 2

2461

13

ND

11 0

NR

Comments and references PG(2, 2); cf. 1.4.4 AG(2, 3) Res(#3); cf. 1.2.9 PG(2, 3); cf. 1.2.3 Res(#7), NR II.8.7; cf. 1.5.14 AG(2, 4) Res(#6); cf. IV.5.19, VIII.4.12 PG(2, 4); cf. IV.5.19, Vill.4.12 Paley; cf. III.9.6, IV.7.1O, VI. 1.12 ill.9.6; cf. Mathon, Phelps and Rosa (1983) 2#1 or dev({O, 1, 3), {O, 2, 3}) mod 7; cf. Nandi (1946a) Res(#13), (Vill.1.6.a); cf. Nandi (1946a) AG(2, 5) Res(#12); cf. MacInnes (1907) PG(2, 5); cf. MacInnes (1907) I.9.7, VI. 1.3; cf. Hussain (1945) I.5.8, ill.9.3; cf. Mathon, Phelps and Rosa (1983) AG2(3, 2), Res(#20), Ext(#l); cf. Gibbons (1976) 'i!Res(#19) 'i!Res(#18) 'i! II.4.lO 'i! II.3.9 PG2(3, 2); cf. Nandi (1946a) Ind(#40); cf. Mathon and Rosa (1977) VII.3.2; cf. Kramer, Magliveras and Mathon (1989) and Spence (1996) 2#3 ordev({O, I, 3, 9), (0,2,5, 6l) mod 13; cf. Pietsch (l994) Ind(#41), VIL1.6, VII.5.2; cf. Gibbons (1976) Connor (1952)

=

0 1

=

=

7 1 0 0

9

(cant.)

Table A1.I. (cant.) v

b

r

k

;.,.

ND

26 27 28 29 30 31 32 33 34 35

49 57 29 19 10 7 28 10 46 16

56 57 29 57 30 21 63 18 69 24

8 8 8 9 9 9 9 9 9 9

7 8 8 3 3 3 4 5 6 6

1 1 2 1 2 3 1 4 1 3

1 1 0 ;:::1.1 x 109 960 10 ;:::145 21

36 37 38 39 40 41 42 43 44 45 46

28 64 73 37 25 19 21 6 16 41 21

36 72 73 37 25 19 70 20 40 82 42

9 9 9 9 9 9 10 10 10 10 10

7 8 9 9 9 9 3 3 4 5 5

2 1 1 2 3 4 1 4 2 1 2

7 1 1 4 78 6 ;:::2 x 106 4 22859

47 48 49 50

11 51 21 36

22 85 30 45

10 10 10 10

5 6 7 8

4 1 3 2

4393

No.

NR

Comments and references AG(2, 7) = Res(#27); cf. Hall (1953, 1954a) PG(2, 7); cf. Hall (1953, 1954a)

;:::7 0

? 18920 0 1

~192

1 ~1O

~5

;:::22998

11 II.4.6 ill.9.5, VII.2.2; cf. Colbourn and Dinitz (1996a) Ind(#54), VII.4.9; cf. Colbourn, Colbourn, Harms and Rosa (1983) 3#1 or dev({O, I, 2), {O, I, 4). {O, 2, 4)) mod 7; cf. Morgan (1977b) VII.3.2; cf. Brouwer (1989) Res(#41), NR II.8.7; cf. van Lint, van Tilborg and Wiekema (1977) would have trivial group, see Siftar and Shita (1991) Res(#40), II. 10.8, VII.5.2; cf. Colbourn and Dinitz (1996a) and van Rees (1989) Res(#39), NR II.8.7; cf. Assmus, Mezzaroba and Salwach (1977) AG(2, 8)=Res(#38); cf. Hall, Swift and Walker (1956) PG(2, 8); cf. Hall, Swift and Walker (1956) VI.2.13; cf. Assmus, Mezzaroba and Salwach (1977) Hall (1986); cf. Denniston (1982) Paley, see VI. 1.12; cf. Gibbons (1976) 1.6.8; cf. Wilson (1974d), Dejter, Franek and Rosa (1996) 2#4; cf. Kageyama (1972), Gronau and Prestin (1982) 2#5, VII.5.2; cf. Pietsch (1994) III.9.6; cf. Mathon(1995) 2#6 or dev({3, 6, 7, 12, 14}. {7, 9, 14, 15, 19}); cf. Topalova (1998a) Ind(#63), VII.5.2; cf. Colbourn and Dinitz (1996a)

? 3809 0

0

Res(#54), NR II.8.7; cf. Spence (1992) l1Res(#53)

51 52 53 54 55

81 91 46 31 12

90 91 46 31 44

10 10 10 10 11

9 10 10 10 3

1 1 2 3 2

7 4 0 151

7

~106

~2

56

12

33

11

4

3

~17172470

:::1

57 58 59 60 61 62 63 64 65 66 67

45 12 45 100 111 56 23 25 13 9 7

99 22 55 110 111 56 23 100 52 36 28

11 11 II 11 11 11 11 12 12 12 12

5 6 9 10 11 11 11 3 3 3 3

1 5 2 1 1 2 5 1 2 3 4

~16

?

68 69 70 71

37 19 13 10

111 57 39 30

12 12 12 12

4 4 4 4

1 2 3 4

72 73 74

25 61 31

60 122 62

12 12 12

5 6 6

2 1 2

~118884

75

21

42

12

6

3

:::236

~II.3.9

11603 ~16

0 0

0

~5

1102 ~1014 ~92714

22521 35

AG(2, 9) = Res(#52); cf. Lam, Kolesova and Thiel (1991) PG(2, 9); cf. Lam, Kolesova and Thiel (1991)

~395

~3 ~423 ~3702

13769944 ~748

Vill.l.14; cf. Spence (1992) Ind(#84); cf. Curran and Vanstone (1988/89), Royle (1989) IX.5.8; cf. Kageyama (1972), Colbourn and Dinitz (1996a) VII.7.2; cf. Mathon and Rosa (1985a) Res(#63), Ext(#7); cf. IY.7.l2, Pietsch (1994) Res(#62), NR 11.8.7; cf. Denniston (1980a) ~Res(#61)

Lam, Thiel and Swiercz (1989) II.7.15; cf. Janko and van Trung (1986) Paley, see V1.1.l2; cf. Spence (l995a) VII.4.6; cf. Wilson (1974d) 2#8, VII.5.2; cf. Mathon and Rosa (1985a) 3#2, VII.5.2; cf. Mathon and Lomas (1992) 4#1 or dev({O, 1,2), {O, 1,3), (O, 1, 4), (O, 2, 4}) mod 7; cf. Gronau and Prestin (1982) VIL3.2; cf. Colboum (1980) VII.5.2; cf. Mathon (1995) 3#3, Ind(#97), VII.5.2; cf. Spence (1991a) 2#10 or dev({O, 1,2,4), (O, 1,3, 6), {O, 1, 4, 6}) mod 10; cf. Denny (1998) 2#11, VII.5.2; cf. Topalova (1998b)

? ~72

2#12 ordev({J,5, 11,24,25,27), {4, 6, 7, 20, 26,30)) mod31; cf. lungnickel and Vedder (1987) dev((0,2, 10, 15, 19,20), {O, 3, 7, 9, 10, 16}) mod 21; cf. Kapralov and Topalova (1998) (cant.)

Table A1.I. (cant.) No.

v

b

r

k

A

ND

12 12 12 12 12 12 12 12 12 12 13

6 6 8 9 10 11 12 12 12 12 3

4 5 4 3 2 1 1 2 3 4

:::111 :::2572156 ? :::3375 0 :::1 :::1 0 :::3752 0

76

16

77

13

78 79 80 81 82 83 84 85 86

22 33 55 121 133 67 45 34 27

32 26 33 44 66 132 133 67 45 34 117

87

40

130

13

4

88 89 90

66 14 27

143 26 39

13 13 13

6 7 9

91 92 93 94 95 96 97 98 99 100

40 66 144 157 79 53 40 27 15 22

52 78 156 157 79 53 40 27 70 77

13 13 13 13 13 13 13 13 14 14

10 11 '12 13 13 13 13 13 3 4

NR

:::1 :::1

:::10 11

:::909

:::107

:::2

1 6 4

:::1 :::17896 :::8071

? 0 68

3 2 1 1 2 3 4 6 2 2

? :::2 ? ? ?;2 0 ?;389 208310 :::685521 ?;7921

0 0 ?

:::21

Comments and references 2#13; cf. Jungnicke1 (1985) Ind(#98), VII.5.2; cf. Pietsch (1994) cf. van Rees (1996), McKay and Radziszowski (1996) Res(#84); cf. Mathon and Spence (1996) llRes(#83) AG(2, 11) = Res(#82) PG(2,11) II II.4.6 VL9.2; cf. Mathon and Spence (1996) II II.3.9 AGI(3, 3); cf. Mathon, Phelps and Rosa (1983), Colbourn, Magliveras and Mathon (1992) PGI (3,3), VII.7.5; cf. A.E.Brouwer (personal cornm.), Colbourn and Dinitz (1998) VIll.1.l0 Res(#98), NR II.8.7; cf. Pietsch (1994) AG2(3, 3), Res(#97); cf. Spence (1991a), Lam and Tonchev (1996) NRII.8.7 Res(#95), NR II.8.7

Aschbacher(1971) II IIA.6 PG2(3, 3); cf. Spence (1991a) Paley, see VI. 1.12; cf. Spence (1995a) 2#14, Ind(#140); cf. Mathon and Rosa (1985a) IX.5.7; cf. Franek, Mathon et aI. (1990)

101 102 103

8 15 36

28 42 84

14 14 14

4 5 6

6 4 2

2310 831 :::::5

104 105 106 107 108

15 85 43 29 22

35 170 86 58

6 7 7 7 7

5 1 2 3 4

:::::117

44

14 14 14 14 14

109 110 111 112 113 114 115 116 117 118 119 120 121 122 123

15 78 169 183 92 31 16 11 7 6 16 61 31 21 16

30 91 182 183 92 155 80 55 35 30 60 183 93 63 48

14 14 14 14 14 15 15 15 15 15 15 15 15 15 15

7 12 13 14 14 3 3 3 3 3 4 5 5 5 5

6 2 1 1 2 2 3 5 6 3 1 2 3 4

124 125

13 11

39 33

15 15

5 5

5 6

4

? :::::2

2#15; cf. Mathon and Rosa (1985a) IX.8.5, Ind(#141); cf. Tiessen and van Rees (1997) Vrn.3.13; cf. Jungnickel (1990d), Wertheimer (1991) Ind(#142); cf. Mathon (1995)

? :::::4 :::::1 :::::3393 :::::57810 0 :::::1 :::::1 0 :::::6 x 10 16 :::::10 13 :::::436800 109 6 :::::6 x 105 :::::10 >1

:::::i09 :::::11

:::::30 :::::127

:::::1

0 :::::6 x 105

VII.1.6, Vill.3.13; cf. Jungnickel (1990d), Abel (1994) VII.5.2 dev({O, 2, 6, 8, 9, 10, 13}, to, 3, 5, 6, 12, 13, 17}) mod 22; cf. Tiessen and van Rees (1997) 2#20, Ind(#143); cf. Denny (1998) jJRes(#I13) AG(2, 13) = Res(#112) PG(2.13) jJ II.3.9 PGI (4,2); cf. Mathon and Rosa (1985a) VII.4.9, VII.5.2; cf. Mathon and Rosa (1985a) VII.3.1, VII.5.2; cf. Mathon and Rosa (1985a) #9 + #31; cf. Colboum and Dinitz (1996a) 3#4; cf. Mathon and Rosa (1985a) 3#5, Ind(#170); cf. Mathon and Rosa (1985a) VII.3.5, VII.3.6; cf. Colboum (1980) VII.5.2 3#6; cf. Colboum and Dinitz (1996a) VII.5.2, Ind(#171); cf. Brouwer and Wilbrink (1984) VII.5.2; cf. Tonchev and Raev (1982a) 3#7, VII.5.2; cf. Jungnickel (1985) (cont.)

