Rank Distance Bicodes and Their Generalization

Rank Distance Bicodes - Cover:Layout 1

4/5/2010

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W. B. Vasantha Kandasamy Florentin Smarandache N. Suresh Babu R.S. Selvaraj

RANK DISTANCE BICODES AND THEIR GENERALIZATION

Svenska fysikarkivet Stockholm, Sweden 2010

Svenska fysikarkivet (Swedish physics archive) is a publisher registered with the Royal National Library of Sweden (Kungliga biblioteket), Stockholm.

Postal address for correspondence: Näsbydalsvägen 4/11, 18331 Täby, Sweden

Peer-Reviewers: Prof. Ion Patrascu Department of Mathematics Fratii Buzesti College, Craiova, Romania Prof. Catalin Barbu, Vasile Alecsandri College, Bacau, Romania Dr. Fu Yuhua, 13-603, Liufangbeili Liufang Street, Chaoyang district, Beijing, 100028, P. R. China

Copyright © Authors, 2010 Copyright © Publication by Svenska fysikarkivet, 2010

Copyright note: Electronic copying, print copying and distribution of this book for non-commercial, academic or individual use can be made by any user without permission or charge. Any part of this book being cited or used howsoever in other publications must acknowledge this publication. No part of this book may be reproduced in any form whatsoever (including storage in any media) for commercial use without the prior permission of the copyright holder. Requests for permission to reproduce any part of this book for commercial use must be addressed to the Author. The Author retains his rights to use this book as a whole or any part of it in any other publications and in any way he sees fit. This Copyright Agreement shall remain valid even if the Author transfers copyright of the book to another party.

ISBN: 978-91-85917-12-9

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CONTENTS

Preface

5

Chapter One

BASIC PROPERTIES OF RANK DISTANCE CODES

7

Chapter Two

RANK DISTANCE BICODES AND THEIR PROPERTIES

25

Chapter Three

RANK DISTANCE m-CODES

3

77

Chapter Four

APPLICATIONS OF RANK DISTANCE m-CODES

131

FURTHER READING

133

INDEX

144

ABOUT THE AUTHORS

150

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PREFACE In this book the authors introduce the new notion of rank distance bicodes and generalize this concept to Rank Distance n-codes (RD n-codes), n, greater than or equal to three. This definition leads to several classes of new RD bicodes like semi circulant rank bicodes of type I and II, semicyclic circulant rank bicode, circulant rank bicodes, bidivisible bicode and so on. It is important to mention that these new classes of codes will not only multitask simultaneously but also they will be best suited to the present computerised era. Apart from this, these codes are best suited in cryptography. This book has four chapters. In chapter one we just recall the notion of RD codes, MRD codes, circulant rank codes and constant rank codes and describe their properties. In chapter two we introduce few new classes of codes and study some of their properties. In this chapter we introduce the notion of fuzzy RD codes and fuzzy RD bicodes. Rank distance m-codes are introduced in chapter three and the property of m-covering radius is analysed. Chapter four indicates some applications of these new classes of codes.

5

Our thanks are due to Dr. K. Kandasamy for proofreading this book. We also acknowledge our gratitude to Kama and Meena for their help with corrections and layout. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE N. SURESH BABU R.S.SELVARAJ

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Chapter One

BASIC PROPERTIES OF RANK DISTANCE CODES

In this chapter we recall the basic definitions and properties of Rank Distance codes (RD codes). The Rank Distance (RD) codes are special type of codes endowed with rank metric introduced by Gabidulin [24, 27]. The rank metric introduced by Gabidulin is an ideal metric for it has the capability of handling varied error patterns efficiently. The significance of this new metric is that it recognizes the linear dependence between different symbols of the alphabet. Hence a code equipped with the rank metric detects and corrects more error patterns compared to those codes with other metric. In 1985 Gabidulin has studied a particular class of codes equipped with rank metric called Maximum Rank Distance (MRD) codes. Throughout this book Vn denotes a linear space of dimension n over the Galois field GF(2N), N > 1. By fixing a basis for Vn over GF(2N), we can represent any element x  Vn as an n-tuple (x1, x2, …, xn) where xi GF(2N).

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Again, GF(2N) can be considered as a linear space of dimension ‘N’ over GF(2). Hence an element xi GF(2N) has a representation as a N-tuple (Di1, Di2, …, Din) over GF(2) with respect to some fixed basis. Hence associated with each x  Vn, (n d N) there is a matrix,

ª D11 ! D1n º «D ! D 2n »» m(x) = « 21 « # # » « » ¬ D N1 ! D Nn ¼

T

where the ith column represents the ith coordinate ‘xi’ of x over GF(2). If we assume xiGF(2N) has a representation as a Ntuple x i D1i u1  D 2i u 2  ...  D Ni u N where u1, u2, …, uN is some fixed basis of the field Fq N regarded as a vector space over Fq. If

A nN denote the ensemble of all N u n matrices over Fq and if A : V n o A nN is a bijection defined by the rule, for any vector x

= (x1, …, xn)  FqnN , the associated matrix denoted by,

Ax

ª a11 a12 ! a1n º «a » « 21 a 22 ! a 2n » « # # # » « » ¬a N1 a N2 ! a Nn ¼

where the ith column represents the ith coordinate xi of x over Fq. Now we proceed on to define the rank of an element x  V n . DEFINITION 1.1: The rank of a vector x FqnN is the rank of the associated matrix A(x). Let r(x) denote the rank of the vector x  FqnN , over Fq.

By the usual properties of the rank of a matrix, it is easy to prove the following inequalities.

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1. 2. 3. 4.

r(x) t 0 for every x  Vn. r(x) = 0 if and only if x = 0. r(x+y) d r(x) + r(y) for every x, y  Vn. r(ax) = | a | r(x) for every a  GF(2) and x  Vn.

Thus the function x o r(x) defines a norm on Vn, (x o r(x) defines a norm on FqnN ) and is called the rank norm. In this book we denote the rank norm by r(x) or wt(x) or by || x ||. The rank norm induces a metric called rank metric (or rank distance) on FqnN . DEFINITION 1.2: The metric induced by the rank norm is defined as the rank metric on Vn( FqnN ) and it is denoted by dR.

The rank distance between x, yVn is the rank of their difference : dR(x, y) = r(x – y). The vector space Vn ( FqnN ) over

Fq N equipped with the rank metric dR is defined as a rank distance space. DEFINITION 1.3: A linear space Vn over GF(2N), N > 1 of dimension n such that n d N, equipped with the rank metric is defined as a rank space or rank distance space.

Now we proceed on to recall the definition of Rank Distance (RD) codes. DEFINITION 1.4: A Rank Distance code (RD-code) of length n over GF(2N) is a subset of the rank space Vn over GF(2N). A linear [n, k] RD code is a linear subspace of dimension k in the rank space Vn. By C[n, k] we denote a linear [n, k] RD-code. DEFINITION 1.5: A generator matrix of a linear [n, k] RD code C is a k u n matrix over GF(2N) whose rows form a basis for C. A generator matrix G of a linear RD code C[n, k] can be brought into the form G = [Ik, Ak, n–k] where Ik is the identity matrix and Ak, n–k is some matrix over GF(2n). This form is called the standard form.

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DEFINITION 1.6: Let G be a generator matrix of the linear RD code C[n, k], then a matrix H of order (n – k) u n over GF(2N) such that GHT = (0) is called a parity check matrix of C[n, k]. Suppose C is a linear [n, k] RD code with G and H as its generator and parity check matrices respectively, then C has two representations 1. C is the row space of G and 2. C is the solution space of H.

We shall illustrate this situation by an example. Example 1.1: Let

G

ª D11 D12 ... D15 º «D » « 21 D 22 ... D 25 » «¬ D 31 D 32 ... D 35 »¼ 3u5

be a generator matrix of the linear [5, 3] Rank Distance code C; over GF(25), here (D11, D12, …, D15), (D21, D22, …, D25), (D31, D32, …, D35) forms a basis of C; Dij  GF(25); 1 d i d 3 and 1 d j d 5. Now as in case of linear codes with Hamming metric, we in case of Rank Distance codes have the concept of minimum distance. We just recall the definition. DEFINITION 1.7: Let C be a rank distance code, the minimum rank distance d is defined by d = min{dR(x, y) | x, y  C, x z y}. In other words, d = min{r(x – y) | x, y  C, x z y}. i.e., d = min{r(x) | x  C and x z 0}. If an RD code C has the minimum – rank distance d then it can correct all errors e  FqnN with rank « d  1» r(e) = « ». ¬ 2 ¼

Let C denote an [n, k] RD – code over Fq N . A generator matrix G of C is a k u n matrix with entries from FqnN whose rows form

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a basis for C. Then an (n–k)un matrix H with entries from FqnN such that GHT = (0) is called the parity check matrix of C.

Result (singleton - style bound) The minimum - rank distance d of any linear [n, k] RD code C Ž FqnN satisfies the following bound: d d n – k + 1. Now based on this, the notion of Maximum Rank Distance; MRD codes were defined in [24, 27]. DEFINITION 1.8: An [n, k, d] RD code C is called Maximum Rank Distance (MRD) code if the singleton – style bound is reached; i.e., if d = n – k + 1.

Now we just briefly recall the construction of MRD code. Let [s] = qs for any integer s. Let g1, …, gn be any set of elements in Fq N that are linearly independent over Fq. A generator matrix G of an MRD code C is defined by

G

ª g1 « >1@ « g1 « > 2@ « g1 « # « >k 1@ «¬ g1

g2 >1@

g2

g>2 @ # 2

g>2

k 1@

!

gn º 1 » ! g>n @ » 2 » ! g>n @ » . # » » k 1 ! g>n @ »¼

It can be shown that the code C given by the above generator matrix G has the rank distance d = n – k + 1. Any matrix of the above form is called a Frobenius matrix with generating vector gc = (g1, g2, …, gn). Gabidulin has proved the following theorem.

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THEOREM 1.1: Let C[n, k] be a linear (n, k, d) MRD-code with d = 2t + 1. Then C[n, k] corrects all errors of rank atmost t and detects all errors of rank greater than t.

Circulant Rank codes were defined by [61]. Consider the Galois field GF(2N) where N > 1. An element DGF(2N) can be denoted by a N-tuple (a0, a1, …, aN–1) as well as by a polynomial a0 + a1x + … + aN–1xN–1 over GF(2). DEFINITION 1.9: The circulant transpose (Tc) of a vector D = (a0, a1, …, aN–1) GF(2N) is defined as D TC = (a0, a1, …, aN–1). If D  GF(2N) has polynomial representation a0 + a1x + … + [GF (2)]( x) then by Di, we denote the vector aN–1xN–1 in ( x N  1) corresponding to the polynomial [(a0 + a1x + … + aN–1xN–1).xi] (mod xN + 1), for i = 0 to N – 1. (Note D0 = D). DEFINITION 1.10: Let f: GF(2N)o[GF(2N)]N be defined as f (D ) (D 0TC , D1TC ,..., D NTC1 ) ; we call f(D) as the ‘word’ generated by D.

MacWilliams F.J. and Sloane N.J.A., [61] defined circulant matrix associated with a vector in GF(2N) as follows. DEFINITION 1.11: A matrix of the form

ª a0 «a « N 1 « # « ¬ a1

a1 ! aN 1 º a0 ! aN 2 »» # # » » a2 ! a0 ¼

is called the circulant matrix associated with the vector (a0, a1, …, aN–1)GF(2N). Thus with each DGF(2N) we can associate a circulant matrix whose ith column represents D iTC , i = 0, 1, 2, …, N – 1. f is nothing but a mapping of GF(2N) on to

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the algebra of all N u N circulant matrices over GF(2). Denote the space f(GF(2N)) by VN. We define norm of a word v  VN as follows: DEFINITION 1.12: The ‘norm’ of a word v  VN is defined as the ‘rank’ of v over GF(2) (By considering it as a circulant matrix over GF(2)).

We denote the ‘norm’ of v by r(v). We just prove the following theorem. THEOREM 1.2: Suppose DGF(2N) has the polynomial representation g(x) over GF(2) such that the gcd(g(x), xN +1) has degree N – k, where 0 d k d N. Then the ‘norm’ of the word generated by D is ‘k’.

Proof: We know the norm of the word generated by D is the rank of the circulant matrix (D T0C , D1TC ,..., D TNC1 ) , where DiTC represents the polynomial [xig(x)] (modxN+1) over GF(2). Suppose the gcd(g(x), xN +1) is a polynomial of degree ‘N – k’, (0 d k d N – 1). To prove that the word generated by ‘D’ has rank ‘k’. It is enough to prove that the space generated by the Npolynomials g(x)(modxN+1), [x˜g(x)](modxN+1), …, [xN–1˜g(x)] (modxN+1) has dimension ‘k’. We will prove that the set of k-polynomials g(x)(modxN+1), [x˜g(x)] (modxN +1), …, [xN–1˜g(x)] (modxN+1) forms a basis for this space. If possible, let a0(g(x)) + a1(x˜g(x)) + a2(x2˜g(x)) + … + ak– k–1 ˜g(x)) { 0(mod(xN + 1)), where aiGF(2). This implies xN 1(x + 1 divides (a0 + a1x + … + ak–1xk–1) ˜g(x). Now if g(x) = h(x) a(x) where h(x) is the gcd(g(x), xN +1), then (a(x), xN + 1) = 1. Thus xN + 1 divides (a0 + a1x + … + ak–1 xk–1)˜g(x) implies that the quotient (x N  1) h(x) k–1 divides (a0 + a1x + … + ak–1x )˜a(x). That is

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ª (x N  1) º « » ¬ h(x) ¼ divides (a0 + a1x + … ak–1xk–1) which is a contradiction as ª (x N  1) º « » ¬ h(x) ¼ has degree k where as the polynomial (a0 + a1x + … + ak–1xk–1) has degree atmost k – 1. Hence the polynomials g(x)mod(xN+1), [x˜g(x)]mod(xN+1), …, [xk–1˜g(x)]mod(xN+1) are linearly independent over GF(2). We will prove that the polynomials g(x)mod(xN+1), [x˜g(x)]mod(xN+1), …, [xk–1˜g(x)]mod(xN+1) generate the space. For this, it is enough to prove that xi˜g(x) is a linear combination of these polynomials for k d i d N – 1. Let xN +1 = h(x)b(x), where b(x) = b0 + b1x + … + bkxk (Note that here b0 = bk =1, since b(x) divides xN + 1). Also, we have g(x) = h(x). a(x). Thus (g(x) ˜ b(x)) xN 1 a(x) i.e., g(x)(b0  b1 x  ...  b k x k ) = 0 mod(xN + 1), a(x) that is ª (g(x) ˜ x k ) º g(x) ˜ (b0  b1x  ...  b k x k 1 ) N = « » mod(x + 1). a(x) a(x) ¬ ¼

(Since bk = 1). That is xk˜g(x) = (b0 + b1x + … + bk–1xk–1 ˜ g(x)) mod(xN + 1). Hence xkg(x) = b0g(x) + b1[x˜g(x)] + … + bk–1 [x k–1 ˜ g(x)] (mod(xN +

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1)), is a linear combination of g(x)mod(xN + 1), [x˜g(x)]mod(xN + 1), …, [xk–1˜g(x)] mod(xN + 1) over GF(2). Now it can be easily proved that xig(x) is a linear combination of g(x)mod(xN + 1), [x˜g(x)]mod(xN + 1), …, [xk– 1 ˜g(x)] mod(xN + 1) for i > k. Hence the space generated by the polynomial g(x)mod(xN + 1), [x˜g(x)] mod(xN + 1), …, [xk–1˜g(x)] mod(xN + 1) has dimension k; i.e., the rank of the word generated by D is k. The following two corollaries are obvious. COROLLARY 1.1: If DGF(2N) is such that its polynomial representation g(x) is relatively prime to xN + 1, then the norm of the word generated by D is N and hence f(D) is invertible.

Proof: Follows from the theorem as gcd(g(x), xN +1) = 1 has degree 0 and hence rank of f(D) is N. COROLLARY 1.2: The norms of the vectors corresponding to the polynomials x +1 and xN–1 + xN–2 + … + x +1 are respectively N – 1 and 1.

Now we proceed on to define the distance function on VN. DEFINITION 1.13: The distance between two words v, u. in VN is defined as d(u, v) = r(u + v).

Now we define circulant code of length N. DEFINITION 1.14: A circulant rank code of length N is defined as a subspace of VN equipped with the above defined distance function. DEFINITION 1.15: A circulant rank code of length N is called cyclic if, whenever (v1, v2, …, vN) is a codeword, then it implies (v2, v3, …, vN, v1) is also a codeword.

Now we proceed on to recall the definition of the new class of codes, Almost Maximum Rank Distance codes (AMRD-codes).

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DEFINITION 1.16: A linear [n, k] RD code over GF(2N) is called Almost Maximum Rank Distance (AMRD) code if its minimum distance is greater than or equal to n – k. An AMRD code whose minimum distance is greater than n – k is an MRD code and hence the class of MRD codes is a subclass of the class of AMRD codes.

We recall the following theorem. THEOREM 1.3: When n – k is an odd integer,

1. The error correcting capability of an [n, k] AMRD code is equal to that of an [n, k] MRD code. 2. An [n, k] AMRD code is better than any [n, k] code in Hamming metric for error correction. Proof: (1) Suppose C be an [n, k] AMRD code such that ‘n – k’ is an odd integer. The maximum number of errors corrected by (n  k  1) (n  k  1) . But is equal to the error C is given by 2 2 correcting capability of an [n, k] MRD code (since n – k is odd). Thus when n – k is odd an [n, k] AMRD code is as good as an [n, k] MRD code. (2) Suppose C be an [n, k] AMRD code such that ‘n – k’ is odd, then, each codeword of C can correct Lr(n) error vectors where (n  k  1) r= 2 and n ªn º Lr(n) = 1  ¦ « » (2 N  1) ... (2 N  2i1 ) . i 1 ¬i ¼

Consider the same [n, k] code in Hamming metric. Let it be C1, then the minimum distance of C1 is atmost (n – k +1). The error correcting capability of C1 is « (n  k  1  1) » (n  k  1) =r « » 2 2 ¬ ¼ 16

(since n – k is odd). Hence the number of error vectors corrected by a codeword is given by r §n· N i ¦ ¨ ¸ (2  1) , i 0 © i ¹ which is clearly less than Lr(n). Thus the number of error vectors that can be corrected by the [n, k] AMRD code is much greater than that of the same code considered in Hamming metric. For any given length ‘n’, a single error correcting AMRD code is one having dimension n – 3 and minimum distance greater than or equal to ‘3’. We give a characterization for a single error correcting AMRD codes in terms of its parity check matrix. This characterization is based on the condition for the minimum distance proved by Gabidulin in [24, 27]. We just recall the main theorem for more about these properties one can refer [82]. THEOREM 1.4: Let H = (Dij) be a 3 u n matrix of rank 3 over GF(2N), n d N which satisfies the following condition. For any two distinct, non empty subsets P1, P2 of {1, 2, …, n} there exists i1, i2{1, 2, 3} such that

§ · § · ¨¨ ¦ D i1 j ˜ ¦ D i2k ¸¸ z ¨¨ ¦ D i2 j ˜ ¦ D i1k ¸¸ kP2 kP2 © jP1 ¹ © jP1 ¹

then, H as a parity check matrix defines a [n, n – 3] single error correcting AMRD code over GF(2N). Proof: Given H is a 3un matrix of rank 3 over GF(2N), so H as a parity check matrix defines a [n, n – 3] RD code C over GF(2N); where C = {x  Vn | xHT = 0}.

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It remains to prove that the minimum distance of C is greater than or equal to 3. We will prove that no non-zero codeword of C has rank less than ‘3’. The proof is by the method of contradiction. Suppose there exists a non-zero codeword x such that r(x)d 2; then, x can be written as x = y˜M where y = (y1, y2); yiGF(2N) and M = (mij) is a 2 u n matrix of rank 2 over GF(2). Thus (y˜M)HT = 0 implies y(MHT) = 0. Since y is non zero y(M˜H T) = 0 implies that the 2 u 3 matrix MHT has rank less than 2 over GF(2N). Now let P1 = {j such that m1j = 1} and P2 = {j such that m2j = 1}. Since M = (mij) is a 2 u n matrix of rank 2, P1 and P2 are disjoint nonempty subsets of {1, 2, …, n}, and

MH

T

§ ¦ D1j ¨ jP1 ¨ ¨ ¦ D1j © jP2

¦D

2j

jP1

¦D jP2

¦D

· ¸ ¸. D3 j ¸ ¦ jP2 ¹ 3j

jP1

2j

But the selection of H is such that there exists i1, i2{1, 2, 3} such that § · § · ¨¨ ¦ Di1 j ˜ ¦ Di2k ¸¸ z ¨¨ ¦ D i2 j ˜ ¦ D i1k ¸¸ . kP2 kP2 © jP1 ¹ © jP1 ¹ Hence in MHT there exists a 2 u 2 submatrix whose determinant is nonzero; i.e., r(MHT) = 2 over GF(2N), this contradicts the fact that rank (MHT) < 2. Hence the result. Analogous to the constant weight codes in Hamming metric, we define the constant rank codes in rank metric. A constant weight code C of length n over a Galois field F is a subset of Fn with the property that all codewords in C have the same Hamming weight [61]. A(n, d, w) denotes the maximum number of vectors in Fn, distance atleast d apart from each other and constant Hamming weight ‘w’.

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Obtaining bounds for A(n, d, w) is one of the problems in the study of constant weight codes. A number of important bounds on A(n, d, w) were obtained in [61]. Here we just define the constant rank codes in rank metric and analyze the function A(n, r, d) which is the analog of the A(n, d, w) and obtain some interesting results. DEFINITION 1.17: A constant rank code of length n is a subset of a rank space Vn with the property that every codeword has same rank. DEFINITION 1.18: A(n, r, d) is defined as the maximum number of vectors in Vn, constant rank r and the distance between any two vectors is atleast d.

(By a (n, r, d) set, we mean a subset of vectors in Vn having constant rank r and distance between any two vectors is at least d). We analyze the function A(n, r, d). THEOREM 1.5: 1. A(n, r, 1) = Lr(n), the number of vectors of rank r in Vn. 2. A(n, r, d) = 0 if r > 0 or d > n or d > 2r.

Proof. (1) is obvious from the fact that Lr(n) is the number of vectors of length n, constant rank r and the distance between any two distinct vectors in the rank space Vn is always greater than or equal to one. (2) Follows immediately from the definition of A(n, r, d). THEOREM 1.6: A(n, 1, 2) = 2n – 1 over any Galois field GF(2N).

Proof: Let V1 denote the set of vectors of rank 1 in Vn. We know for each non zero element DGF(2N) there exists (2n – 1) vectors of rank one having D as a coordinate. Thus the cardinality of V1 is (2N – 1) (2n – 1).

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Now divide V1 into (2n – 1) blocks of (2N – 1) vectors such that each block consists of the same pattern of all non-zero elements of GF(2N). Thus from each block atmost one vector can be chosen such that the selected vectors are atleast rank 2 apart from each other. Such a set we call as a (n, 1, 2) set. Also it is always possible to construct such a set. Hence A(n, 1, 2) = 2n – 1. We give an example of A(n, 1, 2) set for a fixed N and n as follows. Example 1.2: Let N = 3, we use the following notation to define GF(23). Let 0, 1, 2, 3 be the basic symbols. Then GF(23)={0, 1, 2, 3, (12), (13), (23), (123)} (Note that by (ijk), we denote the linear combination of i + j + k over GF(2)).

…

100 200

110 220 …

010 020

…

001 002 …

Suppose n = 3, divide F3 into 23 – 1 blocks of 23 – 1 vectors as follows:

101 202

011 022

111 222

…

…

…

(00) (123) 0(123)0 (123)00 (123)(123)0

(123) 0 (123) 0(123) (123) (123) (123) (123) It is clear from this arrangement that atmost one vector from each block can be selected to form a (3, 1, 2) set. Also, the following set is a (3, 1, 2) set. {001, 020, 300, (12)(12)0, (13)0(13), 0(23)(23), (123)(123)(123)}. Thus A(3, 1, 2) = 7 = 23 – 1. Now we prove another interesting theorem.

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THEOREM 1.7: A(n, n, n) = 2N – 1 over any GF(2N).

Proof: Denote by Vn the set of vectors of rank n in the space Vn. We know that the cardinality of Vn is (2N – 1) (2N – 2) … (2N – 2n–1). By definition in a (n, n, n) set the distance between any two vectors should be n. Thus no two vectors can have a common symbol at a coordinate place i (1 d i d n). This implies that A(n, n, n) d 2N – 1. Now we construct a (n, n, n) set as follows: Select N vectors from Vn such that, 1. Each basis element of GF(2n) should occur (can be as a combination) atleast once in each vector. 2. If the ith vector is choosen (i + 1)th vector should be selected such that its rank distance from any linear combination of the previous i vectors is n. Now the set of all linear combinations of these N vectors over GF(2) will be such that the distance between any two vectors is n. Hence it is a (n, n, n) set. Also the cardinality of this (n, n, n) set is 2N – 1. (We do not count the all zero linear combination). Thus A(n, n, n) = 2N – 1. We illustrate this by the following example. Example 1.3: Consider GF(2N) for any N > 1. As in example 1.2, we represent GF(2N) as a linear combination of the symbols 1, 2, …, N over GF(2). Let n = 2. We construct a (2, 2, 2) set by taking the set of all linear combinations of the N vectors choosen as follows: We have two cases to be considered separately, when N is an odd integer say 2k + 1 and when N is an even integer say 2k.

Case 1. N is an odd integer say 2k + 1. In this case choose the N vectors as 12, 23, 34, …, (2k +1) (12).

21

Case 2. N is an even integer say 2k. In this case choose the N vectors as 1(12), 21, 3(34), 43, …, (2k – 1) ((2k – 1)2k), (2k) (2k – 1). Consider the set of linear combinations of the N-vectors. It can be verified easily that this set is a (2, 2, 2) set. Now we obtain the value of A(n, r, d) for a particular triple, when n = 4, r = 2 and d = 4 in the following theorem. THEOREM 1.8: A(4, 2, 4) = 5 over any Galois field GF(2N).

Proof: Consider V4, the 4-dimensional space over GF(2N). We denote the elements of V4 as 4-tuples (a b c d) where a, b, c, d GF(2N). Denote by V2, the set of vectors of rank 2 in V4. The cardinality of V2 is 35 u (2N–1) (2N–2). (since V2

L 2 (4)

(24  1)(24  2)(2 N  1)(2 N  2) · ¸. (22  1)(22  2) ¹

Thus for each distinct non-zero pair of elements of GF(2N) there are 35 vectors in V2. Let a, b  GF(2N) be such that a z 0, bz0 and a zb. Divide the set of 35 vectors containing a, b and the linear combination (ab) into six blocks as follows: I

II

III

IV

V

VI

00ab 0a0b 0ab0 0aab 0aba 0baa 0ab(ab)

a00b ab00 aa0b ab0a ab0b ab0(ab)

a0ab aba(ab) ab(ab)a ab(ab)(ab) a0ba a0bb

aaab abab abba aaab aaba abaa abbb

aab0 aba0 abb0 aab(ab) abbab

ab(ab)b a0b(ab) ab(ab)0 a0b0

22

From the arrangement of these six blocks it can be verified easily that atmost 5 vectors can be chosen to form any (4, 2, 4) set. For example if we choose a vector of pattern 00ab then no vector of any other pattern from block I can be chosen (otherwise distance between the two is < 4). Now, move to block II. The first pattern in block II cannot be chosen. So select a vector in the second pattern ab00. No other pattern can be selected from block II. Now move to block III. Here also the first pattern cannot be chosen. So choose a vector of the pattern aba(ab). Similarly from block IV select the pattern abab. In the block V, the first four pattern cannot be selected. Hence select the pattern abb(ab). Now move to block VI. But no pattern can be selected from block VI since each pattern is at a distance less than four from one of the already selected patterns. Similarly we can exhaust all the possibilities. Hence a (4, 2, 4) set in this space can have atmost five vectors. Also it is always possible to choose five vectors in different patterns to form a set (4, 2, 4) set. Thus A(4, 2, 4) = 5. Now we proceed on to give a general bound for A(n, n, d). THEOREM 1.9: A(n, n, d) d(2N – 1) (2N – 2) … (2N – 2n–d) over any field GF(2N).

Proof: Let Vn be the set of vectors of rank n in the space Vn. The cardinality of Vn is given by (2N – 1) (2N – 2) … (2N – 2n–1). Now for a (n, n, d) set two vectors should be different atleast by d coordinate places. Thus the cardinality of any (n, n, d) set is less than or equal to (2N – 1) (2N – 2) … (2N – 2n–d). This proves,

23

A(n, n d) d (2N – 1) (2N – 2) … (2N – 2n–d). These AMRD codes are useful for error correction in data storage systems.

24

Chapter Two

RANK DISTANCE BICODES AND THEIR PROPERTIES

In this chapter we define for the first time the notion of Rank Distance Bicodes; RD-Bicodes and derive some interesting results about them. The properties of bivector spaces and bimatrices can be had from the books [91-93]. As the error correcting capability of a code depends mainly on the distance between codewords, not only choosing an appropriate metric is important but also simultaneous working of a pair of system would be advantageous in this computerized world. This is done by introducing the concepts of rank distance bicodes, maximum rank distance bicodes, circulant rank bicodes, RD-MRD bicodes, RD-circulant bicodes, MRD circulant bicodes, RD-AMRD bicodes, AMRD bicodes and so on. We aim to give certain classes of new bicodes with rank metric.

25

Let Vn and Vm be n-dimensional and m-dimensional vector spaces over the field Fq N ; n d N and m d N (m z n). That is Vn = FqnN and Vm = FqmN .

We know V = Vn ‰ Vm is a (m, n) dimensional vector bispace (or bivector space) over the field Fq N . By fixing a bibasis of V = Vn ‰ Vn over Fq N we can represent any element x ‰ y V = (Vn ‰ Vm) as a (n, m)-tuple; (x1, …, xn)‰(y1, …, ym) where xi, yj  Fq N ; 1 d i d n and 1 d j d m. Again Fq N can be

considered as a pseudo false linear bispace of dimension (N, N) over Fq (we say a linear vector bispace V = Vn ‰ Vn to be a pseudo false linear bispace if m = n = N). Hence the elements xi, yj Fq N has a representation as N-bituple (D1i, …, DNi)‰(E1j, …, ENj) over Fq with respect to some fixed bibasis. Hence associated with each x ‰ y  Vn ‰ Vm (n z m) there is a bimatrix

ª a11 ! a1n º ª b11 ! b1m º «a ! a 2n »» «« b 21 ! b 2m »» ‰ m(x) ‰ m(y) = « 21 « # # » « # # » « » « » ¬ a N1 ! a Nn ¼ ¬ b N1 ! b Nm ¼ where the ith ‰ jth column represents the ith ‰ jth coordinate of xi ‰ yj of x ‰ y over Fq. Remark: In order to develop the new notion of rank distance bicodes and trying to give the bimatrices and biranks associated with them we are forced to define the notion of pseudo false bivector spaces.

For example; V

Z52 ‰ Z52 is a false pseudo bivector space over

Z2. Likewise Z37 ‰ Z37 is a pseudo false bivector space over Z3. Also V

Z 578 ‰ Z 578 is a pseudo false bivector space over Z7 . 8

26

However throughout this book we will be using only vector bispaces over Z2 or Z2N unless otherwise specified. Now we see to every x ‰ y in the bivector space Vn ‰ Vm we have an associated bimatrix m(x) ‰ m(y). We now proceed on to define the birank of the bimatrix m(x) ‰ m(y) over Fq or GF(2). DEFINITION 2.1: The birank of an element x ‰ y  (Vn ‰ Vm) is defined as the birank of the bimatrix m(x) ‰ m(y) over GF(2) or Fq. (the birank of the bimatrix m(x) ‰ m(y) is the rank of m(x) ‰ rank of m(y)). We shall denote the birank of x ‰ y by r1(x) ‰ r2(y) = r(x ‰ y), we see analogous to the properties of rank we can in case of the birank of a bimatrix prove the following: (i)

For every x ‰ y  (Vn ‰ Vm) (x  Vn and y  Vm) we have r(x ‰ y) = r1(x) ‰ r2(y) t 0 ‰ 0 (i.e., each r1(x) t 0 and each r2(y) t 0 for every x  Vn and y  Vm).

(ii)

r(x ‰ y) = r1(x) ‰ r2(y) = 0 ‰ 0 if and only if x ‰ y = 0 ‰ 0 i.e., x = 0 and y = 0.

(iii) r((x1 + x2) ‰ (y1 + y2)) d {r1(x1) + r1(x2)} ‰ r2(y1) + r2(y2) for every x1, x2  Vn and y1, y2  Vm. That is we have r((x1 + x2) ‰ (y1 + y2)) = r1(x1 + x2) ‰ r2(y1 + y2) d r1(x1) + r1(x2) ‰ r2(y1) + r2(y2); (as we have for every x1, x2  Vn, r(x1 + x2) d r1(x1) + r1(x2) and for every y1 , y2  Vm; r2 (y1 + y2) d r2 (y1) + (y2)). (iv)

r1(a1x) ‰ r2(a2y) = |a1|r1(x) ‰ |a2|r2(y) for every a1, a2,  Fq or GF(2) and x Vn and y  Vm.

Thus the bifunction x ‰ y o r1(x) ‰ r2(y) defines a binorm on Vn ‰ Vm. DEFINITION 2.2: The bimetric induced by the birank binorm is defined as the birank bimetric on Vn ‰ Vm and is denoted by

27

d R1 ‰ d R2 . If x1 ‰ y1, x2 ‰ y2  Vn ‰ Vm then the birank bidistance between x1 ‰ y1 and x2 ‰ y2 is d R1 ( x1 , x2 ) ‰ d R2 ( y1 , y2 ) r1 ( x1  x2 ) ‰ r2 ( y1  y2 ) (here d R1 (x1, x2) = r1(x1 – x2) for every x1, x2 in Vn, the rank distance between x1 and x2 likewise for y1, y2  Vm). DEFINITION 2.3: A linear bispace Vn ‰ Vm over GF(2N), N > 1 of bidimension n ‰ m such that n d N and m d N equipped with the birank bimetric is defined as the birank bispace. DEFINITION 2.4: A birank bidistance RD bicode of bilength n ‰ m over GF(2N) is a bisubset of the birank bispace Vn ‰ Vm over GF(2N). DEFINITION 2.5: A linear [n1, k1] ‰ [n2, k2] RD bicode is a linear bisubspace of bidimension k1 ‰ k2 in the birank bispace Vn ‰ Vm. By C1[n1, k1] ‰ C2[n2, k2], we denote a linear [n1, k1] ‰ [n2, k2] RD bicode.

We can equivalently define a RD bicode as follows: DEFINITION 2.6: Let Vn and Vm, m z n be rank spaces over GF(2N), N > 1. Suppose P  Vn and Q  Vm be subsets of the rank spaces over GF(2N). Then P ‰ Q Ž Vn ‰ Vm is a rank distance bicode of bilength (n, m) over GF(2N). DEFINITION 2.7: Let C1[n1, k1] be [n1, k1] RD code (i.e., a linear subspace of dimension k1, in the rank space Vn) and C2[n2, k2] be [n2, k2] RD code (i.e., a linear subspace of dimension k2 in the rank space Vm) (m z n); then C1[n1, k1] ‰ C2[n2, k2] is defined as the linear RD bicode of the linear bisubspace of dimension (k1, k2) in the rank bispace Vn ‰ Vm.

Now we proceed onto define the notion of the generator bimatrix of a linear [n1, k1] ‰ [n2, k2] RD bicode.