Table AI.I. (cont.)

!

!

No.

v

b

r

k

).

ND

126 127 128 129

76 26 16 91

190 65 40 195

15 15 15 15

6 6 6 7

j

3 5 1

::::1 ::::1 ::::15 ::::2

130 131

16 32

30 35

15 15

8 9

7 6

::::92436200 ::::104

132 133 134 135 136 137 138 139 140 141 142 143 144

136 46 28 56 91 196 211 106 71 43 36 31 33

204 69 42 70 105 210 211 106 71 43 36 31 176

15 15 15 15 15 15 15 15 15 15 15 15 16

10 10 10 12 13 14 15 15 15 15 15 15 3

1 3 5 3 2 1 1 2 3 5 6 7

? ? ::::3 ::::4 0 0 0 0 ::::8 0 ::::25634 ::::1266891 ::::10 13

145 146 147 148 149

9 49 25 17

48 196 100 68 52

16 16 16 16 16

3 4 4 4 4

4 1 2 3 4

13

16585031 ::::769 ::::19 ::::1 ::::2462

NR

?

::::5

0 0

::::529 ::::9

Comments and references VIII. 1.5 VIII.1.18 VII.5.2, Ind(#176); cf. Mathon (1995) dev({O, 10,27,28,31,43,50), {O, 11,20,25,49,55, 57}, (O, 13,26,39,52,65, 78}) mod 91; cf. Bhat-Nayak, Kane et al. (1983) AG3(4, 2), VII.5.2; cf. Kocay and van Rees (1991) Res(#142); cf. Bussemaker, Mathon and Seidel (1981)

van Lint, Tonchev and Landgev (1990) Res(#140); cf. Haemers (1980) :ilRes(#139) :ilRes(#138) :il II.4.lO :il II.3.9 Haemers (1980) :il II.4.6 II.3.19, V1.9.3; cf. Spence (1995c) PG3(4, 2); cf. Norman (1976) 1.6.11, Vil.3'!, Vil.4.I; cf. Mathon and Rosa (I985b), Tonchev and Vanstone (1992), Buratti and Zuanni (1998) 4#2, VII.5.2; cf. Colboum and Dinitz (1996a) VII.3.5; cf. Colboum and Dinitz (1998) 2#22, VII.5.2 (simple) Ind(#185) 4#3, VII.5.2; cf. Colboum and Dinitz (1996a)

150 151 152 153 154 155 156

9 65 81 21 49 113 57

36 208 216 56 112 226 114

16 16 16 16 16 16 16

4 5 6 6 7 8 8

6 1 1 4 2 1 2

:::::1.25 x 108 :::::2 ? :::::1 :::::73 ? :::::1362

157 158 159 160 161

29 17 145 25 33

58 34 232 40 48

16 16 16 16 16

8 8 10 10 11

4 7 1 6 5

:::::2 :::::11 ? :::::43 :::::19

162 163 164 165 166 167 168 169 l70 172

177 45 65 105 225 241 121 81 61 49 41

236 60 80 120 240 241 121 81 61 49 41

16 16 16 16 16 16 16 16 16 16 16

12 12 13 14 15 16 16 16 16 16 16

1 4 3 2 1 1 2 3 4 5 6

? :::::1 ? ? ? ? ? ? :::::6 :::::12146 :::::115307

173

18

102

l7

3

2

:::::4 x 10 14

:::::1

174

52

221

17

4

:::::206

:::::30

171

:::::1 :::::73

2#24, VII.5.2; cf. Colboum and Dinitz (1996a) VII.7.2; cf. Co1boum (1980) Example VII.1.8 2#26; cf. Jungnickel (1986) 2#27 or dev(±{O, 1,7, 19,23,44,47, 49}) mod 57; cf. Jungnickel and Vedder (1987) VII.5.2; cf. Colboum and Dinitz (1996a) VII.5.2, Ind(#186); cf. Tonchev and Raev (1982b)

0

0

VII.5.2, Ind(#I72); cf. Colboum and Dinitz (1996a) Ind(#171), NR II.8.7; cf. Brouwer and Wilbrink (1984) Ind(#170) NR II.8.7

?

Landjev and Topalova (1997) Brouwer and Wilbrink (1984); cf. KOlmel (1994) Bridges, Hall and Hayden (1981); cf. Colboum and Dinitz (1996a) VIlA.9, Hanani (1974a); cf. Mathon and Rosa (1985a) VII.7.5; cf. Colboum and Dinitz (1998) (cont.)

Table At.t. (cont.) No.

v

b

r

k

A

ND

NR

175 176

35 18

119 51

17 17

5 6

2 5

2:1 2:3

2:1 2:2

177 178 179

35 120 18

85 255 34

17 17 17

7 8 9

3 1 8

2:2 2:94 2:103

2:1 0

180 181 182

52 120 256

68 136 272

17 17 17

13 5 16

4 2 1

2:1 0 2:189

0 0 2:189

183

273

273

17

17

184 185

137 69

137 69

17 17

17 17

2 4

0 2:4

186

35

35

17

17

8

2:108131

2:22

?

Comments and references VII.7.3 dev({oo, 0, 2, 8, 9, 12}, {O, 1, 2, 4, 7, 15J, {O, 1,3,7,8, 12}) mod 17; cf. Kageyama (1983) Abel (1994) Penttila, Royle and Simpson (1996), Seiden (1963) NR II.S.7; cf. Bussemaker, Mathon and Seidel (1981) Ind(#185), NR II.8.7 :1Res(#184) AG(2, 16) = Res(#183); cf. Colboum and Dinitz (1996a) PG(2, 16); cf. Colboum and Dinitz (1996a) :1 II.4.6 Shrikhande and Singhi (1975), Whiteman (1988), Colboum and Dinitz (1998) TPP, see VI.8.2; cf. Spence (J995c)

(

§2. Symmetric designs

981

§2. Symmetric Designs

In this section we provide further information on the existence problem for symmetric designs, going beyond the range covered in § 1. Let us begin by listing the known families of symmetric designs; we remark that Series 3, 4 and 5 below were first obtained in an unpublished manuscript of R. M. Wilson with the title Symmetric group divisible designs. Moreover, some of the following series belong to families of difference sets; these will be discussed in §3.

. qd+l_l k qd-l_l d-l h . NaturaI Senes. v = -q1-' =~ n = q , were q IS a q- 1 ' A = --1-' qprime power and d an integer 2:2. These parameters are realised by the classical designs PGd -1 (d, q), see Proposition 1.2.16. There are many other examples having the same parameters; see, for instance, Theorem XII.2.23 and Corollary X.9.20.

Hadamard Series. v = 4n - 1, k = 2n - 1, A= n - 1. Existence is conjectured for all positive integers n; some constructions were presented in sections 1.9 and V1.8. There is an extensive literature on Hadamard designs and Hadamard matrices; we refer the reader to the books by Wallis, Street and Wallis (1972), Geramita and Seberry (1979) and Agaian (1985) and to the recent survey by Seberry and Yamada (1992). Extensive tables regarding the existence of (various special types of) Hadamard matrices may be found in §IV.24 of Colboum and Dinitz (1996a). In what follows, we only point out a few results. Some particularly important early constructions are due to Williamson (1944, 1947), Baumert and Hall (1965) and Turyn (1972) who constructed the first infinite series of "Williamson matrices". For along time, the smallest open case was n = 67; Hadamard designs of this order as well as examples for n = 103, 134 and 268 are now known, see Sawade (1985). Thus the smallest open c,ase is at present n = 107, and the next ones are n = 167 and 179. Given any odd positive integer t , thereexistsaHadamarddesignofordern = 2S tforalls 2: L210g2 (t-3)J by an impOltant result of Seberry (1976); see also Street and Street (1987) for a proof of her result. Her bound was recently improved to s 2: 4r ~ log2«t - 1)/21 + 2 by Craigen (1995) who also gave several other constructions and a table of best known exponents s for various values of t. A further improvement to s 2: 4L log2i~-I) J + 6 is due to Craigen, Holzmann and Kharaghani (1997). A similar result for regular Hadamard matrices is due to Craigen and Kharaghani (1994). Some powerful new constructions for Hadamard designs were given by Yamada (1989) and Miyamoto (1991). In particular, Yamada constructed such designs for n = 127,151,233 and 466, and Miyamoto obtained the cases n = 219,249, 267, 269, 373 and 445. Dokovic (1992a, b, 1994) settled three

Appendix. Tables

982

further small cases, namely n = 163,239 and 463. According to Table IV.24.33 of Colboum and Dinitz (1996a), there remain 48 odd integers n :::: 1000 for which the existence of a Hadamard matrix of order 4n is still in doubt. In Table A2.1 below we exhibit these open cases; for each value n, the integer in brackets gives the smallest t for which a Hadamard matIix of order 2t n is known.

Table A2.1. Values of n ::: 1000 for which the existence of a Hadamard matrix of order 4n is undecided 107 (3) 347 (3) 523 (3) 653 (3) 787 (3) 907 (3)

167 (3) 359 (4) 537 (3) 659 (4) 789 (3) 917 (3)

179 (3) 419 (4) 571 (3) 669 (3) 823 (3) 919 (3)

191 (3) 443 (3) 573 (3) 719 (4) 839 (4) 933 (4)

223 (3) 479 (4) 599 (5) 721 (3) 853 (3) 947 (3)

251 (3) 487 (3) 631 (3) 739 (5) 859 (3) 955 (3)

283 (3) 491 (5) 643 (3) 749 (4) 863 (3) 971 (5)

311 515 647 751 883 991

(3) (3) (3) (3) (3) (3)

Series 1. Let q be a prime power and d a positive integer. Then there exists a symmetric design with parameters v = qd+1 1 + q d+1 (

q-l

1)

,

d qd+! - 1 k =q . " ' - - - q-l

and

qd _ 1 A=qd. _ _

q-l

(and hence n = q2d). This series is due to Wallis (1971) and can also be realised by the McFarland difference sets; see Corollary II.8.17 and Theorem VI.9.2. Series 2. v = 4t 2 , k = 2t 2 - t, A = t 2 - t and hence n = t 2 ; existence is conjectured for all positive integers t. SymmetIic designs with parameters of this type were first studied by Menon (1962). Examples are known to exist whenever 2t is the order of a Hadamard matrix (see Corollary II.8.l7) and if both 2t - 1 and 2t + 1 are prime powers, cf. Wallis, Street and Wallis (1972), pp. 341-346 and Brouwer and van Lint (1984), where also an example with t = 7 is constructed. Finally, a large class of examples is provided by Hadamard difference sets, see Theorem VI.12.1S. Series 3. Let d be a positive integer. Then there exists a symmetric design with parameters

(and hence n = 32d ). There are corresponding difference sets due to Spence (1977), see Theorem VI.9.3.

983

§2. Symmetric designs

Series 4. Let q and n be prime powers satisfying n = q2 exists a symmetric design with order n and parameters

+ q + 1. Then there

(so that v = kJ... + 1). This series is due to Shrikhande and Singhi (1975); see also Whiteman (1988) or Theorem 17.5.5 of Hall (1986).

Series 5. Let q be a prime power and d a positive integer. Then there exists a symmetric design with parameters . 2q(qd - 1) V=

q-l

+ 1,

k = qd

A = qd-l

and

(and hence n = qd (q + 1)/2). This series is due to Brouwer (1983); see also de Launey (1986) or Theorem 17.5.7 of Hall (1986).

Series 6. Let q be a prime power and d a positive integer, and assume that q - 1 is the order of a projective plane. Then there exists a symmetric design with parameters

v = qd+1 _ q

+ 1,

k = qd

and

A = qd-I

(and hence n = qd _qd-I). This series is due to Mitchell (1979) and Rajkundlia (1978, 1983); generalising the work of Beker and Piper (1977).

Series 7. Let p and q be prime powers satisfying q = (pd+1 - l)/(p - 1) for some positive integer d. Then there exists a symmetric design with parameters k =q 2111-1 p d

an

d

1\.= q 1

2m-2 d( d 1) P pp-l

(and hence n = q2111-2 p2d) for every positive integerm. The smallest interesting case is p = 2,d = 1, q = 3, m = 2; this gives a symmetric (160, 54, 18)-design which is due to Spence, Tonchev and van Trung (1993). This construction was generalised to the family with p = 2 above by Spence (1993a) and to primes p by Jungnickel and Pou (1994); the final generalisation is due to lonin (1998a, 1999a). It can be obtained by a construction combining certain generalised balanced weighing matrices and McFarland difference sets.

Appendix. Tables

984

Series 8. Let p and q be prime powers satisfying q = pd+l + P - 1 for some positive integer d. Then there exists a symmetric design with parameters pd(q2m _ 1) V=

(p - 1)(pd

+ 1) ,

and

(and hence n = q2m-2 p2d) for every positive integer m. The smallest interesting case is p = 2, d = 1, q = 5, m = 2, giving a symmetric (416,166, 66)-design; this construction is also due to louin (1998a, 1999a). It combines generalised balanced weighing matrices and the complements of McFarland difference sets.

Series 9. Let q be a prime power satisfying q = (3 d +1 + 1) /2 for some positive integer d. Then there exists a symmetric design with parameters

(and hence n = 32d q2m-2) for every positive integer m. Again, this construction is due to lonin (1998a, 1999a); this times one combines generalised balanced weighing matrices and Spence difference sets.

Series 10. Let q be a prime power satisfying q = 3d +1 - 2 for some positive integer d. Then there exists a symmetric design with parameters

and 3d (3 2d q 2m-2 - 1)

= --=2-:':(3=-:-d---:1-:)--'(and hence n = 32d q2m-2) for every positive integer m. Again, this construction A

is due to lonin (1998a, 1999a); this time one combines generalised balanced weighing matrices and the complements of Spence difference sets.

Series 11. Let d be a positive integer. Then there exists a symmetric design with parameters

v=

73 d +1 - 64

9

(and hence n = 64· 73 d -

1

).

k = 73 d

and

A = 9· 73 d -

1

This series is due tolonin (1999b).

§2. Symmetric designs

985

Series 12. Let q = 2P - 1 be a Mersenne prime, and let d be a positive integer. Then there exists a symmetric design with parameters

2(q

+ 1)«q + 1)2d -

v=l+

q+

1)

2'

k = q(q

+ 1)2d -

q

q+2

and

(and hence n = q(q + 1)2d~1 /2). For q = 3, this series is due to Flanning (1995); the general case was given by lonin and Shrikhande (1998) in terms of the complementary designs.

Series 13.

v = 22d+4(22d +2 - 1)/3,

k = 22d +I(22d+3 + 1)/3,

A = 22d+I(22d+1 + 1)/3,

n = 22d+2 ,

where d is a non-negative integer. This series is realised by the difference sets due to Davis and Jedwab (1997a); see Theorem VI.9.4.

Series 14. v

= 4q 2d q2d -

1

q2 -1 '

A=q2d~l(q_l)q

k

= 2d~1 (2(q2d -

2d~1

q +1

q

and

+1

1)

+ 1)

'

n=q4d~2,

q+l where q is the square of an odd prime or a power of either 2 or 3. This series is realised by the difference sets of Chen (1997, 1998); see Theorem VI.9.5.

Series 15.

v=

h«2h - 1)2m - 1)

, k = h(2h _ 1)2m-l, h-l A = h(h - 1)(2h - Ipm-2 and n = h2(2h _ 1)2m-2, where m and d are positive integers, h = ±3 ·2d and 12h - 11 is a prime power. This series is due to lonin (1998b, 1999a), who obtained it by using regular Hadamard matrices (resp. Hadamard difference sets). The next two series are due to lonin (1999a); they are constructed by using the Davis-Iedwab difference sets (Series 13) and their complements.

986

Appendix. Tables

Series 16.

where In and d are positive integers and q = 22d~+1 is a prime power. Series 17.

where In and d are positive integers and q = 22d+3

-

3 is a prime power.

The following two series belong to cyclotomic difference sets, see §3. Series 18. v = 4t 2 + 1, k = t 2 , J... = (t2 - 1)/4, n = (3t 2 + 1)/4, where tis an odd positive integer and v a prime power. These designs may be obtained by using the biquadratic residues modulo v as a difference set; see Theorem VI.8.11 or Hall (1986). Series 19. v = 4t 2 + 9, k = t 2 + 3, A = (t 2 + 3)/4, n = 3(t 2 + 3)/4, where t is an odd positive integer and v a prime power. These designs may be obtained by using the biquadratic residues modulo v together with 0 as a difference set; see Theorem VI.8.11 or Hall (1986). Unfortunately, for several of the preceding series one only knows a few examples; indeed, it is an open problem whether or not the series 4, 18 and 19 yield infinitely many symmetric designs. Similarly, it is undecided whether or not infinitely many values of q arise in the series 6, 7, 8, 9, 10 and 12. In what follows, we present two tables of parameters for symmetric designs SJ..(2, k; v), that is, we list parameter triples (v, k, J...) subject to the following three conditions: (a) J...(v - 1) = k(k - 1) (arithmetic condition); (b) the number-theoretic conditions provided by Corollary II.3.9 (Schiitzenberger) and Theorem 11.4.6 (Bruck, Chowla, Ryser), respectively;

------------------.,~ §2. Symmetric designs

987

(c) k 2 - k + 1 > v > 2k + 1 (thus we exclude projectiv e planes and Hadama rd designs, see Remark II.3.12).

In Table A2.2, we list all such triples with n = k - A :::: 30, includin g those for which existenc e is still undecid ed; in Table A2.3, we list all such triples for which a symmet ric design is known to exist in the range 31 :::: n :::: 100. In the column headed ND, we give a lower bound for the number of isomorp hism classes of designs with the respecti ve paramet ers. Explicit descript ions of symmet ric designs withn :::: 25 may be found in TableI.5 .25 ofColbo um and Dinitz (l996a).

Table A2.2. Admissible parameter triples for symmetric designs with n :::: 30 n=k-J. ..

v

k

,\.

4 6 7

16 25 37

6 9 9

2 3 2

3 78 4

7 9

31 56

10 11

3 2

151 2:5

9

45

12

3

2:3752

9 9 10

40 36 41

13 15 16

4 6 6

2:389 2:25634 2:115307

11 11

79 49

13 16

2 5

2:2 2: 12146

12

71

15

3

2:8

12 13 13 14 15 15 16 16 16 16 16

61 81 69 121 71 61 154 115 96 85 78

16 16 17 16 21 25 18 19 20 21 22

4 3 4 2 6 10 2 3 4 5 6

2:6

ND

References and comments

?

2:4 ? 2:2 2:1 ? ? 2:2 2:213964 ::-::3

Series 1, Series 2; Hussain (1945) Series 5, Series 6; Denniston (1982) Series 12, VI.2.13; Assrnus, Mezzaroba and Salwach (1977) Series 4, VIII.1.14; Spence (1992) Theorem II.7.15; Janko and van Trung (1986) Series 1, Series 3; Mathon and Spence (1996) PG2(3, 3); Spence (1991a) Series 2, Series 3; Spence (1995c) Bridges, Hall and Hayden (1981); Spence (1993b), Colbourn and Dinitz (1996a) Aschbacher (1971) Brouwer and Wilbrink (1984); KOlmeI (1994) Haemers (1980), Beker and Haemers (1980) Series 6; Landjev and Topalova (1997) Series 4; Colbourn and Dinitz (1998) Janko and van Trung (l985b) Series 5

Series I; Sane (1982) PGz(3, 4); JungnickeI (1984b)

Janko and van Trung (1985a), Tonchev (1987a) (cant.)

988

Appendix. Tables

Table A2.2. (cant.) n =k-J...

v

k

16

70

24

8

::::12

16

66

26

10

::::19

16

64

28

12

::::8784

18 18 19 19 19 19 20 20 21

191 79 211 155

20 27 21 22 25 28 24 25 26

2 9 2 3

21 21 22 22 22

109 85 201 127 97

23 23 25 25 25

101 85 139 121 131

J...