28

DEFINITION 2.8: A generator bimatrix of a linear [n1, k1] ‰ [n2, k2] RD-bicode C1 ‰ C2 is a k1 u n1 ‰ k2 u n2 bimatrix over GF(2N) whose birows form a bibasis for C1 ‰ C2. A generator bimatrix G = G1 ‰ G2 of a linear RD bicode C1[n1, k1] ‰ C2[n2, k2] can be brought into the form G = G1 ‰ G2 = [ I k1 , Ak1 , n1  k1 ] ‰

[ I k2 , Ak2 un2  k2 ] where I k1 , I k2 is the identity matrix and Aki uni  ki , i = 1, 2 is some matrix over GF(2N). This form of G = G1 ‰ G2 is called the standard form. DEFINITION 2.9: If G = G1 ‰ G2 is a generator bimatrix of C1[n1, k1] ‰ C2[n2, k2] then a bimatrix H = H1 ‰ H2 of order (n1 – k1 u n1, n2 – k2 u n2) over GF(2N) such that GHT = (G1 ‰ G2) (H1 ‰ H2)T = (G1 ‰ G2) ( H1T ‰ H 2T )

= G1 H1T ‰ G2 H 2T =0‰0 is called a parity check bimatrix of C1[n1, k1] ‰ C2[n2, k2]. Suppose C = C1 ‰ C2 is a linear [n1, k1] ‰ [n2, k2] RD code with G = G1 ‰ G2 and H = H1 ‰ H2 as its generator and parity check bimatrices respectively, then C = C1 ‰ C2 has two representation, (i) C = C1 ‰ C2 is a row bispace of G = G1 ‰ G2 (i.e., C1 is the row space of G1 and C2 is the row space of G2) (ii) C = C1 ‰ C2 is the solution bispace of H = H1 ‰ H2; i.e., C1 is the solution space of H1 and C2 is the solution space of H2. Now we proceed on to define the notion of minimum rank bidistance of a rank distance bicode C = C1 ‰ C2. DEFINITION 2.10: Let C = C1 ‰ C2 be a rank distance bicode, the minimum-rank bidistance is defined by d = d1 ‰ d2 where, d1 = min{dR(x, y) | x, y  C1, x z y} and d2 = min{dR(x, y) | x, y  C1, x z y} i.e., d = d1 ‰ d2

29

= min {r1(x) | x  C1 and x z 0} ‰ min {r2(x) | x  C2 and x z 0} . If an RD bicode C = C1 ‰ C2 has the minimum rank bidistance d = d1 ‰ d2 then it can correct all bierrors d = d1 ‰ d2  FqnN ‰ FqmN with birank r(e) = (r1 ‰ r2)(e1 ‰ e2) « d  1» « d  1» = r1(e1) ‰ r2(e2) d « 1 » ‰ « 2 » . ¬ 2 ¼ ¬ 2 ¼

Let C = C1 ‰ C2 denote an [n1, k1] ‰ [n2, k2] RD-bicode over Fq N . A generator bimatrix G = G1 ‰ G2 of C = C1 ‰ C2 is a k1

u n1 ‰ k2 u n2 bimatrix with entries from Fq whose rows form N

a bibasis for C = C1 ‰ C2. Then an (n1 – k1) u n1 ‰ (n2 – k2) u n2 bimatrix H = H1 ‰ H2 with entries from Fq N such that

GHT = (G1 ‰ G2) (H1 ‰ H2)T = (G1 ‰ G2) ( H1T ‰ H 2T ) = G1 H1T ‰ G2 H 2T =0‰0 is called the parity check bimatrix of C = C1 ‰ C2. The result analogous to singleton-style bound in case of RD bicode is given in the following: Result (singleton-style bound) The minimum rank bidistance d = d1 ‰ d2 of any linear [n1, k1] ‰ [n2, k2] RD bicode C = C1 ‰ C2 Ž FqnN ‰ FqmN satisfies the following bounds. d = d1 ‰ d2 d n1 – k1 + 1 ‰ n2 – k2 +1. Based on this notion we now proceed on to define the new notion of Maximum Rank Distance (MRD) bicodes.

30

DEFINITION 2.11: An [n1, k1, d1] ‰ [n2, k2, d2] RD bicode C = C1 ‰ C2 is called a Maximum Rank Distance (MRD) bicode if the singleton-style bound is reached, that is d = d1 ‰ d2 = n1 – k1 + 1 ‰ n2 – k2 +1.

Now we proceed on to briefly give the construction of MRD bicode. Let [s] = [s1] ‰ [s2] = q s1 ‰ q s2 for any two integers s1 and s2. Let {g1, …, gn} ‰ {h1, h2, …, hm} be any set of elements in Fq N that are linearly independent over over Fq. A generator bimatrix G = G1 ‰ G2 of an MRD bicode C = C1 ‰ C2 is defined by G = G1 ‰ G2

ª g1 « >1@ « g1 « > 2@ « g1 « # « >k 1@ 1 ¬« g n

g2 g

>1@ 2

> 2@

g2 # g>n 1

k 1@

!

g n º ª h1 « 1 1 » ! g>n @ » « h1> @ « 2 2 » ! g>n @ » ‰ « h1> @ # » « # » « k 1 k 1 ! g>n 1 @ ¼» ¬« h >m 2 @

h2 h

>1@ 2

h >2 @ # 2

h >m 2

k 1@

!

hm º 1 » ! h >m@ » 2 » ! h >m @ » # » » k 1 ! h >m 2 @ ¼»

It can be easily proved that the bicode C = C1 ‰ C2 given by the generator bimatrix G = G1 ‰ G2, has the rank bidistance d = d1 ‰ d2 = (n1 – k1 + 1) ‰ (n2 – k2 + 1). Any bimatrix of the above from is called a Frobenius bimatrix with generating bivector gC = g C1 ‰ h C2 = (g1, …, gn) ‰ (h1, h2, …, hm). One can prove the following theorem: THEOREM 2.1: Let C[n, k] = C1(n1, k1) ‰ C2(n2, k2) be the linear (n1, k1, d1) ‰ (n2, k2, d2) MRD bicode with d1 = 2t1 + 1 and d2 = 2t2 + 1. Then C[n, k] = C1(n1, k1) ‰ C2(n2, k2), bicode corrects all bierrors of birank atmost t = t1 ‰ t2 and detects all bierrors of birank greater than t = t1 ‰ t2.

31

Consider the Galois field GF(2N), N > 1. An element D1 ‰ E2  GF(2N) ‰ GF(2N) can be denoted by a biN-tuple (a0, …, aN–1) ‰ (b0, b1, …, bN–1) as well as by the bipolynomial a0 + a1 x + … + aN – 1 xN – 1 ‰ b0 + b1 x+ …+ bN – 1 xN – 1 over GF(2). We now proceed on to define the new notion of circulant bitranspose. DEFINITION 2.12: The circulant bitranspose TC

TC11 ‰ TC22 of a

bivector D = D1 ‰ E2 = (a0, …, aN–1) ‰ (b0, b1, …, bN–1)  GF(2N) is defined as

DT

TC11

TC22

D1 ‰ E 2 = (a0, a1, …, aN–1) ‰ (b0, b1, …, bN–1). If D = D1 ‰ E2  GF(2N) ‰ GF(2N) has the bipolynomial C

representation a0 + a1 x + … + aN – 1 xN – 1 ‰ b0 + b1 x+ …+ bN – 1 xN – 1 in ª GF (2)( x) º ª GF (2)( x) º « N »‰« N » «¬ x  1 »¼ «¬ x  1 »¼ then by; Di = D1i ‰ E2i we denote the bivector corresponding to the bipolynomial [(a0 + a1 x + … + aN – 1 xN – 1)˜ xi] mod(xn+1) ‰ [(b0 + b1 x+ …+ bN – 1 xN – 1)˜ xi] mod(xn+1) for i = 0, 1, 2, . . . , N – 1. (Note D0 = D1 ‰ E2 = D). Now we proceed on to define the biword generated by D = D1 ‰ E2. DEFINITION 2.13: Let f = f1 ‰ f2: GF(2N) ‰ GF(2N)o [GF(2N)]N ‰ [GF(2N)]N be defined as, f(D) = f1(D1) ‰ f2(E2) T1

T1

T1

T2

T2

T2

(D 0 C1 , D1 C1 ,! ,D NC1 1 ) ‰ ( E 0 C2 , E1 C2 , ! , E NC21 ) .

32

We call f(D) = f1(D1) ‰ f2(E2) as the biword generated by D = D1 ‰ E2. We analogous to the definition given in MacWilliams and Sloane [61] define circulant bimatrix associated with a bivector in GF(2N)‰GF(2N). DEFINITION 2.14: A bimatrix of the from

ª a0 «a « N 1 « # « ¬ a1

a1 ! aN 1 º ª b0 b1 ! bN 1 º a0 ! aN  2 »» ««bN 1 b0 ! bN  2 »» ‰ # # » « # # # » » « » a2 ! a0 ¼ ¬ b1 b2 ! b0 ¼

is called the circulant bimatrix associated with the bivector (a0, a1, …, aN–1) ‰ (b0, b1, …, bN–1)  GF(2N)‰GF(2N). Thus with each D = D1 ‰ E2 GF(2N)‰GF(2N), we can associate a T1

T2

circulant bimatrix whose ith bicolumn represents D i C1 ‰ E i C2 ; i = 0, 1, 2, …, N – 1. f=f1‰f2 is nothing but a bimapping of GF(2N)‰GF(2N) on to the pseudo false bialgebra of all N u N circulant bimatrices over GF(2). Denote the bispace f(GF(2N))=f1(GF(2N))‰f2(GF(2N)) by VN ‰ VN. We define binorm of a biword v = v1 ‰ v2  VN ‰ VN as follows. DEFINITION 2.15: The binorm of a biword v = v1 ‰ v2  VN ‰ VN is defined as the birank of v = v1 ‰ v2 over GF(2N) (by considering it as a circulant bimatrix over GF(2)).

We denote the binorm of v = v1 ‰ v2 by r(v) = r1(v1) ‰ r2(v2), we prove the following theorem: THEOREM 2.2: Suppose D = D1 ‰ E2  GF(2N) ‰ GF(2N) has the bipolynomial representation g1(x) ‰ h2(x) over GF(2) such that gcd (g1(x), xN + 1) has degree N – k1 and gcd (h2(x), xN + 1)

33

has degree N – k2 where 1 d k1, k2 d N; then the binorm of the biword generated by D = D1 ‰ E2 is k1 ‰ k2. Proof: We know the binorm of the biword generated by D = D1 ‰ E2 is the birank of the circulant bimatrix T1

T1

T1

T2

T2

T2

(D 0C1 , D1 C1 , ... , D NC1 1 ) ‰ (E0C2 , E1 C2 , ... , E NC21 ) T1

T2

where D iTC D1iC1 ‰ E2iC2 represents the bipolynomial [xig1(x)] [mod xN +1] ‰ [xih2(x)] [mod xN +1] over GF(2). Suppose the bigcd (g1(x), xN+1) ‰ (h2(x), xN+1) has bidegree N – k1 ‰N – k2, (0 d k1, k2 d N – 1). To prove that the biword generated by D = D1 ‰ E2 has birank k1 ‰k2. It is enough to prove that the bispace generated by the Nbipolynomials {g1(x) mod (xN + 1), (xg1(x)) [mod xN +1], …, [xN-1g1(x)] mod[xN + 1]} ‰ ^h2(x) mod (xN + 1), (xh2(x)) [mod xN +1], …, [xN-1h2(x)] mod (xN + 1)} has bidimension k1 ‰ k2. We will prove that the biset of k1 ‰ k2 bipolynomials {g1(x) mod (xN + 1), (xg1(x)) mod (xN + 1), …, (xN-1 g1(x)) mod(xN + 1)} ‰ {h2(x) mod (xN + 1), (xh2(x)) (mod xN + 1), …, (xN-1h2 (x)) (mod xN + 1)} forms a bibasis for this bispace. If possible let a0(g1(x)) + a1(xg1(x)) + … + a k1 1 ( x k1 1 g1(x)) ‰b0(h2(x)) + b1(xh2(x)) + … + b k 2 1 (x k 2 1h 2 (x))

{ 0 ‰ 0(mod xN + 1)

where ai, bi  GF(2). This implies xN+1 ‰ xN+1 bidivides (a0 + a1x + … + a k1 1x k1 1 )g1(x)

34

‰ (b0 + b1x + … + b k 2  2 x k 2 1 )h2(x) Now if g1(x) ‰ h2(x) = p1(x) a1(x) ‰ p2(x) b2(x) where, p1(x) ‰ p2(x) is the bigcd {(g1(x), xN + 1) ‰ (h2(x), xN + 1)} then (a1(x), xN + 1) ‰ (b2(x), xN + 1) = 1 ‰ 1. Thus xN + 1 bidivides (a0 + a1x + … + a k1 1x k1 1 )g1(x)

‰ (b0 + b1x + … + b k 2  2 x k 2 1 )h2(x) which inturn implies that the biquotient

x

N

1

p1 (x)

‰ x

N

1

p 2 (x)



bidivides (a0 + a1x + … + a k1 1x k1 1 )a1(x)

‰ (b0 + b1x + … + b k 2  2 x k 2 1 )b2(x) That is

x

N

1

p1 (x)

‰ x

N

1

p 2 (x)



bidivides (a0 + a1x + … + a k1 1x k1 1 ) ‰ (b0 + b1x + … + b k 2  2 x k 2 1 ) which is a contradiction as

x

N

1

p1 (x)

‰ x

N

1

p 2 (x)



has bidegree k1 ‰ k2 where as the bipolynomial (a0 + a1x + … + a k1 1x k1 1 ) ‰ (b0 + b1x + … + b k 2  2 x k 2 1 ) has bidegree atmost k1 –1 ‰ k2 –1. Hence the bipolynomials {g1(x) mod (xN + 1), xg1(x) mod (xN + 1), …,

35

x k1 1 g1(x) mod(xN + 1)} ‰ {h2(x) mod (xN + 1), xh2(x)mod (xN + 1), …, x k 2 1 h2 (x) mod (xN + 1)}

are bilinearly independent over GF(2) ‰ GF(2), i.e.; {g1(x) mod (xN + 1), xg1(x) mod (xN + 1), …, x k1 1 g1(x) mod(xN + 1)} ‰ {h2(x) mod (xN + 1), xh2(x)mod (xN + 1), …, x k 2 1 h2 (x) mod (xN + 1)} bigenerate the bispace. For this it is enough to prove that xig1(x) ‰ xih2(x) is a linear bicombination of these bipolynomials for k1 d i d N –1 and k2 d i d N – 1. Let xN + 1 ‰ xN + 1 = p1(x)q1(x) ‰ p2(x)q2(x) where q1(x) ‰ q2(x) =

c 0  c1 x  ...  c k i x k1 ‰ d 0  d 1 x  ...  d k 2 x k 2 (Note: c0

c k1

1 and d 0

d k 2 1 , since q1(x) ‰ q2(x) bidivides

xN + 1 ‰ xN + 1, i.e., g1(x) divides xN + 1 and g2(x) divides xN + 1). Also we have g1(x) = p1(x) a1(x) and h2(x) = p2(x) b2(x) i.e., g1(x) ‰ h2(x) = p1(x) a1(x) ‰ p2(x) b2(x). Thus § p (x) · § p 2 (x) · xN + 1 ‰ xN + 1 = ¨ g1 (x) ˜ 1 ¸ ‰ ¨ h 2 (x) ˜ ¸ a1 (x) ¹ © b 2 (x) ¹ ©

that is

g1 (x) ˜

p1 (x) { 0 mod(x N  1) a1 (x)

h 2 (x) ˜

p 2 (x) { 0 mod(x N  1) b 2 (x)

and

36

§ p (x) · § p 2 (x) · xN + 1 ‰ xN + 1 = ¨ g1 (x) ˜ 1 ¸ ‰ ¨ h 2 (x) ˜ ¸. a1 (x) ¹ © b 2 (x) ¹ © Now g1 (x)(a 0  a1x  ...  a k1 1x k1 1 ) a1 (x) ª g (x)x k1 º N =« 1 » mod(x  1) (since ak =1). a (x) ¬ 1 ¼

That is x k1 g1 (x) (a 0  a1x  ...  a k1 1x k1 1g1 (x)) mod(x N  1) . Hence, x k1 g1 (x) (a 0 g1 (x)  a1 (xg1 (x))  ...  a k1 1[x k1 1g1 (x)]) mod(x N  1) ; a linear combination of g1(x) mod(xN + 1), [xg1(x)] mod (xN + 1), …, [ x k1 1 g1(x)] mod(xN + 1) over GF(2). Now it can be easily proved that xig1(x) is a linear combination of g1(x) mod (xN+1), xg1(x) mod (xN+1), …, x k1 1g1 (x) mod(x N  1) for i > k1. Similar argument holds good for xih2(x) where i > k2. Hence the bispace generated by the bipolynomial {g1(x) mod (xN + 1), xg1(x) mod (xN + 1), …, x k1 1 g1(x) mod(xN + 1)} ‰ ‰ {h2(x) mod (xN + 1), xh2(x)mod (xN + 1), …, x k 2 1 h2 (x) mod (xN + 1)} has bidimension k1 ‰k2; i.e., the birank of the biword generated by D = D1 ‰ E2 is k1 ‰k2. Now we have the two corollaries to be true. COROLLARY 2.1: If D = D1 ‰ E2 GF(2N) ‰GF(2N) is such that its bipolynomial representation g1(x) ‰ h2(x) is relatively prime to xN+1 ‰ xN+1 (i.e. g1(x) is relatively prime to xN+1 and 37

h2(x) is relatively prime to xN + 1) then the binorm of the biword generated by D = D1 ‰ E2 is (N, N) and f(D) = f1(D1) ‰ f2(E2) is invertible. Proof: Follows from the theorem bigcd(g1(x) ‰ h2(x), xN+1 ‰ xN + 1) = bigcd(g1(x), xN + 1) ‰ (h2(x), xN + 1) =1‰1 has bidegree 0 ‰ 0 and hence birank of f(D) = f1(D1) ‰ f2(E2) is (N, N). COROLLARY 2.2: The binorms of the bivectors corresponding to the bipolynomials x + 1 ‰ x + 1 and xN – 1 + xN – 2 + … + x + 1 ‰ xN – 1 + xN – 2 + … + x + 1 are respectively (N – 1, N – 1) and (1, 1).

Now we proceed on to define the bidistance bifunction on VN ‰ VN. DEFINITION 2.16: The bidistance between two biwords u = u1 ‰ u2 and v = v1 ‰ v2 in VN ‰ VN is defined as d(u, v) = d1(u1, v1) ‰ d2(u2, v2) = r(u + v) = r1(u1 + v1) ‰ r2(u2 + v2).

Now we proceed on to define the new notion of circulant rank bicodes of bilength N = N1 ‰N2. DEFINITION 2.17: Let C1 be a circulant rank code of length N1 which is a subspace of V N1 equipped with the distance function d1(u1, v1) = r1(u1, v1) and C2 be a circulant rank code of length N2 which is the subspace of V N2 equipped with distance function d2(u2, v2) = r2(u2, v2) where V N1 and V N2 are spaces defined over GF(2N) with N1 z N2. C = C1 ‰ C2 is defined as the circulant birank bicode of bilength N = N1 ‰N2 defined as a bisubsapce of V N1 ‰ V N2 equipped with the bidistance bifunction d1(u1, v1) ‰ d2(u2, v2) = r1(u1 + v1) ‰ r2(u2 + v2). 38

DEFINITION 2.18: A circulant birank bicode of bilength N = N1 ‰N2 is called bicyclic if whenever (v11 ...v1N1 ) ‰ (u12 ...u N2 2 ) is a

bicodeword then it implies (v12 v31 ! v1N1 v11 ) ‰ (u22u32 ! u N2 2 u12 ) is also a bicodeword. Now we proceed on to define semi MRD bicode. DEFINITION 2.19: Let C1[n1, k1] be a [n1, k1] RD code and C2[n2, k2, d2] be a MRD code. C1[n1, k1] ‰ C2[n2, k2, d2] is defined as the semi MRD bicode if n1 z n2 and k1 z k2 and C1[n1, k1] is only a RD-code and not a MRD code.