References and comments

ND

Janko and van Trung (1984, 1992), Golemac (1993) van Trung (1982), Bridges (1983); Matulic-Bedenic, Horvatic-Baldasar and Kramer (1995) Series 2; Ding, Houghten, Lam, Thiel and Tonchev (1998)

? ::::1463

Series 6; Held and Pavcevic (1997)

? ? ::::1

301 103 352 253 204

28 36 25 28 33 25 34 27 28 29

6 9 4 5 5 7 15 3 6 11 2 11 2 3 4

25 25 25 25 25

175 156 133 120 112

30 31 33 35 37

5 6 8 10 12

::::2 ::::10 17 ::::1 ? ?

25 25 26 27 27 27 27 27 27 28 28

105 100 127 407 291 177 141 121 111 311 249

40 45 36 29 30 33 36 40 45 31 32

15 20 10 2 3 6 9 13 18 3 4

::::4 ::::1 ? ? ? ? ? ::::1029

Series 18

? ? ::::1 ?

Series 6

::::1 ?

Series 19

? ?

? ? ? ? ? ? Series 1; Sane (1982) PGz(3, 5); Jungnickel (1984b)

VI.8.21

Janko (1999) Series 2

PG3(4, 3); Jungnickel (1984b)

? ? ?

(cant.)

989

§2. Symmetric designs

Table A2.2. (cant.) n=k-J...

28 28 28 28 29 29 30

v

k

J...

171 149 131 113 265 181 239

35 37 40 49 33 36 35

7 9 12 21 4 7 5

ND

References and comments

? ?

?

:::1

Series 5

? ? ?

Table A2.3. Parameters of known symmetric designs with 31 S n S 100 n=k-J...

31 36 36 36 36 37 45 48 49 49 49 54 56 64 64 64 64 64 66 72 73 75 81 81 81 81 81 81

References and comments

v

k

J...

223 176 189 160 144 197 181 253 441 400 196 241 505 640 585 341 320 256 265 721 739 311 891 820 378 364 351 324

37 50 48 54 66 49 81 64 56 57 91 81 64 72 73 85 88 120 121 81 82 125 90 91 117 121 126 153

6 14 12 18 30 12 36 16 7 8 42 27 8 8 9 21 24 56 55 9 9 50 9

Series 13 Series 2 Series 5 Series 6 Series 4 Series 5 Series 1

10

PGz(3,9)

36 40 45 72

Series 1, Series 3

Series 4 IV.8.1O, Higman (1969) Janko (1997) Series 7 Series 2 Series 18 Series 5 Series 6 Series 1 PG2(3,7)

Series 2 Series 6 Series 6 Series 1 PG2(3,8) PG3(4,4)

PG4(5,3)

Series 3 Series 2 (cont.)

Appendix. Tables

990 Table A2.3. (cont.) n =k-A

91 96 100 100 100

v

k

A

References and comments

365 409 621 416 400

169 153 125 166 190

78 57 25 66 90

Series 5 Series 12 Series 6 Series 8 Series 2

§3. Abelian Difference Sets In this section, we provide information on the existence of abelian I difference sets. Let us begin by listing the known series of abelian difference sets. Series 1 (Singer Difference Sets). qd+l

v=

-1

qd-l _

A.=

q -1 '

q-l

1 ,

n=

qd-l,

where q is a prime power and d an integer :::::2; see Theorem VI.l.10. We shall use the notation PG(d, q) for this difference set. Often, there are (many) inequivalent cyclic difference sets with the same parameters, e.g. the GMWdifference sets; see §VI.17. Series 2 (Paley Difference Sets). v = q = 4n - 1,

k

= 2n -

1,

A. = n - 1,

G = (GF(q), +),

where q is a prime power; see Theorem VI.l.12. We will denote this difference set by P(q). Series 3 (Twin Prime Power Difference Sets).

1

Some families of non-abelian difference sets were given in Chapter VI. Kibler (1978) listed non-abelian difference sets with k < 20; see also §IV. 13 of Colboum and Dinitz (1996a).

§3. Abelian difference sets

where both q and q

991

+ 2 are prime powers, and G

= (GF(q), +) EB (GF(q +2), +);

see Theorem VI.8.2. This difference set will be denoted by TPP(q).

Series 4 (Hadamard Difference Sets).

Here G may be any group which is a direct product of an abelian group of order 22d+2 and exponent at most 2d +2 (for some non-negative integer d) with groups of type Z~i where each mi is a power of 3, and groups of type Z!j where the Pj are (not necessarily distinct)odd primes; see Theorem VI.12.1S.

Series 5 (McFarland Difference Sets).

where q is a prime power and d a positive integer. Here G may be any abelian group of order v which contains a subgroup isomorphic to E A (qd+ 1); for q = 4, such difference sets also exist in any abelian group of order v which contains a subgroup of order 2 2d+3 and exponent at most 4. See Theorems VI.9.2 and V1.11.12.

Series 6 (Spence Difference Sets).

where d is a positive integer. Here G may be any abelian group of order v which contains a subgroup isomorphic to E A (3 d + 1); see Theorems VI.9.3 and VI.ll.12. We note that a series of non-abelian difference sets with these parameters which admit -1 as a weak multiplier was constructed by Ma (1989a).

Series 7 (Davis-Jedwab Difference Sets). V = 2 2d +4(22d+2

-1)/3,

k = 22d+l(22d+3

A= 22d+l(22d+l

+ 1)/3,

n = 22d+2 ,

+ 1)/3,

Appendix. Tables

992

where d is a non-negative integer. Here G may be any abelian group of order v which contains a subgroup of order 22d+4 and exponent at most 4, with the exception when the subgroup is Z~ in the case d = 1; see Theorem VI.9.4. Series 8 (Chen Difference Sets).

2d-1 (2(q2d - 1) + 1) q q +1 ' 2d-1 + 1 A=q2d-l(q_1)q and n=q4d-2, q+1

v = 4q2d q2d - 1

k

q2 -1 '

=

where q is the square of an odd prime or a power of either 2 or 3. Here G = K x E, where E is the elementary abelian group of order q2d and where K may be any abelian group of order 4(q2d - 1)/(q2d - 1); see Theorem VI.9.S. Series 9 (Biquadric Residue Difference Sets). Either

v = 4t 2 + 1,

k

= t 2,

A = (t2 -1)/4,

n = (3t 2 + 1)/4

or v = 4t 2 + 9,

k = t 2 + 3,

A = (t2 + 3)/4,

n = 3(t 2 + 3)/4,

where t is an odd positive integer and v a prime power; see Theorem VI.S.l1. Four further series of abelian difference sets constructed by using cyclotomic classes were mentioned in Theorems VI.S.3, VI.S.l1 and VI.S.20; as these constructions seem to yield only very few examples, we will not restate them here. We now give a table of abelian difference sets with 2 ::: n ::: 30

and

k < v /2

which is an updated version of the table given by Jungnickel (1992a). Assmus (19S9) noted that there is much to be said for considering n as the most fundamental parameter for a symmetric design, and we shall follow his approach. The fundamental equation (VI.1.4.a) shows that n(n - 1) is divisible by A; writing n(n - 1) = A/L, the parameters of a symmetric (v, k, A)-design D and its complementary design D' can be written as

(2n

+ A + /L, n + A, A)

and

(2n

+ A + /L, n + /L, /L),

§3. Abelian difference sets

993

respectively. We thus list the possible parameters via the order n, using the divisors A of n(n - 1) in increasing order and stopping with A = n - 1; thus we eliminate complementary designs, begin for each n with the projective plane of this order, and end with the corresponding Hadamard design. We shall list only those parameter quadruples (n, A, v, k) which satisfy (VI.1A.a) and which are not already ruled out by Schiitzenberger's theorem (Corollary II.3.9) or the Bruck-Ryser-Chowla theorem (Theorem IIA.6). For any such set of parameters, we list all abelian groups of the relevant order. Groups are denoted by their type; for instance, (9, 3, 5) stands for Z9 ED Z3 EB Z5. In all cases covered by our table, the existence or non-existence of the respective difference set is decided; the column "Comments and references" contains at least one source for the result in question. Regarding non-existence, many cases can be excluded by the results of Chapter VI. Two frequent arguments are as follows: (a) An entry "t h == -1 mod w" indicates that t would be a multiplier of the difference set in question by Theorem VIA.6; thus the hypothesis of the Mann test (Theorem VI.6.2) would be satisfied for the given values of h and w, leading to a contradiction. (b) An entry "t = m, (x a , yb, zC, ...)" indicates that m would be a multiplier of the difference set in question by Theorem VIA.6 and that the group generated by m has a orbits of size x, b orbits of size y, c orbits of size c, ... on G. The desired contradiction then arises from Theorem VI.2.9, since no possible combination of orbits yields a k-set. In all cases not covered by the results of Chapter VI, the reader is referred to the literature. Quite often a non-existence proof may be found in the book by Lander (1983). This book contains a table covering all cases with k 2': 50; we note that all open cases in this table have been settled by now, see Iiams (1998). An extension of Lander's table for the range 50 < k::: 100 was given by Kopilovich (1989). Most of the open cases in this table have likewise been settled; there remain only three undecided cases with k < 100, see Jungnickel and Pott (1996) and Jungnickel and Schmidt (1997). Recently, a table covering the range 100 < k::: 150 was given by Vera L6pez and Garda Sanchez (1997); ten open cases in their table were settled by Arasu and Sehgal (1998). For explicit listings of small difference sets, the reader should consult Jungnickel and Pott (1996) and Baumert (1971) who listed - up to equivalence - all known cyclic difference sets with k::: 100. If any explicit description is contained in the present book, this will also be indicated.

994

Appendix. Tables

Table A3.I. Admissible parameters for abelian difference sets with n ::: 30 n=k-}"

v

k

}..