These bicodes will be useful in application where one set of code never attains the singleton-style bound were as the other code attains the singleton style bound. The special nature of these codes will be very useful in applications were two types of RD codes are used simultaneously. Next we proceed on to define the notion of semi circulant rank bicode of type I and type II. DEFINITION 2.20: Let C1 = C1[n1, k1] be a RD-code and C2 be a circulant rank code both are subspaces of rank spaces defined over the same field GF(2N). Then C1 ‰ C2 is defined as the semi circulant rank bicode of type I.

These codes find their application where one RD-code which does not attain its single style bound and another need is a circulant rank code. Now we proceed on to define the new concept of semi circulant rank bicode of type II. DEFINITION 2.21: Let [n1, k1, d1] = C1 be a MRD code and C2 be a circulant rank code both are subspaces or rank spaces defined over the same field GF(2N) or Fq N . C1 ‰ C2 is defined to

be semi circulant bicode of type II.

39

We see in semi circulant rank bicodes of type I, we use only RD codes which are not MRD codes and in semi circulant rank bicodes of type II, we use only MRD codes which are never RD codes. These rank bicodes will find their applications in special situations. Next we proceed on to define semicyclic circulant rank bicode of type I and type II. DEFINITION 2.22: Let C1[n1, k1] be a RD-code which is not a MRD code and C2 be a cyclic circulant rank code, both take entries from the same field GF(2N). C1 ‰ C2 is defined to be a semicyclic circulant rank bicode of type I.

These also can be used when simultaneous use of two different types of rank codes are needed. Next we proceed on to define the notion of semicyclic circulant rank bicode of type II. DEFINITION 2.23: Let [n1, k1, d1] = C1 be a MRD-code which is a subspace of VN, VN defined over GF(2N). C2 be a cyclic circulant rank code with entries from GF(2N). C1 ‰ C2 is defined to be the semicyclic circulant rank bicode.

These bicodes also find their applications when two types of codes are needed simultaneously. Next we proceed on to define semicyclic circulant rank bicode. DEFINITION 2.24: Let C1 be a circulant rank code and C2 be a cyclic circulant rank code, C1 ‰ C2 is defined as the semicyclic circulant rank bicode, both C1 and C2 take entries from the same field.

The special semicyclic circulant rank bicodes can be used in communication bichannel having very high error probability for error correction. Now we proceed on to define yet another class of rank bicodes. DEFINITION 2.25: Let C1[n1, k1] and C2[n2, k2] be any two distinct Almost Maximum Rank Distance (AMRD) codes with the minimum distances greater than or equal to n1 – k1 and n2 –

40

k2 respectively defined over GF(2N). Then C1[n1, k1] ‰ C2[n2, k2] is defined as the Almost Maximum Rank Distance bicode (AMRD-bicode) over GF(2N). An AMRD bicode whose minimum bidistance is greater than n1 – k1 ‰ n2 – k2 is an MRD bicode and hence the class of MRD bicodes is a subclass of the class of AMRD bicode. We have an interesting property about these AMRD bicodes. THEOREM 2.3: When n1 – k1 ‰ n2 – k2 is an odd pair of biintegers (i.e., n1 – k1 and n2 – k2 are odd integers);

(i) The error correcting capability of the [n1, k1] ‰ [n2, k2] AMRD bicode is equal to that of an [n1, k1] ‰ [n2, k2] MRD bicode. (ii) An [n1, k1] ‰ [n2, k2] AMRD bicode is better than any [n1, k1] ‰ [n2, k2] bicode in Hamming metric for error correction. Proof: (i) Suppose C = C1 ‰ C2 is a [n1, k1] ‰ [n2, k2] AMRD bicode such that n1 – k1 ‰ n2 – k2 is an odd biinteger (i.e., n1 – k1 z n2 – k2 are odd integers). The maximum number of bierrors corrected by C = C1 ‰ C2 is given by (n1  k1  1) (n 2  k 2  1) ‰ . 2 2 But (n1  k1  1) (n 2  k 2  1) ‰ 2 2 is equal to the error correcting capability of an [n1, k1] ‰ [n2, k2] MRD bicode (Since n1 – k1 and n2 – k2 are odd). Thus when n1 – k1 ‰ n2 – k2 is biodd (i.e., both n1 – k1 and n2 – k2 are odd) the [n1, k1] ‰ [n2, k2] AMRD bicode is as good as an [n1, k1] ‰ [n2, k2] MRD bicode. (ii) Suppose C = C1 ‰ C2 is a [n1, k1] ‰ [n2, k2] AMRD bicode such that n1 – k1 and n2 – k2 are odd. Then, each bicodeword of C can correct L r1 (n1 ) ‰ L r2 (n 2 ) L r (n) error bivectors where

41

r

r1 ‰ r2

(n1  k1  1) (n 2  k 2  1) ‰ 2 2

and L r (n) L r1 (n1 ) ‰ L r2 (n 2 ) n1 ªn º 1  ¦ « 1 » (2 N  1)! (2 N  2i 1 ) ‰ i 1 ¬ i ¼ n2 ªn º 1  ¦ « 2 » (2 N  1)! (2 N  2i 1 ) . i 1 ¬ i ¼

Consider the same [n1, k1] ‰ [n2, k2] bicode in Hamming metric. Let it be D1 ‰ D2 = D, then the minimum bidistance of D is atmost (n1 – k1 +1) ‰ (n2 – k2 + 1). The error correcting capability of D is (n1  k1  1  1) (n 2  k 2  1  1) ‰ r1 ‰ r2 2 2 ([n1 – k1] and [n2 – k2] are odd). Hence the number of error bivectors corrected by a codeword is given by r2 ª n1 º N ªn 2 º N i i (2 1)  ‰ ¦ ¦ «i» « i » (2  1) i 0 ¬ i 0 ¬ ¼ ¼ r1

which is clearly less than L r1 (n1 ) ‰ L r2 (n 2 ) . Thus the number of error bivectors that can be corrected by the [n1, k1] ‰ [n2, k2] AMRD bicode is much greater than that of the same bicode considered in Hamming metric. For a given bilength n = n1‰n2, a single error correcting AMRD bicode is one having bidimension n1 – 3 ‰ n2 –3 and the minimum distance greater than or equal to 3 ‰ 3. We now proceed on to give a characterization of a single error correcting AMRD bicode in terms of its parity check bimatrices. The characterization is based on the condition for the minimum distance proved by Gabidulin in [24, 27].

42

THEOREM 2.4: Let H = H1 ‰ H2 = (D ij1 ) ‰ (D ij2 ) be a 3 u n1 ‰ 3

u n2 bimatrix of birank 3 over GF(2N); n1 d N and n2 d N which satisfies the following condition. For any two distinct, non empty bisubsets P1, P2 where P1 P11 ‰ P21 and P2 P12 ‰ P22 of {1, 2, …, n1} and {1, 2, 3, …, n2} respectively there exists i1 i11 ‰ i21 , i2 i12 ‰ i22  {1, 2, 3} ‰ {1, 2, 3} such that, § · § · 1 1 2 2 ¨¨ ¦ D 1 1 ˜ ¦ D 1 1 ¸¸ ‰ ¨¨ ¦ D 2 2 ˜ ¦ D 2 2 ¸¸ z © j11P11 i1 j1 k11P21 i2 k1 ¹ © j12 P12 i1 j1 k22 P12 i2 k2 ¹ § · § · 1 1 2 2 ¨¨ ¦ D 1 1 ˜ ¦ D 1 1 ¸¸ ‰ ¨¨ ¦ D 2 2 ˜ ¦ D 2 2 ¸¸ . © j11P11 i2 j1 k11P21 i1 k1 ¹ © j12 P12 i2 j1 k22 P12 i1 k2 ¹ then, H = H1 ‰ H2 as a parity check bimatrix defines a [n1, n1– 3] ‰[n2, n2 –3] single bierror correcting AMRD bicode over GF(2N). Proof: Given H = H1 ‰H2 is a 3 un1 ‰3 un2 bimatrix of birank 3 ‰ 3 overGF(2N), so H = H1 ‰H2 as a parity check bimatrix defines a [n1, n1 – 3] ‰[n2, n2 – 3] RD bicode C = C1 ‰ C2 over GF(2N) where, C1 {x  V n1 | xH1T 0} and C2 {x  V n 2 | xH T2 0} . It remains to prove that the minimum bidistance of C = C1 ‰ C2 is greater than or equal to 3 ‰ 3. We will prove that no non zero bicodeword of C = C1 ‰ C2 has birank less than 3 ‰ 3. The proof is by method of contradiction. Suppose there exists a non zero bicodeword x = x1 ‰ x2 such that r1(x1) d 2 and r2(x2) d 2, then x = x1 ‰ x2 can be written as x = x1 ‰ x2 = (y1 ‰ y2) (M1 ‰ M2) where y1 (y11 , y12 ) and y 2 (y12 , y 22 ) ; y11 , y12 , y12 , y 22  GF(2 N ) and M =

M1 ‰ M2 = (m1ij ) ‰ (mij2 ) is a 2 u n1 ‰ 2 u n2 bimatrix of birank 2 ‰ 2 over GF(2). Thus

43

(yM)HT = y1M1H1T ‰ y 2 M 2 H T2 = 0 ‰ 0 implies that y(MH T )

y1 (M1H1T ) ‰ y 2 (M 2 H T2 ) = 0 ‰ 0.

Since y = y1 ‰ y2 is non zero; y(MHT) = 0 ‰ 0; implies y1(M1 H1T ) = 0 and y2(M2 H T2 ) = 0 that is the 2 u 3 bimatrix M1H1T ‰ M 2 H T2 has birank less than 2 over GF(2N). Now let P1 P11 ‰ P21 {j11 such that m11 j1

1} ‰ {j22 such that m12 j2

1}

1

and P2 P12 ‰ P22

{j11 such that m12 j1

1} ‰ {j22 such that m 22 j2

1} .

2

1

2

Since M = M1 ‰ M2 = (m1ij ) ‰ (mij2 ) is a 2 u 2 bimatrix of birank 2 ‰ 2. P1 and P2 are disjoint non empty bisubsets of {1, 2, …, n1} ‰ {1, 2, …, n2} respectively and M1H T § 1 ¨¦D 1 ¨ j11P11 1j1 ¨ ¨ ¨ 1 D 1 ¨¦ 1 1 © j1P2 1j1

§ 2 ¨¦D 2 ¨ j22 P12 1j2 ¨ ¨ ¨ 2 ¨ 2¦2 D1j2 2 © j2 P2

M1H1T ‰ M 2 H T2 1 ·

1

¦ D 2 j ¦ D 3j ¸ 1 1

j11P11

1

¦ D 2j

1 1

j11P21

2

¦ D2j

2 2

j22 P12

2

¦ D2j

2 2

j22 P22

44

1 1

¸ ¸‰ ¸ 1 ¸ ¦ D 3j1 ¸ 1¹ j11P21

j11P11

2 · 2¸ 2¸ j22 P12 ¸. ¸ 2 ¸ ¦ D 3j2 ¸ 2¹ j22 P22

¦ D 3j

But the selection of H = H1 ‰ H2 is such that their exists i11 ,i12 ,i 22 ,i12  {1, 2, 3} such that 1

¦ D i j ˜ ¦ D1i k

j11P11

z

1 1 1 1

1

¦ Di

j11P11

1 1 2 1

j

k1P21

˜

1 2 1

¦ D1i k

k1P21

‰

1 1 1

¦ D2i j ˜ ¦ D2i k

j22 P12

‰

2 2 1 2

2

¦ Di

j22 P12

2 2 2 2

j

k 2 P22

˜

2 2

2

2

¦ Di k

k 2 P22

2 1

. 2

Hence in MHT = M1H1T ‰ M 2 H T2 ,there exists a 2 u 2 subbimatrix whose determinant is non zero; i.e., r  MH T r1  M1H1T ‰ r2  M 2 H T2 = 2‰2 over GF(2); this contradicts the fact that birank of MH T M1H1T ‰ M 2 H T2  2 ‰ 2 . Hence the result. Now using constant rank code, we proceed on to define the notion of constant rank bicodes of bilength n1‰n2. DEFINITION 2.26: Let C=C1‰C2 be a rank bicode, where C1 is a constant rank code of length n1 (a subset of rank space V n1 ) and C2 is a constant rank code of length n2 (a subset of rank space V n2 ) then C is a constant rank bicode of bilength n1‰n2; that is every bicodeword has same birank. DEFINITION 2.27: A(n1, r1, d1) ‰ A(n2, r2, d2) is defined as the maximum number of bivectors in V n1 ‰ V n2 , constant birank r1‰r2 and bidistance between any two bivectors is atleast d1‰d2. (By a (n1, r1, d1) ‰ (n2, r2, d2) biset we mean a bisubset of bivectors of V n1 ‰ V n2 having constant birank r1‰r2 and bidistance between any two bivectors is atleast d1‰d2).

45

We analyze the bifunction A(n1, r1, d1) ‰ A(n2, r2, d2) by the following theorem. THEOREM 2.5: (i) A(n1, r1, d1) ‰ A(n2, r2, d2)

Lr1 ( n1 ) ‰ Lr2 ( n2 ) , the number

of bivectors of birank r1‰r2 in V n1 ‰ V n2 . (ii) A(n1, r1, d1) ‰ A(n2, r2, d2)  0‰0 if r1 > 0 and r2 > 0 or d1 > n1 and d2 > n2 or d1 > 2r1 and d2 > 2r2. Proof: (i) Follows from the fact that L r1 (n1 ) ‰ L r2 (n 2 ) is the

number of bivectors of bilength n1‰n2, constant birank r1‰r2 and bidistance between any two distinct bivectors in a rank bispace V n1 ‰ V n 2 , is always greater than or equal to 1‰1. (ii) Follows immediately from the definition of A(n1, r1, d1) ‰ A(n2, r2, d2). THEOREM 2.6: A(n1, 1, 2) ‰ A(n2, 1, 2) any Galois field GF(2N).

2 n1  1 ‰ 2 n2  1 over

Proof: Denote by V1 ‰ V2 the set of bivectors of birank 1‰1 in V n1 ‰ V n 2 . We know for each non zero element D1 ‰ D2  GF(2N) there exists (2n1  1) ‰ (2n2  1) bivectors of birank 1 ‰ 1 having D1‰D2 as a coordinate. Thus the cardinality ofV1‰V2 is (2 N  1)(2n1  1) ‰ (2 N  1)(2n 2  1) .Now bidivide

V1‰V2 into (2n1  1) ‰ (2n 2  1) blocks of (2 N  1) ‰ (2 N  1) bivectors such that each block consists of the same pattern of all nonzero bielements of GF(2N) ‰ GF(2N). Then from each biblock almost one bivector can be chosen such that the selected bivectors are atleast rank 2 apart from each other. Such a biset we call as a (n1, 1, 2) ‰ (n2, 1, 2) biset. Also it is always possible to construct such a biset. Thus A(n1, 1, 2) ‰ A(n2, 1, 2) 2n1  1 ‰ 2n 2  1 .

46

THEOREM 2.7: A(n1, n1, n1) ‰ A(n2, n2, n2) = 2N – 1 ‰ 2N – 1 (i.e. A(ni, ni, ni) = 2N – 1; i = 1, 2); over any GF(2N).

Proof: Denote by Vn1 ‰ Vn 2 ; the biset of all bivectors of birank

n1‰n2 in the bispace V n1 ‰ V n 2 . We know the bicardinality of Vn1 ‰ Vn 2 is (2N – 1) (2N – 2) … (2 N  2n1 1 ) ‰ (2N – 1) (2N – 2) … (2 N  2n 2 1 ) , by the definition of a (n1, n1, n1) ‰ (n2, n2, n2) biset the bidistance between any two bivectors should be n1‰n2. Thus no two bivectors can have a common symbol at a coordinate place i1 ‰ i2; (1di1dn1, 1di2dn2). This implies that A(n1, n1, n1) ‰ A(n2, n2, n2) d 2N – 1 ‰ 2N – 1. Now we construct a (n1, n1, n1) ‰ (n2, n2, n2) biset as follows. Select N bivectors from Vn1 ‰ Vn 2 such that 1. Each bibasis bielement of GF(2 N ) ‰ GF(2 N ) should occur (can be as a bicombination) atleast once in each bivector. 2. If the (i1th ,i 2th ) bivector is chosen [(i1  1) th ,(i 2  1) th ] bivector should be selected such that its birank bidistance from any bilinear combination of the previous (i1, i2) bivectors is n1‰n2. Now the set of all bilinear combination of these N‰N bivectors over GF(2)‰GF(2), will be such that the bidistance between any two bivectors is n1‰n2. Hence it is a (n1, n1, n1) ‰ (n2, n2, n2) biset. Also the bicardinality of this (n1, n1, n1) ‰ (n2, n2, n2) biset is 2N – 1 ‰ 2N – 1 (we do not count all zero bilinear combination); thus A(n1, n1, n1) ‰ A(n2, n2, n2) = 2N – 1 ‰ 2N – 1. Recall a [n, 1] repetition RD code is code generated by the matrix G = (1 1 1 … 1) over Fq N . Any non zero codeword has rank 1. DEFINITION 2.28: A [n1, 1] ‰ [n2, 1] repetition RD bicode is a bicode generated by the bimatrix G = G1 ‰ G2 = (11 … 1) ‰ (1 1 … 1) (G1zG2) over F2 N . Any non zero bicodeword has birank

1‰1.

47

We proceed onto define the notion of covering biradius. DEFINITION 2.29: Let C=C1‰C2 be a linear [n1, k1]‰[n2, k2] RD bicode defined over F2 N . The covering biradius of

C=C1‰C2 is defined as the smallest pair of integers (r1, r2) such that all bivectors in the rank bispace F2nN1 ‰ F2nN2 are within the rank bidistance r1 ‰ r2 of some bicodeword. The covering biradius of C=C1‰C2 is denoted by t(C1)‰t(C2). In notation, t(C)=t(C1)‰t(C2) max ­ min ½ ^r ( x  c )`¾ ® x1  F2nN1 ¯c1  C1 1 1 1 ¿

‰

max x2  F2nN2

­ min ½ ^r2 ( x2  c2 )`¾ . ® ¯c2  C2 ¿

THEOREM 2.8: The linear [n1, k1]‰[n2, k2] RD bicode C=C1‰C2 satisfies t(C)=t(C1)‰t(C2) d n1 – k1 ‰ n2 – k2.