Group type

2 3 3

7 13

3

4

4

21

5 5

1 1 2 1

(7) (13) (11) (3,7)

Yes Yes Yes Yes

PG(2, 2), VI.1.3 PG(2, 3), VI. 1.3 P(ll), ill.9.6 PG(2, 4), ill.9.6

4

16

6

2

(16) (8,2) (4,4) (4,2,2) (2,2,2,2)

No Yes Yes Yes Yes

V1.14.20 Series 4 Series 4 Series 4 Series 4, VI. 1.3

4 5 5 6

15 31 19 25

7 6 9 9

3 1 4 3

(3,5) (31) (19) (25) (5,5)

Yes Yes Yes No No

PG(3, 2), TPP(3), VI.2.1O PG(2, 5), VI.2.1O P(19) VI.6.4, VI.5.9 VI.6.4, VI.5.9

6 7 7 7

23 57 37 31

11 8 9 10

5 1 2 3

(23) (3, 19) (37) (31)

Yes Yes Yes No

P(23), VI.2.13 PG(2, 7), VI.2.1O VI.2.13 VI.2.13

7

27

13

6

(27) (9,3)

No No

8 8 9 9

73 31 91 56

9 15 10 11

1 7 1 2

9

45

12

3

(3,3,3) (73) (31) (7, 13) (8,7) (4,2,7) (2,2,2,7) (9,5) (3,3,5)

Yes Yes Yes Yes No No No No Yes

VI.14.26 Bozikov (1985), Arasu (1988b), Wei (1990) P(27) PG(2,8) PG(4, 2), P(31) PG(2,9) VI.6.4 VI.6.4 VI.6.4 VI. 14.20 Series 5 (q = 3, d = 1)

9

40

13

4

9

36

15

6

(8,5) (4,2,5) (2,2,2,5) (4,9) (2,2,9) (4,3,3) (2,2,3,3)

Yes No No No No Yes Yes

PG(3,4) VI.6.4 VI.6.4 VI. 14.20 V1.14.20 Series 6 (d = 1) Series 6 (d = 1), VI.9.S

9 10 10 10 11

35 111 41 39 133

17 11 16 19 12

8 1 6 9 1

(5,7) (3,37) (41) (3, 13) (7, 19)

Yes No No No Yes

11 11

79 49

13 16

2 5

(79) (49) (7,7)

No No No

TPP(5) VI.6.12, VI.7.19 2 1O =-lmod41 VI.5.8 PG(2,11) t = 11, (11, 392)

11

Existence

Comments and references

t = 11, (11,3 2 ,212) VI.14.29

(cont.)

§3. Abelian difference sets

995

Table A3.1. (cont.) v

k

11 12

43 157

21 13

10 (43) 1 (157)

12 12 12 13

71 61 47 183

15 16 23 14

3 4 11 1

13

81

16

3

n=k-J...

13 13 14

69 51 121

17 25 16

14 15

55 241

27 16

15 15 15 16 16 16

71 21 61 25 59 29 273 17 154 18 115 19

)..

Group type

P(43) VI.7.14, VI.7.19

(71) (61) (47) (3,61)

No No Yes Yes

t = 2, (11, 35 2 ) 35 == -1 mod 61 P(47) PG(2,13)

(81) (27, 3) (9,9)

No No No

(9,3,3)

No

(3,3,3, 3)

No

Lander (1983), p.220 Lander (1983), p.220 Bozikov (1985), Arasu (1986) Bozikov (1985), Arasu (1986) Lander (1983), p.183 t = 13, (1 3 ,11 6)

(3,23) (3,17) (121) (11, 11) 13 (5, 11) 1 (241)

16

96

20

4

16 16 16 16

85 78 70 66

21 22 24 26

5 6 8 10

16

64 28

16

63

31

17 17 18

307 67 343

18 33 19

No No No No No No

132 == -1 mod 17 255 == -1 mod 121 2 55 == -1 mod 121 22 == -1 mod 5 VI.6.13, VI.7.19

(71) (61) (59) (3,7, 13) (2,7, 11) (5,23)

No No Yes Yes No No

t = 3, (11, 35 2 )

(32,3) (16,2,3) (8,4,3) (8,2,2,3) (4,4,2,3) (4,2,2,2,3) (2,2, 2, 2, 2, 3) (5,17) (2,3, 13) (2,5,7) (2,3, 11)

No No No No No Yes Yes Yes No No No

VI.I4.20 VI.l4.20 Arasu and Sehgal (l995a) Arasu, Davis et al. (1996) Arasu and Sehgal (1995b) Series 5 (q = 4, d = 1) Series 5 (q = 4, d = 1) PG(3,4) VI.6.4 Lander (1983), p.221 VI.6.4

No No Yes Yes Yes Yes Yes No

VI.l3.7 V1.13.7 VI.l2.2 PG(5,2) TPP(7) PG(2,17) P(67) VI.6.12, VI.7.19

(64) (32,2) all other types 15 (9,7) (3,3,7) 1 (307) 16 (67) 1 all types 12

Comments and references

Yes No

4 12 2

6 10 14 1 2 3

Existence

35 == -1 mod 61 P(59) PG(2,16) VI.6.4 V1.l4.26

(cont.)

Appendix. Tables

996

TableA3.1. (cont.) n=k-A

v

k

A

20 27 35 20 21 22 25 28 37

2 9 17 1 2

19 19 19 19

191 79 71 381 211 155 101 85 75

3 6 9 18

20 20 20

421 139 121

21 24 25

4 5

20

79

39

21 21 21 21 22

131 109 85 83 201

22 22 22 23 23

127 97 87 553 301

Group type

Existence

Comments and references

No No Yes Yes No

t = 3, (11, 95 2) 3 39 == -1 mod 79 P(71) PG(2,19) t = 19, (11,15 14 )

19

(191) (79) (71) (3, 127) (211) (5,31) (101) (5, 17) (3,25) (3,5,5) (421) (139) (121) (11, 11) (79)

No Yes No No No No No No No Yes

26 28 36 41 25

5 7 15 20 3

(131) (109) (5, 17) (83) (3,67)

No Yes No Yes No

19==-lmod5 Series 9 19 == -1 mod 5 19==-lmod5 19==-lmod5 VI.7.14, VI.7.19 269 == -1 mod 139 25 == -1 mod 11 25 == -1 mod 11 P(79) t = 7, (11, 65 2)

28 33 43 24 25

6 11 21 1 2

(127) (97) (3,29) (7,29) (7,43)

No No No Yes No

23 23 24 24 25 25 25 25

103 34 91 45 601 25 95 47 651 26 352 27 253 28 204 29

11 22 1 23 1 2 3 4

(103) (7,13) (601) (5, 19) (3,7,31) all types (11,23) all types

No No No No Yes No No No

25

175

30

5

(25,7) (5,5,7)

No Yes

25

156

31

6

(4,3, 13) (2,2,3, 13)

Yes No

25 25

133 120

33 35

8 10

(7,19) (8,3,5)

Yes No

(4,2,3,5)

No

(2, 2, 2, 3, 5)

No

18 18 18 19 19

Series 9 7 2 == -1 mod 5 P(83) 2==-lmod3 t = 11, (11, 67 2) 224 == -1 mod 97 2==-lmod3 PG(2,23) t = 23, (11, 32, 23 2 , 694 ) Lander (1983), p.221 33 == -1 mod 13 VI.7.14, VI.7.19 32 == -1 mod 5 PG(2,25) VI.2.15 VI.2.14 VI.6.2: s = 6, U = 34 > k, 58 == -1 mod 34 VI.14.16: 53 == -1 mod 7 Series 5 (q = 5, d = 1)

PG(3,5) VI.6.2: p = 5 > s = 3, u = 26, 52 == -1 mod 26 VI.8.21 VI.14.21: s = 4, u* = 30, m=5 VI.14.21: s = 4, u* = 30, m=5 VI.14.16: 5 == -1 mod 6 (cont.)

§4. Small Steiner systems

997

Table A3.1. (cont.) n=k-A

v

k

A Group type Existence Comments and references

112 37 12 all types 10540 15 (3,5,7) 100 45 20 (4,25) (2,2,25) (4,5,5) (2,2,5,5) 99 49 24 (9, 11) (3,3,11) 703 27 1 (19,37) 127 36 10 (127) 103 51 25 (103) 757 28 1 (757) 407 29 2 (11,37) 291 30 3 (3,97) 177 33 6 (3,59)

No No No No No No No Yes No No Yes Yes No No No

141 36 9 (3,47) 121 40 13 (121) (11,11) 111 45 18 (3,37) 107 53 26 (107) 813 29 1 (3, 271) 311 31 3 (311) 249 32 4 (3, 83) 171 35 7 (9,19) (3,3, 19)

No Yes No No Yes No No No No No

28 28 28

149 37 9 (149) 131 40 12 (131) 111 55 27 (3,37)

No No No

29 29 29 29 30 30

871 265 181 115 239 119

1 4 7 28 5 29

Yes No No No No No

25 25 25

25 26 26 26 27 27 27 27

27 27 27 27 28 28 28 28

30 33 36 57 35 59

(13,67) (5,53) (181) (5, 23) (239) (7)(17)

V1.6.4 VI.l4.2I: 53;: -1 mod 21 VI.l4.22 VI.l4.22 VI.l3.12 VI.l3.12 VI.l4.25: qm = 11, rt = 9 TPP(9)

VI.6.12 t = 13, (11,63 2) P(103) PG(2,27)

VI.6.2: 39 ;: -1 mod 37 VI.6.2: 324 ;: -1 mod 97 V1.5.l4: s = t = 3, U = 59, x + 29y + 29z = 33, x 2 + 29y2 + 29z 2 = 45 V1.5.15 PG(4,3)

Lander (1983), p.221 VI.14.16: 39 ;: -1 mod 37 P(107)

VI.7.14, VI.7.19 t = 2, (11, 155 2) VI.6.2: s = 1, 241 ;: -1 mod 249 VI.6.2: s = 1,2 19 ;: -1 mod 171 VI.6.2: s = 1,2 19 ;: -1 mod 171 t = 7;: 2142 mod 149, (11,742) VI.6.2: s = 1,265 ;: -1 mod 131 VI.5.14: s = 3, t = 2, x + 36y = 55, x 2 + 36y2 = 109 PG(2,29)

VI.6.2: 29 == -1 mod 5 t = 29, (11, 15 12 ) VI.6.2: 29 ;: -1 mod 5 t = 2 ;: 362 mod 239, (11, 1192) VI.6.2: 24 ;: -1 mod 17

§4. Small Steiner Systems

In this section we present a list of all triples (t, k, v) satisfying the arithmetic conditions (I.3.2.a) for the existence of an S(t, k; v) in the range 7 :::: v :::: 28. Relying on the original source Doyen and Rosa (1980), we also include the

998

Appendix. Tables

number ND of non-isomorphic systems S(t, k; v) or a lower bound (if such a bound is known to us). The notation Der( ... ) and Ext( ... ) refers to point derivations and extensions, respectively. If the existence of certain systems can be excluded by other results, we give a reference; we also include references with regard to the entries in column "ND". If the existence question is unsettled, this is denoted by a question mark in column "Existence". For a more extensive table, see §I.4.3 of Colboum and Dinitz (l996a). Table A4.1. Steiner systems with n

::s 28

v

S(t,k; v)

Existence

7 8 9 10 12

S(2, 3; 7) S(3, 4; 8) S(2, 3; 9) S(3,4; 10) S(4, 5; 11) S(5, 6; 12)

Yes Yes Yes Yes Yes Yes

1 1 1 1 1 1

13

S(2, 3; 13)

Yes

2

14 15

S(2,4; 13) S(3,4; 14) S(2, 3; 15)

Yes Yes Yes

S(4, 5; 15)

No

111.9.6; cf. Mathon, Phelps and Rosa (1983) 1 PG(2, 3); 1.4.5 4 ill.8.12; cf. Mendelsohn and Hung (1972) 80 1.5.8, VI.9.3; cf. Mathon, Phelps and Rosa (1983) 0 Mendelsohn and Hung (1972)

16

S(2,4; 16)

Yes

AG(2, 4) = Res S(2, 5; 21); IV.5.19,

S(2,6; 16) S(3,4; 16) S(5, 6; 16)

No Yes No

17

S(3, 5; 17) 5(4,5; 17) 5(6,7; 17)

Yes ? No

18

S(4,6; 18) S(5, 6; 18) S(7, 8; 18)

No ? No

19

S(2, 3; 19)

Yes

S(6,7; 19) S(8,9; 19)

? No

20

S(3, 4; 20) S(7, 8; 20) S(8, 9; 19)

Yes ? No

21

S(2, 3; S(2, 5; S(2, 6; S(4, 5; S(8, 9;

Yes Yes No ? ?