Proof is direct. THEOREM 2.9: The covering biradius of a [n1, 1]‰[n2, 1] repetition RD-bicode over F2 N is [n1 – 1]‰[n2 – 1].

Direct from theorem 2.8 as k1 = k2 = 1. Next we proceed on to define the Cartesian biproduct of two linear RD-bicodes. The Cartesian biproduct of two linear RD-bicodes C C1[n11 , k11 ] ‰ C 2 [n12 , k12 ] and D D1[n12 , k12 ] ‰ D 2 [n 22 , k 22 ] over F2N is given by CuD=C1uD1‰C2uD2. ^(a , b ) | a  C1 , b11  D1` ‰ ^(a12 , b12 ) | a12  C2 , b12  D2 ` . 1 1

1 1

C u D is a

1 1

^(n

1 1

 n12 ) ‰ (n12  n 22 ),(k11  k12 ) ‰ (k12  k 22 )` linear

RD bicode (We assume = (n11  n12 ) d N and (n12  n 22 ) d N ).

48

Now the reader is left with the task of proving the following theorem: THEOREM 2.10: If C=C1‰C2 and D=D1‰D2 be two linear RD bicodes then t(C u D) d (t(C1) + t(D1) ‰ t(C2) + t(D2)).

Hint for the proof. If C=C1‰C2 and D=D1‰D2 then C u D = {C1 u D1} ‰ {C2 u D2} and t(C u D ) = (C1 u D1) ‰ (C2 u D2) d {t(C1) + t(D1)} ‰ {t(C2) + t(D2)}. . Now we proceed on to define notion of bidivisble linear RD bicodes. We just recall the definition of divisible linear RD codes. Let C(n, k, d) be a linear RD code over Fq N , n d N and N > 1 with length n, dimension k and minimum distance d. If there exists m > 1 an integer such that m for all 0 z c  C then the code r(c;q) C is defined to be divisible. (r(x; q) denotes the rank norm of x over the field Fq). DEFINITION 2.30: Let C=C1(n1, k1, d1)‰C2(n2, k2, d2) be a linear RD bicode over Fq N , n1 d N, n2 d N and N > 1. If there

exists (m1, m2) (m1 > 1 and m2 > 1) such that m1 m and 2 r  c1 ; q r  c2 ; q for all c1  C1 and for all c2  C2 then we say the bicode C is bidivisible. THEOREM 2.11: Let C=[n1, 11, n1]‰[n2, 12, n2] (n1zn2) be a MRD-bicode for all n1 dN and n2 d N; C is a bidivisible bicode.

Proof: Since there cannot exist bicodewords of birank greater than (n1, n2) in an [n1, 1, n1]‰[n2, 1, n2] MRD-bicode.

49

DEFINITION 2.31: Let C1 = [n1, k1] be a linear RD-code and C2 = (n2, k2, d2) linear divisible RD-code defined over GF(2N). Then the RD-bicode C=C1‰C2 is defined to be a semidivisible RD-bicode. DEFINITION 2.32: Let C1 = (n1, k1, d1) be a MRD code which is not divisible and C2=(n2, k2, d2) a divisible RD code defined over GF(2N); then the bicode C=C1‰C2 is defined to be a semidivisible MRD bicode. DEFINITION 2.33: Let C1 be a circulant rank code and C2=(n2, k2, d2) a divisible RD code defined over GF(2N) then C=C1‰C2 is defined to be semidivisible circulant bicode. DEFINITION 2.34: Let C1 be a AMRD code and C2=(n2, k2, d2) be a divisible RD code defined over GF(2N) then C=C1‰C2 is defined as a semidivisible AMRD bicode.

To show the existence of non divisible MRD bicodes, we proceed on to define certain concepts analogous to the ones used in MRD codes. DEFINITION 2.35: Let C1 be a (n2, k2, d2) MRD code and C2=(n2, k2, d2) MRD code over Fq N ; n1 dN and n2 dN; (n2 z

n2). As1 (n1 , d1 ) ‰ As2 (n2 , d 2 ) be the number of bicodewords with rank norms s1 and s2 in the linear (n1, k1, d1) MRD code and (n2, k2, d2) MRD code respectively. Then the bispectrum of the MRD bicode C1‰C2 is described by the formulae; A0(n2, d2)‰A0(n2, d2) = 1‰1 Ad1  m1 (n1 , d1 ) ‰ Ad2  m2 (n2 , d 2 )

ª n1 º j1  m1 ª d1  m1 º  m1 -j1  m1 -j1 -1  Q j1 1  1 « d  m » ¦  1 «d  j »q 2 1 ¼ j1 0 ¬ 1 ¬ 1 1¼ m1

50

ª n2 º m2 j m ‰«  1 2 2 ¦ » ¬ d 2  m2 ¼ j2 0

ª d 2  m2 º  m2 -j2  m2 -j2 -1 j2 1  Q  1 «d  j »q 2 2 ¼ ¬ 2

m1 = 0, 1, …, n1 – d1, m2 = 0, 1, …, n2 – d2, where ª n1 º «m » ¬ 1¼

q q

ª n2 º «m » ¬ 2¼

q q

and

n1

m1

n2

m2

 1  q n1  q ...  q n1  q m1 1

 1  q m1  q ... q m1  q m1 1  1  q n2  q ...  q n2  q m2 1

 1  q m2  q ...  q m2  q m2 1

with Q = qN. Using the bispectrum of a MRD bicode we prove the following theorem: THEOREM 2.12: All C1[n1, k1, d1]‰C2[n2, k2, d2] MRD bicodes with d1 < n1 and d2 < n2 (i.e., with k1 t 2 and k2 t 2) are non bidivisible.

Proof: This is proved by making use of the bispectrum of the MRD bicodes. Clearly A d1 (n1 ,d1 ) ‰ A d2 (n 2 ,d 2 ) z 0 ‰ 0. If the existence of a bicodeword with birank d1 + 1 ‰ d2 + 1 is established then the proof is complete as bigcd{(d1, d1 + 1) ‰ (d2, d2 + 1)} = 1 ‰ 1. So the proof is to show that A d1 1 (n1 ,d1 ) ‰ A d2 1 (n 2 ,d 2 ) is non zero (i.e., A d1 1 (n1 ,d1 ) z 0 and A d2 1 (n 2 ,d 2 ) z 0 ). Now A d1 1 (n1 ,d1 ) ‰ A d2 1 (n 2 ,d 2 ) = ª n1 º ª d1  1º 2 « d  1» (  « d » [(Q –1) + (Q – 1)] ‰ ¬ 1 ¼ ¬ 1 ¼

51

ª n 2 º ªd 2  1º 2 « d  1» ( « d » [(Q –1) + (Q – 1)] = ¬ 2 ¼ ¬ 2 ¼ § ª n1 º ª d1  1º · « d  1»  Q  1  ¨ Q  1  « d » ¸ ¬ 1 ¼ ¬ 1 ¼¹ © § ª n º ªd  1º · ‰ « 2 »  Q  1  ¨ Q  1  « 2 » ¸ . ¬d 2  1¼ ¬ d2 ¼ ¹ © Suppose that ª d1  1º ªd  1º ‰  Q  1  « 2 » » ¬ d1 ¼ ¬ d2 ¼

 Q  1  «

0‰0 .

i.e., qN  1

q di 1  1 q 1

i.e., q 1

q di  1 q N 1

Clearly q di  1 1. q N 1 For if q di  1 t1 q N 1 then qN–1 < q di – 1, which is not possible as di < ni d N; i = 1, 2. Thus q – 1 < 1 which implies q < 2 a contradiction. Hence A d1 1 (n1 ,d1 ) ‰ A d2 1 (n 2 ,d 2 ) is non zero. Thus except C1(n1, 1, n1) ‰ C2(n2, 1, n2) MRD bicodes all C1[n1, k1, d1] ‰ C2[n2, k2, d2] MRD bicodes with d1 < n1 and d2 < n2 are non divisible.

52

Now we finally define the notion of fuzzy rank distance bicodes. Recall von Kaenel [99] introduced the idea of fuzzy codes with Hamming metric. He analysed the distance properties for symmetric error model. Hall and Gur Dial [41] did it for asymmetric and unidirectional error models. Here we define fuzzy RD bicodes. The study of coding theory resulted from the encounter of noise in communication channels which transmit binary digital data. If a signal 0 or 1 is transmitted electronically it may be distorted into the other signal. A problem occurs when a message in the form of an n-tuple is transmitted, distorted in the channel and received as a new n-tuple representing a different message. If both 1 o 0 and 0 o 1 transitions (or errors) appear in a received word with equal probability then the channel is called symmetric channel and the errors are called symmetric error. In an ideal asymmetric channel only one type of error can occur and the error type is know as apriori. Such errors are known as asymmetric. If both 1 o 0 and 0 o 1 errors can occur in the received words, but in any particular word all error are of one type, then they are called unidirectional errors. DEFINITION [41, 99]: Let F2n denote the n-dimensional vector

space of n-tuples over F2. Let u, v  F2n where u = (u1, …, un) and v = (v1, …, vn). Let p represent the probability that no transition is made and q represent the probability that a transition of the specified type occurs, so that p + q = 1. A fuzzy word fu is the fuzzy subset of F2n defined by fu = {(v, fu(v)) | v  F2n } where fu(v) is the membership function. (i) For the symmetric error model with the Hamming distance, d ¦ ui  vi , f u (v) p n  d q d . i 1

(ii)

For unidirectional error model, ­0 if min  k , k z 0 fu  v ® m d d 1 2 ¯ p q otherwise

where,

53

n

k1

¦ max  0, u

i

 vi

i 1

and n

¦ max  0, v  u

k2

i

i

i 1

d

­k1 if k2 ® ¯k2 if k1

0 0

­ n °¦ ui if k2 0 °° i 1 n m ® n  ¦ ui if k1 0 ° i 1 ° max  6ui , n  6ui if k1 °¯

k2

0.

For the asymmetric error model f u (v )

­0 if min  k1 , k2 z 0 ® md d ¯ p q otherwise

where d = k1 and m = 6ui for asymmetric 1 o 0 error model and d = k2 and m = n – 6ui for asymmetric 0 o 1 error model. DEFINITION 2.36: Asymmetric distance da between u and v is defined as da(u, v) = max(k1, k2), for u, v  F2n . DEFINITION 2.37: The generalized Hamming distance between fuzzy sets is a metric in the set f n = {fu: u  F2n } that is

d ( fu , f v )

¦

f u ( z )  f v ( z ) for fu, fv  f n.

zF2n

Note that if u  F2n represents a received word and C is a codeword then fc(u) is the probability that c was transmitted.

54

THEOREM 2.13: Let u, v  F2n be such that dH(u, v) = d. If p

zq and p z0, 1 then d ( f u , f v )

d

§d ·

i 0

© ¹

¦¨ i ¸

p i q d i  p d i q i

for a

symmetric error model. THEOREM 2.14: Let u, v  F2n be such that da(u, v) = da. If p z

q and p z 0, 1 then d ( fu , f v ) 2(1  q da ) for an asymmetric error model. These two theorems show that the distance between the fuzzy words is dependent only on the Hamming or asymmetric distance (as the case may be) between the base codewords and not on the dimension of the code space. On the other hand it is not so with the unidirectional error model. Now, we proceed onto recall the definition of fuzzy RD codes and their properties. For more please refer [77, 96]. DEFINITION 2.38: Let Vn denote the n-dimensional vector space of n-tuples over F2 N , n d N and N > 1. Let u, v  Vn where u =

(u1, u2, …, un) and v = (v1, v2, …, vn) with each ui, vi  F2n . A fuzzy RD word fu is the fuzzy subset of Vn defined by fu = {(v, fu(v)) | v  Vn} where fu(v) is the membership function. DEFINITION 2.39: For the symmetric error model, assume p to represent the probability that no transition (i.e., error) is made and q to represent the probability that a rank error occurs so that p + q = 1. Then fu(v) = pn–rqr where r = r(u – v; 2) = ||u – v||. DEFINITION 2.40: For the unidirectional and asymmetric error models assume q to represent the probability that 1o 0 transition or 0 o 1 transition occurs. n

Then fu (v)

–f

ui

(vi ) where f ui (vi ) inherits its definition

i 1

from the above equation for unidirectional and asymmetric

55

error models respectively, since each ui or vi itself in an N-tuple over F2. That is since ui, vi  F2 N each ui or vi itself is an Ntuple from F2. Let ui = (ui1, ui2, …, uiN) and vj = (vj1, vj2, …, vjN) where uis, vjt  F2, 1 d s, t d N. Then for unidirectional error model, ­ 0 if min  k ,k z 0 fui  vi ® mi  di di i1 i 2 q otherwise ¯p where, n

¦ max  0,u

ki1

is

 vis

is

 uis

s 1

and n

¦ max  0,v

ki 2

s 1

di

mi

­ki1 if ki 2 ® ¯ki 2 if ki1

0 0

­ n °¦ uis if ki 2 0 °° s 1 n ® n  u if k 0 is i1 ° ¦ s 1 ° °¯max  6uis ,n  6uis if ki1

. ki 2

For the asymmetric error model ­ 0 if min  k ,k z 0 fui ( vi ) ® mi  di di i1 i 2 q otherwise ¯p where di = ki1 and N

mi

¦u

is

s 1

56

0.

for asymmetric 0 o 1 error model di = ki2 and n

n  ¦ uis

mi

s 1

for asymmetric 0 o 1 error model. DEFINITION 2.41: Let f n = {fu: u  Vn}. Let \: Vn o f n be defined as \(u) = fu. Clearly \ is a bijection. Let C[n, k, d] be an RD code which is a subspace of Vn of dimension k and minimum rank distance d. Then \(C) Ž f n is a fuzzy RD code. If c  C then fc is a fuzzy RD codeword of \(C). For any RD code \(C) its minimum distance is defined as,

d min (\ ( C )) min ^d( f a , f b ) : f a z f b ` f a , fb \  C

where d  f a , f b

¦

zV

n

f a  z  f b  z is a metric in f n.

DEFINITION 2.42: If u  Vn represents a received codeword and c  C then fc(u) gives the probability that c was transmitted. Let T(u) = {fa | a  C, fa(u) t fb(u), b  C}. A code for which |T(u)| = 1 for all u  Vn is said to be uniquely decodable. In such a case u is decoded as \ –1(T(u)).

Now we proceed on to define the notion of fuzzy RD bicodes. DEFINITION 2.43: Let V n1 ‰ V n2 denote (n1, n2) dimensional vector bispace of (n1, n2)-tuples over F2 N ; n1 d N and n2 d N, N

> 1. Let u1 , v1 V n1 , u2 , v2 V n2 where u1 (u11 ,..., u1n1 ), v1

u2 with fu1 ‰u2

(v11 ,..., v1n1 ) ,

(u12 ,..., un22 ) and v2

ui1 , ui2 , vi1 , vi2  F2N .

A

(v12 ,..., vn22 )

fuzzy

RD

bicodeword

f u11 ‰ f u22 is a fuzzy bisubset of V n1 ‰ V n2 defined by,

fu1 ‰u2

f u11 ‰ f u22

57

= {(v1 , f u11 (v1 )) | v1 V n1 } ‰ {(v2 , f u22 (v2 )) | v2 V n2 } where fu11 (v1 ) ‰ f u22 (v2 ) is the membership bifunction. DEFINITION 2.44: For the symmetric error bimodel assume p1‰p2 to represent the biprobability that no transition (i.e., error) is made and q1‰q2 to represent the biprobability that a birank error occurs so that p1+q1‰p2+q2=1‰1. Then fu11 (v1 ) ‰ f u22 (v2 ) p1n1  r1 q1r1 ‰ p2n2  r2 q2r2

where r1 = r1(u1 – v1, 2) = ||u1 – v1|| and r2 = r2(u2 – v2, 2) = ||u2 – v2||. DEFINITION 2.45: For unidirectional and asymmetric error bimodels assume q1‰q2 to represent the probability that (1 o 0) ‰ (1 o 0) bitransition or (0 o 1) ‰ (0 o 1) bitransition occurs. Then

fu11  v1 ‰ f u22  v2

n1

i 1

where

f

1 u1

 v1 ‰ f  v2 2 u2

n2

– f v ‰ – f v 1 ui1

1 i

i 1

2 ui2

2 i

inherits its definition from the

unidirectional and asymmetric bimodels respectively since each ui1 ‰ ui2 or vi1 ‰ vi2 itself is an N-bituple over F2. That is since ui1 , ui2 , vi1 , vi2  F2N each ui1 ‰ ui2 or vi1 ‰ vi2 itself is an N-tuple from F2. 1 ui1 (ui11 , ui12 ,..., uiN ) , v1j ui2

(v1j1 , v1j 2 ,..., v1jN ) ,

(ui21 , ui22 ,..., uiN2 ) , and v 2j

(v 2j1 , v 2j 2 ,..., v 2jN )

where uis1 , uis2 , v1jt , v 2jt  F2N , 1 d s, t d N. Then for unidirectional error bimodel,

58

f

1 ui1

v ‰ f v 1 i

2 ui2

2 i

­°0 ‰ 0 if min  ki11 ,ki12 ‰ min  ki12 ,ki22 z 0 ‰ 0 ® m1  d 1 d 1 2 2 2 °¯ p1 i i q1 i ‰ p2mi  di q2di otherwise

where N

ki11

¦ max 0,u

N

 vis1 , ki12

1 is

s 1

¦ max 0,v

1 is

s 1

N

¦ max 0,u

ki12

2 is

 vis2

2 is

 uis2 ;

s 1

 uis1

and N

ki22

¦ max 0,v s 1

where

mi1

di1

­°ki11 if ki12 ® 1 1 °¯ki 2 if ki1

0 0

di2

­°ki12 if ki22 ® 2 2 °¯ki 2 if ki1

0 0

­ N 1 1 ° ¦ uis if ki 2 0 s 1 ° ° N ® N  u 1 if k 1 0 ¦ is i1 ° s 1 ° 1 1 1 °¯max  6uis ,N  6uis if ki1

ki12

0

­ N 2 2 ° ¦ uis if ki 2 0 s 1 ° ° N ® N  u 2 if k 2 0 ¦ is i1 ° s 1 ° 2 2 2 °¯max  6uis ,N  6uis if ki1

ki22

0

and

mi2

59

for the asymmetric error bimodel

­°0 ‰ 0 if min  ki11 ,ki12 ‰ min  ki12 ,ki22 z 0 ‰ 0 fu11  vi1 ‰ f u22  vi2 ® 1 1 1 2 2 2 i i °¯ p1mi  di q1di ‰ p2mi  di q2di otherwise where d i1

ki11 , d i2

ki21 and N

mi1

N

¦ uis1 , mi2

¦u

2 is

s 1

s 1

for asymmetric (1 o 0) ‰ (1 o 0) error bimodel and d i1 ki11 , d i2 ki12 , N

mi1

N  ¦ uis1 s 1

and N

mi2

N  ¦ uis2 s 1

for the asymmetric (1o0)‰0o1) error bimodel.