11

ND

Comments and references

0 >31300 0

PG(2, 2); 1.404 AG2(3, 2); (1.3A.a), II.8.11 AG(2, 3); 1.4.5

Ext S(2, 3; 9); 111.6.9, IV.2.8 Ext S(3, 4; 10); ill.8.8, IV. 1.2, IV.2.6 Ext S(4, 5; 11); 111.8.8, IY.1.2, IV.2.6

VillA. 12 11.2.7 AG2(4, 2); cf. Lindner and Rosa (1978) Ext S(4, 5; 15) Ext S(2, 4; 16),111.6.9; cf. Witt (1938a)

0

Ext S(5, 6; 16)

0

Witt (1938a)

0

Ext S(6, 7; 17)

::::1.1 x 109 III.9.5, VII.2.2; cf. Colbourn and Dinitz (1996a)

21) 21) 21) 21) 21)

0 >10 17

Ext S(7, 8; 18) 111.6.17; cf. Lindner and Rosa (1976)

0 Ext S(8, 9; 19) ::::2160980 1.6.8; cf. Wilson (1974d) 1 PG(2, 4); IV.5.19, VIllA. 12 0 11.2.7

(cant.)

§5. Infinite series of Steiner systems

999

Table A4.1. (eOlu.) v

S(t,k; v)

Existence

ND

Comments and references

S(3, 4; 22)

Yes

2:119

S(3, 6; 22) S(3, 7; 22) S(5, 6; 22) S(9, 10; 22) S(4, 5; 23) S(4, 7; 23) S(6, 7; 23) S(10, 11; 23) S(5, 6; 24) S(5, 8; 24) S(7, 8; 24) S(l1, 12; 24)

Yes No

1 0

IX.lO.5, Der S(4, 5; 23); cf. Diener (1980), Phelps (1991), Kreher and Frenz (1992) Ext S(2, 5; 21); II1.8.9, ry.J.6, IV.8.1 Ext S(2, 6; 21)

0 1

Der S(5, 6; 24) Ext S(3, 6; 22); III.8.9, IV. 1.6, ry.8.9

S(2, 3; 25) S(2, 4; 25)

Yes Yes

S(2, 5; 25) S(6, 7; 25) S(8, 9; 25)

Yes

26

S(3, S(3, S(3, S(7, S(9,

Yes Yes Yes ?

27

S(2, 3; 27)

22

23

24

25

28

4; 26) 5; 26) 6; 26) 8; 26) 10; 26)

S(4, 5; 27) S(4, 6; 27) S(8, 9; 27) S(10, 11; 27) S(2, 4; 28) S(3, 4; 28) S(5, 6; 28) S(5, 7; 28) S(9, 10; 28) S(ll, 12; 28)

? ? Yes Yes ? ? Yes Yes ? ?

2:3 1

2: 10 14 18

III. 8. 10; cf. Grannell and Griggs (1979) Ext S(4, 7; 23); III.8.9, Iy'1.6, ry.4.18

VII.4.6; cf. Wilson (1974d) VII.3.2; cf. Kramer, Magliveras and Mathon (1989) and Spence (1996) AG(2, 5); cf. MacInnes (1907)

? ? 2:1 2:1 I

IX.lO.l! Der S(4, 6; 27); cf. Hanani (1979) III.6.9, Ext S(2, 5; 25)

2: 1011

AGI (3,3); cf. Mathon, Phelps and Rosa (1983)

? Yes

? Yes

? ? Yes Yes ? Yes ? ?

2:1

>145

2:4~102o 2:1

Der S(5, 7; 28)

VII.3.2; cf. Brouwer (1989) III.6.9; cf. Lindner and Rosa (1978) Denniston (1976)

§S. Infinite Series of Steiner Systems In his classical paper Witt (l938b) gave a list of Steiner systems Set, k; v) which we now reproduce in an updated form; we shall restlict ourselves to the (few) infinite series that are known. The present state of knowledge regarding

Appendix. Tables

1000

Steiner S-designs was discussed in §III.8 (in particular, see Remark ill.8.Il). For more information, including base blocks for the known Steiner S-designs, we refer to §I.4 of Colbourn and Dinitz (1996a).

Table AS.t. Infinite series of Steiner systems Series AG

PG

SG KW

UG AK D

W H STS SQ8 B4 B5 Bk

References

Comments

Parameters S(2, q; qn)

affine geometries AGl(n, q), q a prime power, n::;:2 projective geometries S(2, q + 1; qn+ PGl (n, q), ... + q + 1) q a prime power, n::;:2 spherical geometries, S(3, q + 1; qn + 1) q a prime power, n::;:2 q a prime power, n::;:2, S(3, q + 1; qn + 1) not isomorphic to Series SG S(2, q + 1; q3 + 1) unitals, q a prime power S(3,6; (4m + 2)/3) m::;:3 an integer S(2, 2r; 2r+, + 2r - 2') r < s positive integers S(2, 2e - 1 ; 2e - 1 (2 e - 1» e ::;: 2 an integer Infinitely many examples S(3, 5; v), S(3, 6; v) S(2, 3; v) v == I or 3 (mod 6) 8(3,4; v) v == 2 or 4 (mod 6) 8(2,4; v) v == 1 or 4 (mod 12) 8(2,5; v) t' == 1 or 5 (mod 20) 8(2, k; v) v == 1 (mod k - 1), tI(v - 1) == 0 (mod k(k - 1», t' sufficiently large

1.2.13 1.2.16 III.6.9 Il.7.25 VIII.5.28 Il.7.24 VlII.5.21 VIII.5.17,5.34.(d) Hanani (1979) VIl.4.7, IX.5.1 IX. 10.2 IX.5.6 IX.7.8 XU.S

Tables AS.2 and AS.3list the 33 open cases for the existence of a Steiner system S(2, 6; v) and the 22 open cases for the existence of a Steiner system S(2, 7; v), respectively; see Theorems IX.l2.2 and IX.12.3. These tables are followed by Tables A5.4 and A5.5 for Steiner systems S(2, 8; v) and resolvable quadruple systems RS(3, 4; v); see Theorems IX. 12.4 and IX.12.18, respectively. Table AS.2. Values ofvforwhich the existence of S(2, 6; v) is open 46 291 471 771

51 316 496 796

61 321 501 801

81 346 526

166 351 561

226 376 591

231 406 616

256 411 646

261 436 651

286 441

676

1001

§6. Remark on t-designs with t :::: 3 Table AS.3. Values of v for which the existence of S (2, 7; v) is open

85 799 1645

127 805 1765

133

211

925

1135 1974

1807

253 1387 2479

505 1435 2605

589 1555

715 1638

Table AS.4. Values of v for which the existence of S(2, 8; v) is open 113

169

176

225

281

1065 1457 2185

1121

1128

1177

1464 2241

1513 2577

1520 2913

1233 1569 3305

337 1240 1576 1417

393 1296 1737 3473

624 1345 1793 3753

736 1401 1905

785 1408 1961

Table AS.5. Values of v for which the existence ofRS(3, 4; v) is open

220 1076 2380

235 1100 2540

292 1252 2740

364 1316 2812

460 1820 3620

596 2236 3820

676 2308 6356

724

2324

§6. Remark on (-designs with ( :::::: 3 Many constructions for (not necessarily simple) t-designs with t:::: 3 were presented in Chapter ID. Some other (both recursive and direct) constructions for (farnilies of) both simple and non-simple t-designs were mentioned in §III.8. Teirlinck's general existence theorem for simple t-designs and the recent discovery of simple 7- and 8-designs with small parameters discussed there are particularly important. We will not give any tables of t-designs with t :::: 3 here but refer the reader to the literature. Hanani, Hartman and Kramer (1983) give an excellent survey of 3-designs including extensive tables (for v :::: 32). Kreher and Radziszowski (1987a) give a table of t-designs with t:::: 4 and v :::: 14. An influential detailed study (including tables) of t -designs with t :::: 3 is due to Driessen (1978). Finally, we refer the reader to Sections 1.3.2 and 1.3.3 of Colboum and Dinitz (1996a) for a list of the known infinite families of simple t-designs and a table of simple t-designs with t ::: 3 and v :::: 30, respectively; many further examples were constructed by Bluskov (1997).

Appendix. Tables

1002

§7. Orthogonal Latin Squares

In this final section, we give two tables concerning mutually orthogonal Latin squares: a table of upper bounds on the numbers nk, where nk denotes the largest order for which there exists no set of k mutually orthogonal Latin squares, and a table of the best-known lower bounds on the maximum number N(n) of MOLS of order n, in the range n :s 100. Since one has N(n) = n - I for prime powers n, we only include non-prime power values in the latter table. Most of the bounds quoted here are proved in the present text; we then only give the quotation from the preceding chapters. If a better bound is known, we give a reference and include the bound proved by us in square brackets. For a table of bounds on N (n) up to order 10000, see Section 11.2.6 of Col bourn and Dinitz (1996a); this is also the source for the following table of upper bounds for the numbers nk. Table A7.1. Upper bounds on the numbers nk k

2 7 12

~

k

~

k

~

k

~

k

6 180 7286

3 8 13

10 2774 7288

4 9 14

22 3678 7874

5 10 15

62 5804 8360

6 11 30

75 7222 52502

Table A7.2. Lower bounds jar the number oj MOLS ojorders .:s: 100 n

6 10 12 14

N(n)?:.

References

n

N(n)?:.

References

=1

§X.J3 VIII.4.9, VIII.4.27 VIII. 3. 13 VIII.4.17

36 38 39 40

6 [5] 4 5 [4] 7

VIIl.3.17, Abel (1998) VIII.4.17 VIII.4.17, Abel (1998) VIII.3.17

VIII.3.17 VIII.4.23 VIII.4.17 VIIl.3.!7

42 44 45 46

VIII.4.32 VIlI.3.17 VIII.3.17 VIIl.4.30

48

5 5 6 4 7 [5]

15 18 20 21

2 5 3 4 3 4 5

22 24 26 28

3 5 4 5

VIII.4.23 VIII.3.17 VIII.4.32 VIII.3.17

30 33 34 35

4 5 4 5

VIII.4.32 VIII.3.17 VIII.4.l7 VIII.3.l7

6

50 51 52

5 5 [3]

54

5 [4]

Wojtas (1997) VIIl.3.17 VIII.4.30 VlII.3.17 Colbourn and Dinitz (1996a), 1.7.8 Colbourn and Dinitz (1998), X.1.3 (cant.)

§7. Orthogonal Latin squares

1003

Table A7.2. (cant.) n

N(n)?:.

References

n

55 56 57 58 60 62 63 65 66 68 69 70 72 74 75 76

5 [4] 7 [6] 7 5 4 4 6 7 5 5 6 [5] 6 7 5 5 6 [5]

77 78

6 6

Mills (1977b), 1.7.8 Mills (1977b), 1.7.8 IX.1.19 X.3.l3 1.7.8 X.3.6 I.7.8 VIIIA.27 IXAA IX.4A Zhu (1984), IXA.4 X.1.3 I.7.8 IXAA IXAA Colboum, Yin and Zhu (1995), IXAA 1.7.8 X.1.3

N(n)?:.

References

80

9 [7]

82

8

Abel and Cheng (1994), (X.1.3.g) VIII.4.30

84 85 86 87 88 90 91 92 93 94 95 96 98 99 100

6 6 6 6 7 6 7 6 6 6 6 7 6 8 8

X.3.6 X.3.6 X.3.6 X.3.l3 I.7.8 X.3.9 X.3.l1 X.3.13 X.3.13 X.3.13 X.3.l3 X.3.l3 X.3.l3 I.7.8 VIII.4.30

Notation and Symbols

We begin by explaining the notation used in organising the material presented in this book. Theorems, lemmas, definitions, etc. are numbered consecutively within each section; e.g., II.4.S refers to item no. S in Section 4 of Chapter II (which happens to be a lemma). Within Chapter II, it would be quoted just by giving the number 4.S. Formulae are numbered according to the item where they occur by appending lower case letters; e.g., the formulae in IIA.S are labelled (4.5.a), (4.S.b), ... , (4.S.k). If one of these formulae need be quoted in another chapter, we prefix the number II again: so (1I.4.S.a) would be the result. The end or absence of a proof is always indicated by a •. Finally, in defining a quantity x in terms of a known quantity y, we often write x := y (so the term being defined is the one left of the colon).

General Symbols Here we collect some general notation concerning sets, numbers, algebraic structures, etc., that is more or less standard. The more specific notation concerned with incidence structures, automorphism groups, etc., will be treated separately.

Sets

AUB AnB A\B AxB A+B

the union of A and B the intersection of A and B A without the elements in B the cartesian product of A and B the symmetric difference (A U B)\(A 100S

n B) of A and B

Notation and symbols

1006

2A

No

the power set of A the set of t-subsets of A the set of ordered t-tuples from A the set of ordered t-tuples with distinct entries from A A. is a subset of B A is a proper subset of B the set of positive integers the set of non-negative integers

N~ A~

{xEN:a:sx:sb} A nN~ for A C N

\A\

the cardinality of A the set of primes (resp. prime powers)

(~) N A(t)

AS;B ACB N

P,P*

Lists

See Definition III.9.2 Maps

J:A-+B J:xl-+y xl or J(x)

Sf TI- 1 Xx, Xx

Oij X

XI Xw Tr Trq" /q'

J is a mapping from A to B

J maps x onto y the image of x under y (usually, mappings are written as exponents) {f(x) : XES} for J : A -+ Band S S A {x E A : J(x) E T} for J : A -+ Band T S; B characteristic functions, cf. (II.2.9.b) the Kronecker 0, i.e. oij = I for i = j and 0 otherwise (quadratic) character principal character additive character, cf. (VI.3.17.a) absolute trace, cf. (VI.3.17.b) relative trace, cf. V1.12.9 Numbers

I:7=lai 1 ai [x] or LxJ

D7_ rxl

C)

ala2·· . all the largest integer :sx (for x E JR.) the smallest integer 2:x (for x E JR.) the number of t-subsets of an x-set

Notation and symbols

1007

the number of ordered t-tuples with distinct entries from an x-set, i.e. x(x - 1) ... (x - t + 1) x factorial, i.e. x(x - 1) ... 2· 1 (for x E N) the number of i-dimensional subspaces of the n-dimensional vector space over GF(q) , cf. Lemma 1.2.14 a cyclotomic number, cf. Definition VI.8.8 a divides b / strictly divides b

xl

(i, j)m

alb pi lib

Matrices

AT J Jm,n

In I diag(xj, ... , x n )

the transpose of A a matrix with all entries equal to 1 the (m, n)-matrix with all entries equal to 1 the identity matrix of order n an identity matrix the diagonal matrix of size n with diagonal entries Xl,·· -,Xn

circ (a 11 , AI8lB

••. ,

al n )

a circulant matrix, cf. Definition VIIIA.29 the Kronecker product of A and B, cf. (1.9.6.a)

Rings and fields

Z

Zn Q R C ~m Q(~m) Z(~m)

GF(q) K*

KO V(n, q)

Rf

the ring (or group) of integers the ring (or group) of residues modulo n the field of rational numbers the field of real numbers the field of complex numbers a primitive m-th root of unity in C the m-th cyclotomic field the ring of algebraic integers in Q(~m) the finite field of order q (q a prime power) the multiplicative group of K (K a field) the multiplicative group of non-zero squares in K (K a field) the n-dimensional vector space over GF(q) the group ring of rover R (R a ring, r a group), see Definition VI.3.1

1008

Notation and symbols

Permutation groups the symmetric group on n letters the alternating group on n letters the stabiliser of x, cf. Definition III.3.9 the pointwise stabiliser of B, cf. Definition III.3.9 the setwise stabiliser of B, cf. Definition III.3.9 the image of x under g, cf. Definition III.3.1 (for a given action of G) the orbit of x under G, cf. Definition III.3.9 the Galois group of Q(~m) over Q

Groups in general

H-:sG H<JG gil [g,

hJ

G GxH G$H expG GIN

AB Tg

CG(g) CG(H) Nc(H)

C(G) O(G)

H is a subgroup of G H is a normal subgroup of G h-1ghforg,h E G the commutator of g and h, i.e. ghg-1h- 1 character group of an abelian group G, cf. Remark VI.3.3 the direct product of G and H the direct sum of G and H (for additively written groups) the exponent of G the factor group of G by N (for N <J G) the complex product of A and B (for A, B -:s G) the right translation by g, cf. Exercise V.l.l the centraliser of g in G (for g E G) the centraliser of H in G (for H -:s G) the normaliser of H in G (for H -:s G) the centre of G the maximal normal subgroup of odd order in G, cf. Lemma X.12.1

Particular groups the cyclic group of residues mod n the elementary abelian group of order s (s a prime power) the Mathieu group of order v (v = 11, 12,21,22,23,24), cf. Chapter IV the extension of Mv by a group of order 2 (v = 21, 22), cf. Theorem IV.S.14

Notation and symbols H(n+I,q) e(n + 1, q) T(n,q)

1009

cf. Proposition Ill.S.I2 cf. Notation 11I.6.3 cf. Definition Ill.S.7

For the classical groups GL(n, q), SL(n, q), r L(n, q) and the corresponding affine and projective groups (denoted by prefixing an A or a P, respectively), see Section Ill.5.

Special Symbols

Incidence structures in general (P,B, I)

15 D*

D1+ .. ·+D", mD

D'xD" Dx BD

(D) D(B)

D(B, p) H(D, B) dev D devS D (U)

N(EID) D'(p) Dip D'(B)

incidence structure with point set P, block set Band incidence relation I, cf. Definition I.1.1 complementary structure of D, cf. Definition I. 1. 10 the dual structure of D, cf. Definition 104.9 sum of incidence structures, cf. Definition 1.5.2 multiple of D, cf. Definition 1.5.2 direct product of nets, cf. Definition I. 7 .21 derived structure at x, cf. Definition 11.1.2 residual structure with respect to B, cf. Lemma II.1. 7 and Remark 11.1.8 point residue at x, cf. Lemma II. 1. 7 and Remark H.l.8 reduced structure of D, cf. Definition Ill. 2. 12 structure induced on B, cf. Proposition II.S.14 cf. Definition IV. 7.1 Hussain structure of D at B, cf. Definition IV.7.1 development of D, cf. Definition VI. 1.5 development of S, cf. Definition VII. 1.3 group constructible structure defined by U, cf. Definition X.9.3 net induced by the extension E of D, cf. Theorem X.I1A local structure at p, cf. Lemma XI1.2.7 local structure of a linear space, cf. Definition XII.4.1 reduced structure of D(B), cf. Lemma XII.S.1

Types of incidence structures S).(2, k; v) S).(2, K; v) S).(t, K; v)

2-design, cf. Definition 1.2.9 pairwise balanced design, cf. Definition 1.2.17 cf. Definition 1.3.1

1010 S),.(t,k;v) Set, k; v) STS(v) SQS(v) RS),.(t, k; v) GD)..[K, G; v] TD),.[k, g] RTD),.[k, g] (s, r; JL)-net STD),.(g) S"(2, k; v) AiL(s) LS)..(t, k; v) PS),.(t, k; v) NRS(2, k; v) NRB(v, k)

Notation and symbols t-design, cf. Definition 1.3.1 Steiner system, cf. Definition 1.3.1 Steiner triple system, cf. Definition 1.304 Steiner quadruple system, cf. Definition 1.304 resolvable t-design, cf. Notation I.S.6 divisible design, cf. Definition 1.6.1 transversal design, cf. Notation 1.7.4 resolvable transversal design, cf. Notation 1.704 net, cf. Notation 1.704 symmetric transversal design, cf. Definition 1.7.17 linear space with subspace, cf. Notation 1.8.10 affine 2-design, cf. Theorem 11.8.17 large set of t-designs S)..(t, k; v), cf. Remark VIII.8.S partial t -design, cf. remarks following Theorem 1X.11. 7 nearly resolvable Steiner system S(2, k; v), cf. Remark IX.12.8 nearly resolvable design Sk-I (2, k; v), cf. Remark IX.12.1O

Particular incidence structures AG(n, q) AGd(n, q) PG(n, q) PGd(n, q)

Wi

Ex

affine geometry, cf. Definition 1.2.12 design of points and d-dimensional subspaces of AG(n, q), cf. Definition 1.2.12 projective geometry, cf. Definition 1.2.1S design of points and d-dimensional subspaces of PG(n, q), cf. Definition 1.2.1S Witt design (i = 9, 10, II, 12,21,22,23,24), cf. Notations IV. 1.3 and IY.1.6 design induced on a dodecad X of E (E an S(S, 8; 24», cf. Lemma IVA.3 complete graph, cf. Example II.S.7 complete bipartite graph, cf. Example 11.7.2