The minimum bidistance of a fuzzy RD bicode. DEFINITION 2.46: Let f n1 ‰ f n2 { f u11 u1 V n1 } ‰ { f u22 u2 V n2 } .

Let

\ \ 1 ‰\ 2 :V n ‰ V n o f n ‰ f n defined by \ ( u1 ) ‰\ ( u2 ) f u1 ‰ f u2 . Clearly \1 is a bijection 1

2

1

1

2

2

and \2 is a bijection. Let C1[n1, k1, d1]‰C2[n2, k2, d2] be an RD bicode which is a subbispace of V n1 ‰ V n2 of bidimension k1 ‰ k2 and minimum rank bidistance d1 ‰ d2. Then \ 1 ( C1 ) ‰\ 2 ( C2 ) Ž f n1 ‰ f n2 is a fuzzy RD bicode. If c = c1 ‰ c2 C1‰C2 then f c1 ‰ f c2 is a fuzzy bicodeword of \ 1 ( C1 ) ‰\ 2 ( C2 ) .

60

For any fuzzy RD bicode \ 1 ( C1 ) ‰\ 2 ( C2 ) , its minimum bidistance is defined as, d min\ 1 ( C1 ) ‰ d min\ 2 ( C2 )

^

min d f a11 , f b11

^



`

f a11 , f b11 \ 1  C1 ; f a11 z f b11 ‰

`

min d( f a22 , fb22 ) f a22 , f b22 \ 2  C2 ; f a22 z f b22 . where









d f a11 , f b11 ‰ d f a22 , fb22 =

¦

z1

V n1

f a11  z1  fb11  z1 ‰

¦

z2

is a bimetric in f

n1

‰f

n2

V n2

f a22  z2  f b22  z2

.

DEFINITION 2.47: If u1 ‰ u2  V n1 ‰ V n2 represents a received biword and c1 ‰ c2  C1 ‰ C2 then f c11  u1 ‰ f c22  u2 gives the

biprobability that (c1 ‰ c2) was transmitted.

^f ^f

T1(u1) ‰ T 2(u2) =

`

1 a1

| a1  C1 ; f a11  u1 t fb11  u1 , b1  C1 ‰

2 a2

| a2  C2 ; f a22  u2 t fb22  u2 , b2  C2 .

`

A bicode for which |T1(u1) ‰ T2(u2)| = |T1(u1)| ‰ |T2(u2)| =1‰1 n1 for all u1  V and u2  V n2 ; is said to be uniquely bidecodable. In such a case u1 ‰ u2 is bicoded as

\ 11 (T1 (u1 )) ‰\ 21 (T 2 (u2 )) . The notions related to m–covering radius of RD–codes can be analogously transformed from the notion of RD–bicodes. PROPOSITION 2.1: If C11 ‰ C21 and C12 ‰ C22 are RD bicodes

with C11 ‰ C21 Ž C12 ‰ C22 (i.e., C11 Ž C12 and C21 Ž C22 ) then

61

tm1 (C11 ) ‰ tm2 (C21 ) t tm1 (C12 ) ‰ tm2 (C22 ) [ tm1 (C11 ) t tm1 (C12 ) and

tm2 (C21 ) t tm2 (C22 ) ]. Proof: Let S1 Ž V n1 with |S1| = m1 and S2 Ž V n 2 with |S2| = m2 cov(C12 ,S1 ) ‰ cov(C 22 ,S2 )

= min ^cov(x1 ,S1 ) x1  C12 ` ‰ min ^cov(x 2 ,S2 ) x 2  C22 ` d min ^cov(x1 ,S1 ) x1  C11` ‰ min ^cov(x 2 ,S2 ) x 2  C12 ` cov(C11 ,S1 ) ‰ cov(C12 ,S2 ) . Thus t m1 (C12 ) ‰ t m2 (C 22 ) d t m1 (C11 ) ‰ t m2 (C12 ) . PROPOSITION 2.2: For any RD bicode C1 ‰ C2 and a pair of positive integers (m1, m2), tm1 (C1 ) ‰ tm2 (C2 ) d tm1 1 (C1 ) ‰ tm2 1 (C2 ) .

Proof: Let S1 Ž V n1 with |S1| = m1 and S2 Ž V n 2 with |S2| = m2, V n1 ‰ V n 2 is the rank bispace where S1 ‰ S2 is a bisubset of V n1 ‰ V n 2 . Now t m1 (C1 ) ‰ t m2 (C 2 ) =

=

max{cov(C1, S1) | S1 Ž max{cov(C2, S2) | S2 Ž max{cov(C1, S1) | S1 Ž max{cov(C2, S2) | S2 Ž t m1 1 (C1 ) ‰ t m2 1 (C 2 ) .

V n1 ; |S1| = m1} ‰ V n 2 ; |S2| = m2} d V n1 ; |S1| = m1 + 1} ‰ V n 2 ; |S2| = m2 + 1}

PROPOSITION 2.3: For any biset of positive integers {n1, m1, k1, K1} ‰ {n2, m2, k2, K2}; tm1 [n1 , k1 ] ‰ tm2 [n2 , k2 ] d tm1 1[n1 , k1 ] ‰ tm2 1[ n2 , k2 ]

and tm1 (n1 , K1 ) ‰ tm2 (n2 , K 2 ) d tm1 1 (n1 , K1 ) ‰ tm2 1 (n2 , K 2 ) .

62

Proof: Given C1[n1, k1] ‰ C2[n2, k2] RD bicode, with C1 Ž V n1 and C2 Ž V n 2 . Now t m1 [n1 , k1 ] ‰ t m2 [n 2 , k 2 ]

min{ t m1 (C1 ) C1 Ž V n1 ;dim C1

k1} ‰

min{t m2 (C2 ) C2 Ž V n 2 ;dim C2

k2} d

min {t m1 1 (C1 ) C1 Ž V ;dim C1

k1 } ‰

min{t m2 1 (C2 ) C2 Ž V ;dim C2

k2}

n1

n2

t m1 1[n1 , k1 ] ‰ t m2 1[n 2 , k 2 ] . Similarly we have t m1 (n1 , K1 ) ‰ t m2 (n 2 , K 2 ) d t m1 1 (n1 , K1 ) ‰ t m2 1 (n 2 , K 2 ) . That is t m1 (n1 , K1 ) ‰ t m2 (n 2 , K 2 )

min{t m1 (C1 ) C1 Ž V n1 ; C1

K1 } ‰

min{t m2 (C2 ) C2 Ž V n 2 ; C2

K2}

min{t m1 1 (C1 ) C1 Ž V n1 ; C1

K1 } ‰

min{t m2 1 (C2 ) C2 Ž V ; C2

K2}

n2

d t m1 1 (n1 , K1 ) ‰ t m2 1 (n 2 , K 2 ) . PROPOSITION 2.4: For any biset of positive integers {n1, m1, k1, K1} ‰ {n2, m2, k2, K2}; tm1 [n1 , k1 ] ‰ tm2 [n2 , k2 ] t tm1 [n1 , k1  1] ‰ tm2 [n2 , k2  1]

and tm1 (n1 , K1 ) ‰ tm2 (n2 , K 2 ) t tm1 (n1 , K1  1) ‰ tm2 (n2 , K 2  1) . Proof: Given C = C1 ‰ C2 is a RD bicode hence a bisubspace of V n1 ‰ V n 2 . Consider t m1 [n1 , k1  1] ‰ t m2 [n 2 , k 2  1]

63

min{t m1 (C1 ) C1 Ž V n1 ;dim C1

min{t m2 (C2 ) C2 Ž V ;dim C2 n2

k1  1} ‰

k 2  1} d

min{t m1 (C1 ) C1 Ž V ;dim C1

k1} ‰

min{t m2 (C2 ) C2 Ž V n 2 ;dim C2

k2} .

n1

(since for each C1 ‰ C2 Ž C12 ‰ C22; t m1 (C12 ) ‰ t m2 (C22 ) d t m1 (C1 ) ‰ t m2 (C2 )) t m1 (n1 , k1 ) ‰ t m2 (n 2 , k 2 ) . Similarly, t m1 [n1 , K1  1] ‰ t m2 [n 2 , K 2  1] d t m1 (n1 , K1 ) ‰ t m2 (n 2 , K 2 ) . Using these results and the fact k1m1 [n1 , t1 ] denotes the smallest dimension of a linear RD code of length n1 and m1-covering radius t1 and k1m1 [n1 , t1 ] denotes the least cardinality of the RD codes of length n1 and m1-covering radius t1. The following results can be easily proved. Result 1: For any biset of positive integers {n1, m1, t1} ‰ {n2, m2, t2} and k m1 (n1 , t1 ) ‰ k m2 (n 2 , t 2 ) d k m1 1 (n1 , t1 ) ‰ k m2 1 (n 2 , t 2 ) and K m1 (n1 , t1 ) ‰ K m2 (n 2 , t 2 ) d K m1 1 (n1 , t1 ) ‰ K m2 1 (n 2 , t 2 ) . Result 2: For any biset of positive integers {n1, m1, t1} ‰ {n2, m2, t2} we have k m1 [n1 , t1 ] ‰ k m2 [n 2 , t 2 ] t k m1 [n1 , t1  1] ‰ k m2 [n 2 , t 2  1] and K m1 (n1 , t1 ) ‰ K m2 (n 2 , t 2 ) t K m1 (n1 , t1  1) ‰ K m2 (n 2 , t 2  1) . We say a bifunction f1 ‰ f2 is a non-decreasing function in some bivariable say (x1 ‰ x2) if both f1 and f2 happen to be a nondecreasing function in the same variable x1 and x2 respectively.

64

With this understanding we can say the (m1, m2)- covering biradius of a fixed RD bicode C1 ‰ C2, t m1 [n1 , k1 ] ‰ t m2 [n 2 , k 2 ] , t m1 (n1 , K1 ) ‰ t m2 (n 2 , K 2 ) , k m1 [n1 , t1 ] ‰ k m2 [n 2 , t 2 ] and K m1 (n1 , t1 ) ‰ K m2 (n 2 , t 2 ) , are non decreasing bifunctions of (m1, m2). The relationship between the multicovering biradii of two RD bicodes and bicodes that are built using them are described. Let Ci C1i ‰ Ci2 , i = 1, 2 be a [n11 , k11 ,d11 ] ‰ [n12 , k12 ,d12 ] , [n12 , k12 ,d12 ] ‰ [n 22 , k 22 ,d 22 ]

RD

bicodes

over

F2N

with

n11 , n12 , n12 , n 22 , n11  n12 , n12  n 22 d N .

PROPOSITION 2.5: Let C 1 C11 ‰ C21 and C 2 C12 ‰ C22 be RD bicodes described above. C C 1 u C 2 C11 u C12 ‰ C21 u C22

^( x

1

| y1 ) x1  C11 , y1  C12 ` ‰ ^( x2 | y2 ) x2  C21 , y2  C22 ` .

Then C 1 u C 2 is a [n11  n21 ‰ n12  n22 , k11  k21 ‰ k12  k22 , min{ d11 ,d 21 } ‰ min{ d12 ,d 22 }] rank distance bicode over F2 N and tm1 (C11 u C21 ) ‰ tm2 (C12 u C22 ) d

tm1 (C12 )  tm1 (C21 ) ‰ tm2 (C12 )  tm2 (C22 ) . Proof: Let S1 Ž V n1  n 2 and S2 Ž V n1  n 2 1

1

2

2

and

S1 {s11 , !, s1m1 } and S2

{s12 ,...,s 2m2 }

with n11

s1i

(x1i y1i ) and si2

(x 2i y 2i )

n12

n12

2

x1i  V , y1i  V , x 2i  V and y 2i  V n 2 . Let S11 {x11 ! x1m1 }, S12 {y11 ! y1m1 } ,

S12

{x 21 ! x1m2 } and S22

65

{y 21 ! y 2m2 } .

Now t m1 (C11 ) being the m1-covering radius of (C11 ) there exists c11  C11 such that S11 Ž B1t

1 m1 (C1 )

(C11 ) . This implies r1 (x1i  c11 ) d

t m1 (C1 ) for all x1i  S11 . The same argument is true for (C12 ) .

Now consider (C12 ) , this code has m2 covering radius t m2 (C12 ) such that there exists c12  C12 such that S12 Ž B1t

m2

(C12 )

(C12 ) . This

implies r2 (x 2i  c12 ) d t m2 (C 2 ) for all x 2i  S12 . Now C (C11 | C12 ) ‰ (C12 | C 22 ) C1 ‰ C2 . Here r1 (s1i  C1 ) r1 ((x1i | y1i )  (C11 | C12 )) r1 (x1i  C11 | y1i  C12 ) d r1 (x1i  C11 )  r1 (y1i  C12 )

d t m1 (C11 )  t m1 (C12 ) . Similarly we have, r2 (s 2i  C 2 ) r2 ((x 2i | y 2i )  (C12 | C 22 )) r2 (x 2i  C12 | y 2i  C 22 ) d r2 (x 2i  C12 )  r2 (y 2i  C 22 ) d t m2 (C12 )  t m2 (C 22 ) .

Thus t m (C)

t m1 (C11 u C12 ) ‰ t m2 (C12 u C22 ) d

t m1 (C11 )  t m1 (C12 ) ‰ t m2 (C12 )  t m2 (C 22 ) . For any positive integer r, the r-fold repetition RD code C1 is the code C = {(c | c | … | c) | c  C1} where the code word c is concatenated r-times. This is a [rn1, k1, d1] rank distance code. Note that here n1 d N is choosen so that rn1 d N. We proceed on to define (r, r)-fold repetition of RD bicode. DEFINITION 2.48: For any (r, r) (r any positive integer), the (r, r) repetition of RD bicode C1 ‰ D1 is the bicode C = {(c | c | … | c) | c  C1} ‰ D = {(d | d | … | d) | d  D1} where the bicode word c ‰ d is concatenated r-times this is a [rn1, k1, d1] ‰ [rn2, k2, d2] rank distance bicode word with ni d N and rni d N; i = 1,

66

2. Thus any bicode word in C ‰ D would be of the form (c | c | … | c) ‰ (d | d | … | d) where c  C1 and d  D1. We can also define (r1, r2) fold repetition bicode (r1 z r2). DEFINITION 2.49: Let C1 ‰ D1 be a [n1, k1, d1] ‰ [n2, k2, d2] RD-code. Let C = {(c | c | … | c) | c  C1} be a r1-fold repetition RD code C1 and D = {(d | d | … | d) | d  D1} be a r2-fold repetition RD code C2 (r1 z r2). Then C ‰ D is defined as the (r1, r2)-fold repetition bicode.

We prove the following interesting result. PROPOSITION 2.6: For an (r, r) fold repetition RD-bicode C ‰ D, tm1 (C ) ‰ tm2 ( D) tm1 (C1 ) ‰ tm2 ( D1 ) .

Proof: Let S1 {v1 ! v m1 } Ž V n1 be such that cov(C1, S1) = t m1 (C1 ) . Let S2

{u1 ! u m2 } Ž V n 2 be such that cov(D1, S2) =

t m2 (D1 ) . Let v1i

(vi | vi | ! | vi ) . Let S11 {v11 | v12 | ! | v1m1 } be a set

of m1-vectors of length rn1 each. A r-fold repetition of any RD code word retains the same rank weight. Hence (C,S11 ) t m1 (C1 ) . Since t m1 (C) t cov(C,S11 ) , it follows that t m1 (C) t t m1 (C1 )

------

I

Conversely let S1 = {v1 … v m1 } be a set of m-vectors of length rn1 with vi

(v1i | v1i | ... | v1i ) ; v1i  V n1 . Then there exists c

 C1 such that d R1 (c, v1i ) d t m1 (C1 ) for every i (1 d i d m1). This

implies d R1 ((c | c | ...| c), vi ) d t m1 (C1 ) for every i (1 d i d m1). Thus t m1 (C) d t m1 (C1 )

------

From I and II, t m1 (C)

t m1 (C1 ) .

67

II

On similar lines we can prove, t m2  D t m2  D1 where cov(D1 ,S2 )

t m2 (D1 ) .

Hence t m1 (C) ‰ t m2 (D)

t m1 (C1 ) ‰ t m2 (D1 ) .

Multi-covering bibounds for RD-bicodes is discussed and a few interesting properties in this direction are given. The (m1, m2) covering biradius t m1 (C1 ) ‰ t m2 (C 2 ) of a RD-bicode C = C1 ‰ C2 is a non-decreasing bifunction of m1 ‰ m2 (proved earlier). Thus a lower bi-bound for t m1 (C1 ) ‰ t m2 (C 2 ) implies a bibound for t m1 1 (C1 ) ‰ t m2 1 (C 2 ) . First bibound exhibits m1 ‰ m2 t 2 ‰ 2 then situation of (m1, m2)-covering biradii is quite different for ordinary covering radii. PROPOSITION 2.7: If m1 ‰ m2 t 2 ‰ 2 then the (m1, m2)covering biradii of a RD bicode C = C1 ‰ C2 of bilength (n1, n2) ªn º ªn º is atleast « 1 » ‰ « 2 » . «2» «2»

Proof: Let C = C1 ‰ C2 be a RD bicode of bilength (n1, n2) over GF(2N). Let m1 ‰ m2 t 2 ‰ 2, let t1, t2 be the 2-covering biradii of the RD code C = C1 ‰ C2. Let x = x1 ‰ x2  V n1 ‰ V n 2 . Choose y y1 ‰ y 2  V n1 ‰ V n 2 such that all the (n1, n2) coordinates of x  y (x1  y1 ) ‰ (x 2  y 2 ) are linearly independent, that is d R (x, y) d R (x1 ‰ x 2 , y1 ‰ y 2 ) d R1 (x1 , y1 ) ‰ d R 2 (x 2 , y 2 )

(R

R 1 ‰ R 2 and d R1

d R1 ‰ d R 2 )

= n1 ‰ n2 . Then for any c c1 ‰ c 2  C C1 ‰ C2 , d R (x,c)  d R (c, y)

68

d R1 (x1 ,c1 )  d R1 (c1 , y1 ) ‰ d R 2 (x 2 ,c 2 )  d R 2 (c2 , y 2 ) t d R1 (x1 , y1 ) ‰ d R 2 (x 2 , y 2 )

= n1 ‰ n2 . This implies that one of d R1 (x1 ,c1 ) ‰ d R 2 (x 2 ,c2 ) and d R1 (c1 , y1 ) ‰ d R 2 (c 2 , y 2 ) is at least atleast

n1 n 2 ‰ (that is one of d R1 (x1 ,c1 ) and d R1 (c1 , y1 ) is 2 2

n1 and one of d R 2 (x 2 ,c 2 ) and d R 2 (c 2 , y 2 ) is at least 2

n2 ) and hence 2

t

ªn º ªn º t1 ‰ t 2 t « 1 » ‰ « 2 » . «2» «2»

Since t is non decreasing bifunction of m1 ‰ m2 it follows that ªn º ªn º t m1 (C1 ) ‰ t m2 (C 2 ) t « 1 » ‰ « 2 » «2» «2» for m1 ‰ m2 t 2 ‰ 2. t m (C)

Bibounds of the multi-covering biradius of V n1 ‰ V n 2 can be used to obtain bibounds on the multi covering biradii of arbitary bicodes. Thus a relationship between (m1, m2)-covering biradii of an RD bicode and that of its ambient bispace V n1 ‰ V n 2 is established THEOREM 2.15: Let C = C1 ‰ C2 be any RD-code of bilength n1 ‰ n2 over F2 N ‰ F2 N . Then for any pair of positive integers

(m1, m2); tm1 1 (C1 ) ‰ tm2 2 (C2 ) d t11 (C1 )  tm1 1 (V n1 ) ‰ t12 (C2 )  tm2 2 (V n2 ) .

69

Proof: Let S S1 ‰ S2 Ž V n1 ‰ V n 2 (i.e., S1 Ž V n1 and S2 Ž V n 2 ) with |S| = |S1| ‰ |S2| = m1 ‰ m2. Then there exists u = u1 ‰ u2  V n1 ‰ V n 2 such that cov(u,S) cov(u1 ,S1 ) ‰ cov(u 2 ,S2 ) d t1m1 (V n1 ) ‰ t1m1 (V n 2 ) . Also there is a c c1 ‰ c 2  C1 ‰ C 2 such that

d R (c, u )

d R1 (c1 , u 1 ) ‰ d R 2 (c 2 , u 2 ) d t 11 (C1 ) ‰ t 12 (C 2 ) .

Now cov(c, S) = cov(c1, S1) ‰ cov(c2, S2)

^

`

^

max d R1 (c1 , y1 ) y1  S1 ‰ max d R 2 (c 2 , y 2 ) y 2  S2

^ max ^d

` S `

`

d max d R1 (c1 , u1 )  d R1 (u1 , y1 ) y1  S1 ‰ R2

(c 2 , u 2 )  d R 2 (u 2 , y 2 ) y 2

2

d R1 (c1 , u1 )  cov(u1 ,S1 ) ‰ d R 2 (c 2 , u 2 )  cov(u 2 ,S2 )

d t11 (C1 )  t1m1 (V n1 ) ‰ t12 (C2 )  t 2m2 (V n 2 ) . Thus for every S S1 ‰ S2 Ž V n1 ‰ V n 2 with |S| = m = |S1| ‰ |S2| = m1 ‰ m2 one can find a c c1 ‰ c 2  C1 ‰ C 2 such that, cov(c, S) = cov(c1, S1) ‰ cov(c2, S2) d t11 (C1 )  t1m1 (V n1 ) ‰ t12 (C2 )  t 2m2 (V n 2 ) . Since cov(c, S) = cov(c1, S1) ‰ cov(c2, S2) min ^cov(a1 ,S1 ) a1  C1` ‰ min ^cov(a 2 ,S2 ) a 2  C2 `

^

` ^

`

d t11 (C1 )  t1m1 (V n1 ) ‰ t12 (C 2 )  t 2m2 (V n 2 ) ; for all S S1 ‰ S2 Ž V n1 ‰ V n 2 with |S| = |S1| ‰ |S2| = m1 ‰ m2, it follows that t1m1 (C1 ) ‰ t 2m2 (C 2 ) max ^cov(C1 ,S1 ) S1 Ž V n1 ; S1

70

m1` ‰

max ^cov(C 2 ,S2 ) S2 Ž V n 2 ; S2

^

` ^

m2 `

`

d t11 (C1 )  t1m1 (V n1 ) ‰ t12 (C 2 )  t 2m2 (V n 2 ) . PROPOSITION 2.8: For any pair of integers (n1, n2); n1 ‰ n2 t 2 ‰ 2, t21 (V n1 ) ‰ t22 (V n2 ) d n1 – 1 ‰ n2 – 1; where V n1 F2nN1 ,

F2nN2 ; n1 d N and n2 d N.

V n2

Proof: Let

x1

( x 11 , x 12 ,..., x 1n1 ) , y1

( y11 , y12 ,..., y1n1 )  V n1

and

x2

( x 12 , x 22 ,..., x 2n 2 ) , y 2

( y12 , y 22 ,..., y 2n 2 )  V n 2 .

Let u

u1 ‰ u 2  V n1 ‰ V n 2

u1

(x11u12 u13 ! u1n1 1 y1n1 )

u2

(x12 u 22 u 32 ! u n2 2 1 y n2 2 ) .

where and

Thus u = u1 ‰ u2 bicovers x1 ‰ x2 and y1 ‰ y 2  V n1 ‰ V n 2 with in a biradius n1 – 1 ‰ n2 – 1 as d R1 (u1 , x1 ) d n1  1 and d R 2 (u 2 , x 2 ) d n 2  1 and d R1 (u1 , y1 ) d n1  1 and d R 2 (u 2 , y 2 ) d n 2  1 .

Thus for any pair of bivectors x1 ‰ x2 and y1 ‰ y2 in V n1 ‰ V n 2 there always exists a bivector namely u = u1 ‰ u2 which bicovers x1 ‰ x2 and y1 ‰ y2 within a biradius n1 – 1 ‰ n2 – 1. Hence t12 (V n1 ) ‰ t 22 (V n 2 ) d (n1  1 ‰ n 2  1) . Now we proceed on to describe the notion of generalized sphere bicovering bibounds for RD bicodes. A natural question is for a given t1 ‰ t2, m1 ‰ m2 and n1 ‰ n2 what is the smallest RD bicode whose m1 ‰ m2 bicovering biradius is atmost t1 ‰ t2.

71

As it turns out even for m1 ‰ m2 t 2 ‰ 2, it is necessary that t1 n n ‰ t2 be atleast 1 ‰ 2 . Infact the minimal t1 ‰ t2 for which 2 2 such a bicode exists is the (m1, m2) bicovering biradius of C1 ‰ C2 = F2nN1 ‰ F2nN2 . Various external values associated with this notion are t (V n1 ) ‰ t 2m2 (V n 2 ) the smallest (m1, m2)-covering biradius 1 m1

among bilength n1 ‰ n2 RD bicodes t1m1 (n1 , K1 ) ‰ t 2m2 (n 2 , K 2 ) , the smallest (m1, m2) covering biradius among all (n1 , K1 ) ‰ (n 2 , K 2 ) RD bicodes. K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) is the smallest bicardinality of bilength n1 ‰ n2 RD bicode with m1 ‰ m2 covering biradius t1 ‰ t2 and so on. It is the latter quality that is studied in the book for deriving new lower bibounds. From the earlier results K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) is undefined if n1 n 2 ‰ . 2 2 When this is the case, it is accepted to say K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) = f ‰ f. t1 ‰ t 2 

There are other circumstances when K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) is undefined. For instance K12Nn1  n1 , n1  1 ‰ K 22Nn2  n 2 , n 2  1 = f ‰ f . Also K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) = f ‰ f , m1 ! V  n1 , t1

and

m2 ! V  n 2 , t 2 ,

since in this case no biball of biradius t1 ‰ t2 covers any biset of m1 ‰ m2 distinct bivectors. More generally one has the fundamental issue of whether K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 )

72

is bifinite for a given n1, m1, t1 and n2, m2, t2. This is the case if and only if t 1m1 (V n1 ) d t 1 and t 2m 2 (V n 2 ) d t 2 , since t 1m1 (V n1 ) ‰ t 2m 2 (V n 2 ) lower bibounds the (m1, m2) covering biradii of all other bicodes of bidimension n1 ‰ n2. When t1 ‰ t2 = n1 ‰ n2 every bicode word bicovers every bivector, so a bicode of size 1 ‰ 1 will (m1, m2) bicover V n1 ‰ V n 2 for every m1 ‰ m2. Thus K1m1 (n1 , n1 ) ‰ K 2m2 (n 2 , n 2 ) = 1 ‰ 1 for every m1 ‰ m2. If t1 ‰ t2 is n1 – 1 ‰ n2 – 1 what happens to K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) ? When m1 = m2 = 1, K11 (n1 , n1  1) ‰ K12 (n 2 , n 2  1) d 1  L n1 (n1 ) ‰ 1  L n 2 (n 2 ) . For 0 ‰ 0 = (0 … 0) ‰ (0 … 0) will cover all bivectors of birank binorm less than or equal n1 – 1 ‰ n2 – 1 within biradius n1 – 1 ‰ n2 – 1. That is 0 ‰ 0 = (0, 0, …, 0) ‰ (0, 0, …, 0) will bicover all binorm n1 – 1 ‰ n2 – 1 bivectors within the biradius n1 – 1 ‰ n2 – 1. Hence remaining bivectors are rank n1 ‰ n2 bivectors. Thus 0 ‰ 0 = (0, 0, …, 0) ‰ (0, 0, …, 0) and these birank-(n1 ‰ n2) bivectors can bicover the ambient bispace within the biradius n1 – 1 ‰ n2 – 1. Therefore K11 (n1 , n1  1) ‰ K12 (n 2 , n 2  1) d 1  L n1 (n1 ) ‰ 1  L n 2 (n 2 ) . PROPOSITION 2.9: For any RD bicode of bilength n1 ‰ n2 over F2 N ‰ F2 N

K m1 1 ( n1 , n1  1) ‰ K m2 2 (n2 , n2  1) d m1 Ln1 (n1 )  1 ‰ m2 Ln2 (n2 )  1 provided m1 ‰ m2 is such that m1 Ln1 (n1 )  1 ‰ m2 Ln2 (n2 )  1 d V n1  V n2 .

73

Proof: Consider a RD-bicode C = C1 ‰ C2 such that |C| = |C1| ‰ |C2| = m1L n1 (n1 )  1 ‰ m 2 L n 2 (n 2 )  1 . Each bivector in V n1 ‰ V n 2 has L n1 (n1 ) ‰ L n 2 (n 2 ) rank complements, that is from each bivector

v1 ‰ v 2  V n1 ‰ V n 2 ; there are

L n1 (n1 ) ‰ L n 2 (n 2 )

bivectors at rank bidistance n1 ‰ n2. This means for any set S1 ‰ S2 Ž V n1 ‰ V n 2 of (m1, m2) bivectors there always exists a c1 ‰ c2  C1 ‰ C2 which bicovers S1 ‰ S2 birank distance n1 – 1 ‰ n2 – 1. Thus, cov(c1, S1) ‰ cov(c2, S2) d n1 – 1 ‰ n2 – 1 which implies cov(C1, S1) ‰ cov(C2, S2) d n1 – 1 ‰ n2 – 1. Hence K1m1 (n1 , n1  1) ‰ K 2m2 (n 2 , n 2  1) d m1L n1 (n1 )  1 ‰ m 2 L n 2 (n 2 )  1.

By bounding the number of (m1, m2) bisets that can be covered by a given bicode word, one obtains a straight forward generalization of the classical sphere bibound. THEOREM 2.16: (Generalized Sphere Bound for RD bicodes) For any (n1, K1) ‰ (n2, K2) RD bicode C = C1 ‰ C2,

§ V (n1 , tm1 (C1 )) · § V (n2 , tm2 (C2 )) · § 2 Nn1 2 K1 ¨ ¸‰K ¨ ¸ t ¨¨ m1 m2 © ¹ © ¹ © m1

· § 2 Nn2 ¸¸ ‰ ¨¨ ¹ © m2

Hence for any n1, t1 and m1, n2, t2 and m2 ª 2 Nn1 º ª 2 Nn2 º « » « » « m1 ¼» « m2 ¼» ¬ ¬ 1 2 ‰ K m1 (n1 , t1 ) ‰ K m2 ( n2 , t2 ) t ªV (n1 , t1 ) º ªV (n2 , t2 ) º « m » « m » ¬ ¼ ¬ ¼ 1 2

74

· ¸¸ . ¹

where

V (n1 , t1 ) ‰ V (n1 , t1 )

t1

t2

i1 0

i2 0

¦ L1i1 (n1 ) ‰ ¦ L2i2 (n2 ) ,

number of bivectors in a sphere of biradius t1 ‰ t2 and L1i1 (n1 ) ‰ L2i2 (n2 ) is the number of bivectors in V n1 ‰ V n2 whose rank binorm is i1 ‰ i2. Proof: Each set of (m1, m2) bivectors in V n1 ‰ V n 2 must occur in a sphere of biradius t m1 (C1 ) ‰ t m2 (C2 ) around at least one code biword. Total number of such bisets is V n1 ‰ V n 2 , choose m1 ‰ m2, where

V n1 ‰ V n 2

2

N n1

‰2

N n2

.

The number of bisets of (m1, m2) bivectors in a neighborhood of biradius t m1 (C1 ) ‰ t m2 (C2 ) is V(n1 , t m1 (C1 )) ‰ V(n 2 , t m2 (C2 )) . Choose m1 ‰ m2. There are K code biwords. Hence § V(n1 , t m1 (C1 )) · § V(n 2 , t m2 (C2 )) · § 2 Nn1 · § 2 Nn2 2 ‰ K1 ¨ K ¸¸ ‰ ¨¨ ¸ ¨ ¸ t ¨¨ m1 m2 © ¹ © ¹ © m1 ¹ © m 2 Thus for any n1 ‰ n2, t1 ‰ t2 and m1 ‰ m2, ª 2 Nn1 º ª 2 Nn2 º « » « » « m1 ¼» « m 2 ¼» ¬ ¬ 1 2 K m1 (n1 , t1 ) ‰ K m2 (n 2 , t 2 ) t ‰ . ª V(n1 , t1 ) º ª V(n 2 , t 2 ) º « m » « m » 1 2 ¬ ¼ ¬ ¼

75

· ¸¸ . ¹

COROLLARY 2.3: If

ª 2 Nn1 º ª 2 Nn2 º N n § V ( n1 , t1 ) · N n2 § V ( n2 , t2 ) · « »‰« »!2 1¨ ¸‰2 ¨ ¸, © m1 ¹ © m2 ¹ ¬« m1 ¼» ¬« m2 ¼» then K m1 1 (n1 , t1 ) ‰ K m2 2 (n2 , t2 ) f ‰ f .

76

Chapter Three

RANK DISTANCE m-CODES

In this chapter we introduce the new notion of rank distance mcodes and describe some of their properties. DEFINITION 3.1: Let C1 = C1[n1, k1], C2 = C2[n2, k2], …, Cm = Cm[nm, km], be m distinct RD codes such that Ci = Ci[ni, ki] z Cj = Cj[nj, kj] if i z j and Ci = Ci[ni, ki] Œ Cj = Cj[nj, kj] or Cj = Cj[nj, kj] Ž Ci = Ci[ni, ki] for 1 d i, j d m if i z j; be subspaces of the rank spaces V n1 ,V n2 ,...,V nm over the field GF(2N) or Fq N where n1, n2, …, nm d N, i.e., each ni d N for i = 1, 2, …, m. C = C1[n1, k1] ‰ C2[n2, k2] ‰ … ‰ Cm[nm, km] is defined as the Rank Distance m-code (m t 3). If m = 3 we call the Rank Distance 3code as the Rank Distance tricode. We say V n1 ‰ V n2 ‰ ... ‰ V nm to be (n1, n2, …, nm) dimensional vector m-space over the field Fq N . So we can say C1 ‰ C2 ‰ … ‰ Cm is a m-subspace of the

77

vector m-space V n1 ‰ V n2 ‰ ... ‰ V nm . We represent any element of V n1 ‰ V n2 ‰ … ‰ V nm by x1 ‰ x2 ‰ … ‰ xn = 1 1 1 ( x1 , x2 ,..., xn1 ) ‰ ( x12 , x22 ,..., xn22 ) ‰ ... ‰ ( x1m , x2m ,..., xnmm ) , where xiij  Fqn ; 1 d j d m and 1 d ij d nj; j = 1, 2, 3, …, m. Also Fq N can be considered as a pseudo false m-space of dimension

( N , !, N ) over Fq.  

m  times

Thus elements xi, yi  Fq N has N - m-tuple representation as

(D11 j , D 12 j ,..., D1Nj1 ) ‰ (D12j , D 22 j ,..., D 2Nj2 ) ‰ ... ‰ (D1mj , D m2i ,..., D mNjm ) over Fq with respect to some m-basis. Hence associated with each x1 ‰ x2 ‰ … ‰ xm  V n1 ‰ V n 2 ‰ ... ‰ V n m (ni z nj if i z j, 1 d i, j d m) there is a m-matrix. m1 (x1 ) ‰ ... ‰ m m (x m ) 2 2 ª a111 ! a11n1 º ª a11 º ! a1n 2 « 1 » « » 1 2 2 « a 21 ! a 2n1 » « a 21 ! a 2n 2 » « »‰« »‰ … ‰ # # # # « » « » « a1N1 " a1Nn » « a 2N2 " a 2Nn » 1 ¼ 2 ¼ ¬ ¬

where

i1th ‰ i 2th ‰ ... ‰ i mth

m-column

m m ª a11 º ! a1n m « m » m « a 21 ! a 2n m » « » # » « # « a mNm " a mNn » m ¼ ¬

represents

the

i ‰ i ‰ ... ‰ i m–coordinate of x ‰ x ‰ ... ‰ x of x ‰ x2 ‰ … ‰ xm over Fq. It is important and interesting to note in order to develop the new notion rank distance m-codes (m t 3) and while trying to give m- matrices and m- ranks associated with them we are forced to define the notion of pseudo false m- vector spaces. For example, Z52 ‰ Z52 ‰ ... ‰ Z52 is a false pseudo m-vector space th 1

th 2

th m

1 i1

78

2 i2

m im

1

over Z2. Z95 ‰ Z95 ‰ Z95 ‰ Z95 ‰ Z95 ‰ Z95 is a false pseudo 6vector space over Z5. However we will in this book use only m-vector spaces over Z2 or Z2N and denote it by GF(2) or GF(2N) or F2N , unless otherwise specified. Now we see every x1 ‰ … ‰ xm in the mvector space V n1 ‰ V n 2 ‰ ... ‰ V n m have an associated m-matrix m1(x1) ‰ … ‰ mm(xm). We proceed to define m- rank of the m- matrix over Fq or GF(2). DEFINITION 3.2: The m-rank of an element x1 ‰ x2 ‰ … ‰ xm  V n1 ‰ V n2 ‰ ... ‰ V nm is defined as the m-rank of the m-matrix m(x1) ‰ m(x2) ‰ … ‰ m(xm) over GF(2) or Fq [i.e., the m-rank of m(x1) ‰ m(x2) ‰ … ‰ m(xm) is the rank of m(x1) ‰ rank of m(x2) ‰ … ‰ rank of m(xm)]. We shall denote the m-rank of x1 ‰ x2 ‰ … ‰ xm by r1(x1) ‰ r2(x2) ‰ … ‰ rm(xm), we can in case of m-rank of a m-matrix prove the following; (i) For every x1 ‰ x2 ‰ … ‰ xm  V n1 ‰ V n2 ‰ ... ‰ V nm (xi  V ni , 1 d i d m) we have, r(x1 ‰ … ‰ xm) = r1(x1) ‰ … ‰ rm(xm) t 0 ‰ 0 ‰ … ‰ 0 (i.e., each ri(xi) t 0 for every xi V ni ; i = 1, 2, 3, …, m). (ii) r(x1 ‰ x2 ‰ … ‰ xm) = r1(x1) ‰ r2(x2) ‰ … ‰ rm(xm) 0 ‰ 0 ‰ … ‰ 0 if and only if xi = 0 for i = 1, 2, 3, …, m. r[( x11  x21 ) ‰ ( x12  x22 ) ‰ ... ‰ ( x1m  x2m )] d r1 ( x11 )  r1 ( x21 ) ‰ r2 ( x12 )  r2 ( x22 ) ‰ ... ‰ rm ( x1m )  rm ( x2m )

for every x1i , x2i V ni ; i = 1, 2, 3, …, m. That is we have, (iii)

r[( x11  x21 ) ‰ ( x12  x22 ) ‰ ... ‰ ( x1m  x2m )] r1 ( x11  x21 ) ‰ r2 ( x12  x22 ) ‰ ... ‰ rm ( x1m  x2m )

d r1 ( x11 )  r1 ( x12 ) ‰ r2 ( x12 )  r2 ( x22 ) ‰ ... ‰ rm ( x1m )  rm ( x2m ) (as we have for every x1i , x2i V ni ; ri ( x1i  x2i ) d ri ( x1i )  ri ( x2i ) for i = 1, 2, 3, …, m).