Subsets in incidence structures

(B) (p) (Q), (C)

trace of B, cf. Definition II. 1.2 set of blocks on p, cf. Definition 11.1.1 cf. Definition II.l.1

Notation and symbols pq pqr

1011

line through p and q, cf. Definition XII.2.3 plane through p, q and r, cf. Definition XII.2.3 Parameters

v k

number of points, cf. Definition II.2.9 block size, cf. Definition II.2.9 J.... joining number, cf. Definition 11.2.9 r replication number, cf. Theorem II.2.W number of blocks, cf. Theorem II.2.1 0 number of blocks through an s-set, cf. Theorem 1.3.2 and Corollary II.1.4 intersection number of non-parallel blocks, cf. Definition I1.S.1 /h s size of parallel classes, cf. Lemma II.S.2 n order of a symmetric design or a difference set, cf. Lemmas II.3.l1 and VI 1.4 v,k,J.... parameters of difference sets, cf. Definition VI.lO.l m,n,k,J.... parameters of relative difference sets, cf. Definition VI.l 0.1 n,k,J....,/h parameters of strongly regular graphs, cf. Definition II.9.1 b;x;,y number of blocks through an x-set missing a y-yet, cf. Theorem II.6.1 niCB) intersection numbers, cf. Remark II.6.I1 intersection numbers, cf. Remark II.6.11 p-rank of an incidence structure deficiency of net, cf.Definitions X.7.l and X. 7.1 0 point class size, cf. Definition 1.6:1 i cf. Lemma X.7.5 Pjk

Subsets ofN S;..(2, K), B(K, J....), B(k, J....), B(K), B(k) S;..(t, k), Set, k) GD;..(K, G), GD;..(k, g) RS;..(t, k), RB(K, J....), RBCk, J....), RB(k) TD;..(k), TD(k), RTD;..(k), RTDCk) Bu(k) TD*Ck) R'K,;.., Rk,;'" R'K, R'"n, Rk

FK(u)

cf. Notation II.2.19 cf. Notation II.3.1 cf. Definition 1.6.1 cf. Notation II.5.6 cf. Notation II.7A cf. Notation II.S.lO cf. Definition IX.2.10 cf. Definition IX.3.7 cf. Notation IX.3.15

1012

R*k D).(k), D(K)

Notation and symbols cf. Notation IX.6.2 cf. Notation VII.5.3 Numbers

N(g) N*(g)

number of MOLS of order g, cf. Notation VillA.7 number of idempotent M 0 LS of order g, cf. Definition VillA. 10 cf. Remark XA.2

Groups and group rings AutD Aut(E,H)

S

Aa A

== B (mod P)

A(t)

the full automorphism group of D, cf. Definition ill.2.10 automorphism group of a Hussain structure, cf. Definition IV.7.7 sum in group ring, cf. VI.(3.l.b) cf. VI.(3.1.d) cf. Lemma VI.3.7 cf. VI.(3.1.c)

Miscellaneous H",R: b.;jBt

aB 08 (d l , ..• , dk)

GR(g, J....)

OAA(t,k,s),OAA(k,s) (s, r; /-t)-PCP

parallel, cf. Definition 1.204 cf. Notation VII.5.1 cf. Observation Vill.l.2 cf. Definition ill.9.7 cf. Definition ill.9.7 cf. Definition ill.9.7 generalised Hadamard matrix, cf. Definition Vill.3A cf. Definition VillA.1 cf. Definition X.9.3

Bibliography Verachtet mir die Meister nicht, und ehrt rnir ihre Kunst. .. (Wagner)

The following bibliography is not intended to be encyclopedic. In fact, it only contains items referred to in the main text. Aagedal, H., Beth, T., Miiller-Quade, J. and Schmid, M. (1996). Algorithmic design of diffractive optical systems for information processing. Workshop on physics and computation PhysComp96 Boston (eds. T. Toffoli, M. Biafore and J. Leao). New England Complex Systems Institute, pp. 1-6. Abatangelo, V. and Larato, B. (1989). A characterization of Denniston's maximal arcs. Geom. Ded. 30, 197-203. Abatangelo, V., Larato, B. and Rosati, L. A. (1990). Unitals in planes derived from Hughes planes. 1. Comb. In! System Sc. 15, 151-155. Abduhl-Elah, M. S., AI-Dhahir, M. W. and Jungnickel, D. (1987). 83 in PG (2, q). Arch. Math. 49, 141-150. Abe, T. and Yoshiara, S. (1993). On Suzuki's construction of 2-designs over GF(q). Science Rep. Hirosaki Univ. 40, 119-122. Abel, R. J. R. (1991). Four mutually orthogonal Latin squares of order 28 and 52. 1. Comb. Th. (A) 58, 306--309. Abel, R. J. R. (1994). Forty-three balanced incomplete block designs. 1. Comb. Th. (A) 65,252-267. Abel, R. J. R. (1996). Some new BIBDs with A = 1 and 6 ::: k ::: 10.1. Comb. Des. 4, 27-50. Abel, R. J. R. (1998). Personal communication. Abel, R. J. R. and Cheng, Y. W. (1994). Some new MOLS of order 2 n p for p a prime power. Australasian 1. Comb. 10, 175-186. Abel, R. J. R., Ge, G., Greig, M. and Zhu, L. (1998). Resolvable balanced incomplete block designs with a block size of 5. Preprint. Abel, R. J. R. and Greig, M. (1997). Some new RBIBDs with block size 5 and PBDs with block sizes == 1 mod 5. Australasian 1. Comb. 15, 177-202. Abel, R. J. R. and Greig, M. (1998). Balanced incomplete block designs with block size 7. Designs, Codes and Cryptography 13, 5-30. Abel, R. J. R. and Mills, W. H. (1995). Some new BIBDs with k = 6 and A = 1.1. Comb. Des. 3,381-391. Abel, R. J. R. and Todorov, D. T. (1993). Four MOLS of orders 20, 30, 38 and 44. 1. Comb. Th. (A) 64,144--148.

1013

1014

Bibliography

Abraham, C. T. and Ghosh, S. P. (1968). Application of finite geometry in file organization for records with multiple-valued attributes. IBM J. Res. Dev. 12, 180-187. Abraham, C. T., Ghosh, S. P. and Ray-Chaudhuri, D. K. (1968). File organization schemes based on finite geometries.lnf. Control 12, 143-163. Adams, P., Brayant, D. E. and Khodkar, A. (1996). On the existence of super-simple designs with block size 4. Aequat. Math. 51, 230-246. Addelman, S. and Kempthome, O. (1961). Some main effect plans and orthogonal arrays of strength two. Ann. Math. Stat. 32, 1167-1176. Agaian, S. S. (1985). Hadamard matrices and their applications. Springer, Berlin. Agraval, H. (1964). On the bounds of the numbers of common treatments between blocks of semiregular group divisible designs. J. Amel: Statist. Assoc. 59, 867-871. Ahrens, R. W. and Szekeres, G. (1969). On a combinatorial generalization of 27 lines associated with a cubic surface. J. Austral. Math. Soc. 10, 485-492. Aigner, M. (1990). Diskrete Mathematik. In: Ein Jahrhundert Mathematik 1890-1990 (eds. G. Fischer, F. Hirzebruch, W. Scharlau and W. T6mig). Friedr. Vieweg & Sohn, Braunschweig, pp. 83-112. Ajoodani-Namini, S. (1996). Extending large sets of t-designs. J. Comb. Th. (AJ 76, 139-144. Albert, A. A. (1961). The finite planes of Ostrom. Lecture Notes, Univ. of Chicago. Alltop, W. O. (1969). An infinite class of 4-designs. 1. Comb. Th. 6, 320-322. Alltop, W. O. (1971). 5-designs in affine space. Pacific J. Math. 39,547-551. Alltop, W. O. (1972). An infinite class of 5-designs. J. Comb. Th. (AJ 12, 390-395. Alltop, W. O. (1975). Extending t-designs. J. Comb. Th. (AJ 18,177-186. Alon, N. (1985). Expanders, sorting in rounds and superconcentrators of limited depth. Proc. 17thACM symposium on the Theory of Computing, pp. 98-102. Alon, N. (1986). Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica 6, 207-219. Alon, N., Babai, L. and Suzuki, H. (1991). Multilinear polynomials and Frankl-RayChaudhuri-Wilson type intersection theorems. J. Comb. Th. (AJ 58, 165-180. Alquaddoomi, S. and Scholtz, R. A. (1989). On the nonexistence of Barker arrays and related matters. IEEE Trans. Inform. Th. 35, 1048-1057. Alspach, B. and Heinrich, K. (1990). Matching designs. Australasian J. Comb. 2, 39-55. Alspach, B., Heinrich, K. and Liu, G. (1992). Orthogonal factorizations of graphs. In: Contemporary design theory: A collection of surveys (eds. J. H. Dinitz and D. R. Stinson). Wiley, New York, pp. 13-40. Andersen, L. D., Hilton, A. J. W. and Mendelsohn, E. (1980). Embedding partial Steiner triple systems. Proc. London Math. Soc. (3J 41, 557-576. Anderson, I. (1990). Combinatorial designs: Construction methods. Ellis Horwood, Chichester. Anderson, I. (1995). Cyclic designs in the 1850s; the work of Rev. R. R. Anstice. Bull. ICA 15,41-46. Anderson, I. (1996). Some 2-rotational and cyclic designs. J. Comb. Des. 4,247-254. Andre. J. (1954). Uber nichtdesarguessche Ebenen mit transitiver Translationsgruppe. Math. Z. 60, 156--186. Andre, 1. (1955). Projektive Ebenen tiber Fastk6rpem. Math. Z. 62, 137-160. Anstice, R. R. (1852). On a problem in combinatorics. Cambridge and Dublin Math. J. 7,279-292. Anstice, R. R. (1853). On a problem in combinatorics (continued). Cambridge and Dublin Math. J. 8, 149-154. Arasu, K. T. (1986). (81, 16, 3) abelian difference sets do not exist. J. Comb. Th. (AJ 43, 350-353.

Bibliography

1015

Arasu, K. T. (1987a). Number-theoretic consequences on the parameters of a symmetric design. Discr. Math. 65, 1-4. Arasu, K. T. (1987b). On abelian difference sets. Arch. Math. 48,491-494. Arasu, K. T. (1987c). On Lander's conjecture for the case A = 3.1. Comb. Math. Comb. Compo 1,5-11. Arasu, K. T. (l987d). On Wilbrink's theorem. 1. Comb. Th. (AJ 44, 156-158. Arasu, K. T. (1987e). Validity of Lander's conjecture for A = 3 and k :