79

(iv) r1(a1x1) ‰ r2(a2x2) ‰ … ‰ rm(amxm) = |a1|r1(x1) ‰ |a2|r2(x2) ‰ … ‰ |am|rm(xm) for every a1, a2, …, am  Fq or GF(2) and for every xi  V ni ; i = 1, 2, 3, …, m. Thus the m-function x1 ‰ x2 ‰ … ‰ xm o r1(x1) ‰ r2(x2) ‰ … ‰ rm(xm) defines a m-norm on V n1 ‰ V n2 ‰ ... ‰ V nm . DEFINITION 3.3: The m-metric induced by the m- rank m- norm is defined as the m- rank m-metric on V n1 ‰ V n2 ‰ ... ‰ V nm and is denoted by d R1 ‰ d R2 ‰ ! ‰ d Rm . If x11 ‰ x12 ‰ ... ‰ x1m , y11 ‰ y12

‰ … ‰ y1m V n ‰ V n ‰ ! ‰ V n then the m-rank m-distance 1

m

2

between x11 ‰ x21 ‰ ... ‰ xm1 and y11 ‰ y21 ‰ ... ‰ ym1 is d R1 ( x11 , y11 ) ‰ d R2 ( x21 , y21 ) ‰ ... ‰ d Rm ( xm1 , ym1 ) r1 ( x11  y11 ) ‰ r2 ( x21  y21 ) ‰ ... ‰ rm ( xm1  ym1 ) for every xi1 , yi1  V ni , i = 1, 2, 3, …, m (Here d Ri ( xi1 , yi1 ) ri ( xi1  yi1 ) for every xi1 , yi1 V ni ; for i = 1, 2, 3, …, m). DEFINITION 3.4: A linear m-space V n1 ‰ V n2 ‰ ... ‰ V nm over GF(2N), N > 1 of m-dimension n1 ‰ n2 ‰ … ‰ nm such that ni d N for i = 1, 2, 3, …, m equipped with the m-rank m-metric is defined as the m-rank m-space. DEFINITION 3.5: A m-rank m-distance RD m-code of m-length n1 ‰ n2 ‰ … ‰ nm over GF(2N) is a m-subset of the m-rank mspace V n1 ‰ V n2 ‰ ! ‰ V nm over GF(2N). DEFINITION 3.6: A linear [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] RD-m-code is a linear m-subspace of m-dimension k1 ‰ k2 ‰ … ‰ km in the m-rank m-space V n1 ‰ V n2 ‰ ! ‰ V nm . By C1[n1, k1] ‰ C2[n2, k2] ‰ … ‰ Cm[nm, km] we denote a linear [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] RD m-code.

We can equivalently define a RD m-code as follows:

80

DEFINITION 3.7: Let V n1 ,V n2 ,! ,V nm be rank spaces ni z nj if i

z j over GF(2N), N >1. Suppose Pi Ž V n , i = 1, 2, 3, …, m be subset of the rank spaces over GF(2N). Then P1 ‰ P2 ‰ …‰ Pm Ž V n ‰ V n ‰ ... ‰ V n is a rank distance m-code of m-length i

1

2

m

(n1, n2, …, nm) over GF(2N). DEFINITION 3.8: A generator m-matrix of a linear [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] RD-m-code C1 ‰ C2 ‰ … ‰ Cm is a k1 u n1 ‰ k2 u n2 ‰ … ‰ km u nm, m-matrix over GF(2N) whose mrows form a m-basis for C1 ‰ C2 ‰ … ‰ Cm. A generator m-matrix G = G1 ‰ G2 ‰ … ‰ Gm of a linear RD m-code C1[n1, k1] ‰ C2[n2, k2] ‰ … ‰ Cm[nm, km] can be brought into the form G = G1 ‰ G2 ‰ … ‰ Gm [ I k1 , Ak1 un1  k1 ]

‰[ I k2 , Ak2 un2  k2 ] ‰... ‰ [ I km , Akm unm  km ] where I k1 , I k2 ,..., I km is the identity matrix and Aki uni  ki is some matrix over GF(2N); i = 1, 2, 3, …, m; this form of G = G1 ‰ G2 ‰ … ‰ Gm is called the standard form. DEFINITION 3.9: If G = G1 ‰ G2 ‰ … ‰ Gm is a generator mmatrix of C1[n1, k1] ‰ C2[n2, k2] ‰ … ‰ Cm[nm, km] then a mmatrix H = H1 ‰ H2 ‰ … ‰ Hm of order (n1 – k1 u n1, n2 – k2 u n2, …, nm – km u nm) over GF(2N) such that,

GHT = G1 ‰ G2 ‰ … ‰ Gm)(H1 ‰ H2 ‰ … ‰ Hm)T = G1 ‰ G2 ‰ … ‰ Gm) ( H 1T ‰ H 2T ‰ ... ‰ H mT ) G1 H 1T ‰ G2 H 2T ‰ ... ‰ Gm H mT =0‰0‰…‰0 is called a parity check m-matrix of C1[n1, k1] ‰ C2[n2, k2] ‰ … ‰ Cm[nm, km]. Suppose C = C1 ‰ C2 ‰ … ‰ Cm is a linear [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] RD m-code with G = G1 ‰ G2 ‰ … ‰ Gm and H = H1 ‰ H2 ‰ … ‰ Hm as its generator and parity check m-matrix respectively, then C = C1 ‰ C2 ‰ … ‰ Cm has two representations.

81

(a) C = C1 ‰ C2 ‰ … ‰ Cm is a row m-space of G = G1 ‰ G2 ‰ … ‰ Gm (i.e., Ci is the row space of Gi for i = 1, 2, 3, …, m). (b) C = C1 ‰ C2 ‰ … ‰ Cm is the solution m-space of H = H1 ‰ H2 ‰ … ‰ Hm (i.e.; Ci is the solution space of Hi for i = 1, 2, 3, …, m). Now we proceed on to define the notion of minimum rank mdistance of the rank distance m-code C = C1 ‰ C2 ‰ … ‰ Cm. DEFINITION 3.10: Let C = C1 ‰ C2 ‰ … ‰ Cm be a rank distance m-code, the minimum rank m-distance d = d1 ‰ d2 ‰ … ‰ dm is defined by xi , yi  Ci °½ °­ di min ®d Ri ( xi , yi ) ¾ xi z yi ¿° ¯° i = 1, 2, 3, …, m. That is d = d1 ‰ d2 ‰ … ‰ dm min{ r1 ( x1 ) x1  C1 and x1 z 0 } ‰ min{ r2 ( x2 ) x2  C2 and x2 z 0 } ‰! ‰ min{ rm ( xm ) xm  Cm and xm z 0 } .

If an RD m-code C = C1 ‰ C2 ‰ … ‰ Cm has the minimum rank m-distance d = d1 ‰ d2 ‰ … ‰ dm then it can correct all merrors e = e1 ‰ e2 ‰ … ‰ em  FqnN1 ‰ FqnN2 ‰ ! ‰ FqnNm with mrank r(e) = r1 ‰ r2 ‰ … ‰ rm e1 ‰ e2 ‰ … ‰ em r1 ( e1 ) ‰ r2 ( e2 ) ‰ ... ‰ rm ( em ) « d  1» « d  1 » « d2  1 » d« 1 ‰« ‰ ... ‰ « n ». » » ¬ 2 ¼ ¬ 2 ¼ ¬ 2 ¼

Let C = C1 ‰ C2 ‰ … ‰ Cm denote an [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] RD m-code over Fq N . A generator m-matrix G = G1 ‰ G2 ‰ … ‰ Gm of C = C1 ‰ C2 ‰ … ‰ Cm is a k1 u n1 ‰ k2 u n2 ‰ … ‰ km u nm, m-matrix

82

with entries from Fq N whose rows form a m-basis for C = C1 ‰ C2 ‰ … ‰ Cm. Then an (n1 – k1) u n1 ‰ (n2 – k2) u n2 ‰ … ‰ (nm – km) u nm m-matrix H = H1 ‰ H2 ‰ … ‰ Hm with entries from Fq N such that, GHT = G1 ‰ G2 ‰ … ‰ Gm)(H1 ‰ H2 ‰ … ‰ Hm)T G1 ‰ G2 ‰ … ‰ Gm ) ( H 1T ‰ H 2T ‰ ! ‰ H mT ) G1 H 1T ‰ G2 H 2T ‰ ! ‰ Gm H mT =0‰0‰…‰0 is called the parity check m-matrix of the RD m-code C = C1 ‰ C2 ‰ … ‰ Cm.

The result analogous to Singleton–Style bound in case of RD mcode is given in the following. Result: (Singleton–Style bound). The minimum rank m-distance d = d1 ‰ d2 ‰ … ‰ dm of any linear [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] RD m-code C = C1 ‰ C2 ‰ … ‰ Cm Ž FqnN1 ‰ FqnN2 ‰ ! ‰ FqnNm satisfies the following bounds d = d1 ‰ d2 ‰ … ‰ dm d n1 – k1 + 1 ‰ n2 – k2 + 1 ‰ … ‰ nm – km + 1.

Based on this notion we now proceed on to define the new notion of Maximum Rank Distance m-codes. DEFINITION 3.11: An [n1, k1, d1] ‰ [n2, k2, d2] ‰ … ‰ [nm, km, dm] RD-m-code C = C1 ‰ C2 ‰ … ‰ Cm is called a Maximum Rank Distance (MRD) m-code if the Singleton Style bound is reached that is d = d1 ‰ d2 ‰ … ‰ dm = n1 – k1 + 1 ‰ n2 – k2 + 1 ‰ … ‰ nm – km + 1.

Now we proceed on to briefly give the construction of MRD mcode. Let [s] = [s1] ‰ [s2] ‰ … ‰ [sm] = q s1 ‰ q s2 ‰ ! ‰ q sm for any m integers s1, s2, …, sm. Let {g1 , !, g n1 } ‰ {h1 , !, h n 2 }

83

‰... ‰ {p1 , !, p n m } be any m-set of elements in Fq N that are

linearly independent over Fq. A generator m-matrix G = G1 ‰ G2 ‰ … ‰ Gm of an MRD m-code C = C1 ‰ C2 ‰ … ‰ Cm is defined by G = G1 ‰ G2 ‰ … ‰ Gm ª g1 « >1@ « g1 « > 2@ « g1 « « # «g>k1 1@ ¬ 1

g2 g>2 @ 1

g>2 @ 2

#

g>2 1

k 1@

!

g n1 º ª h1 » « 1 1 ! g>n1@ » « h1> @ » « 2 2 ! g>n1@ » ‰ « h1> @ » « # » « # k 1 k 1 ! g>n11 @ »¼ «¬ h1> 2 @ ª p1 « >1@ « p1 « ‰! ‰ « p1>2@ « « # « p>k m 1@ ¬ 1

p2 p>2 @ 1

p>2 @ 2

#

p>2 m

k 1@

h2 h >2 @ 1

h >2 @ 2

#

h >2 2

k 1@

!

h n2 º » 1 ! h >n 2@ » » 2 ! h >n 2@ » » # » k 1 ! h >n 22 @ »¼

!

pnm º » 1 ! p>n m@ » » 2 ! p>n m@ » . » # » k 1 ! p>n mm @ »¼

It can be easily proved that the m-code C = C1 ‰ C2 ‰ … ‰ Cm given by the generator m-matrix G = G1 ‰ G2 ‰ … ‰ Gm has the rank m-distance d = d1 ‰ d2 ‰ … ‰ dm. Any m-matrix of the above form is called a Frobenius m-matrix with generating m-vector g c g c1 ‰ g c2 ‰ ... ‰ g cm (g1 ,g 2 ,!,g n1 ) ‰ (h1 , h 2 ,!, h n 2 ) ‰ ... ‰ (p1 , p 2 ,!, p mn ) . Interested reader can prove the following theorem: THEOREM 3.1: Let C[n, k] = C1[n1, k1] ‰ C2[n2, k2] ‰ … ‰ Cm[nm, km] be the linear [n1, k1, d1] ‰ [n2, k2, d2] ‰ … ‰ [nm, km, dm] MRD m-code with di = 2ti + 1 for i = 1, 2, 3, …, m. Then C[n, k] = C1[n1, k1] ‰ C2[n2, k2] ‰ … ‰ Cm[nm, km] m-code

84

corrects all m-errors of m-rank atmost t = t1 ‰ t2 ‰ … ‰ tm and detects all m-errors of m-rank greater than t = t1 ‰ t2 ‰ … ‰ tm. Consider the Galois field GF(2N) ; N > 1 . An element D D 1 ‰ D 2 ‰ ... ‰ D m  GF (2 N ) ‰ GF (2 N ) ‰ ... ‰ GF (2 N )  

m  times

can be denoted by m-N-tuple (a01 , a11 ,..., aN1 1 ) ‰ ( a02 , a12 ,..., aN2 1 ) ‰ ... ‰ (a0m , a1m ,..., aNm1 )

as well as by the m-polynomial a01  a11 x  ...  aN1 1 x N 1 ‰ a02  a12 x  ...  aN2 1 x N 1 ‰... ‰ a0m  a1m x  ...  aNm 1 x N 1 over GF(2) We now proceed on to define the new notion of circulant mtranspose. DEFINITION 3.12: The circulant m-transpose TC TC11 ‰ TC22 ‰ ... ‰ TCmm

of a m-vector

D = D1 ‰ D2 ‰ … ‰ Dm =

1 0

1 1

(a , a ,..., a

1 N 1

) ‰ (a02 , a12 ,..., aN2 1 ) ‰ ! ‰ (a0m , a1m ,..., aNm 1 )

 GF(2N) is defined as, DT

C

TC11

TCmm

TC22

D1 ‰ D 2 ‰ ! ‰ D m

=

(a , a ,..., a ) ‰ (a , a ,..., a ) ‰ ! ‰ (a , a1m ,..., anmm ) 1 0

If D

1 1

1 n1

2 0

2 1

2 n2

m 0

( 2 N ) ‰ GF ( 2 N ) ‰ ! ‰ GF (2 N ) D 1 ‰ D 2 ‰ ... ‰ D m  GF  

m  times

has the m-polynomial representation a01  a11 x  ...  aN1 1 x N 1 ‰ a02  a12 x  ...  aN2 1 x N 1 ‰... ‰ a0m  a1m x  ...  aNm 1 x N 1 in GF (2)[ x] GF (2)[ x] GF (2)[ x] ‰ N ‰ ... ‰ N N ( x  1) ( x  1) ( x  1)

85

then by D = D1 ‰ D2 ‰ … ‰ Dm , we denote the m-vector corresponding to the m-polynomial [ a01  a11 x  ...  aN1 1 x N 1 ˜ x i ] mod( x N  1 ) ‰ [ a02  a12 x  !  aN2 1 x N 1 ˜ x i ] mod( x N  1 ) ‰ ! ‰ [ a0m  a1m x  !  aNm 1 x N 1 ˜ x i ] mod( x N  1 )

for i = 0, 1, 2, 3, …, N –1 (D = D1 ‰ D2 ‰ … ‰ Dm = DR). We now proceed onto define the m-word generated by D = D1 ‰ D2 ‰ … ‰ Dm. DEFINITION 3.13: Let f = f1 ‰ f2 ‰ … ‰ fm : GF( 2 N ) ‰ GF( 2 N ) ‰ ! ‰ GF( 2 N )    o m  times

[ GF( 2 N )] N ‰ [ GF( 2 N )] N ‰ ! ‰ [ GF( 2 N )] N  

m  times

be defined as f(D) = f1(D1) ‰ f2(D2) ‰ … ‰ fm(Dm) (D

TC11 10

T1

T1

T2

T2

T2

,D 11C1 ,! ,D 1NC11 ) ‰ ( D 20C2 ,D 21C2 ,! ,D 2 CN2 1 ) ‰ ! Tm

Tm

Tm

‰( D m0Cm ,D m1Cm ,! ,D mNCm1 ) . We call f(D) = f1(D1) ‰ f2(D2) ‰ … ‰ fm(Dm) as the m-code word generated by D = D1 ‰ D2 ‰ … ‰ Dm.

We analogous to the definition of Macwilliams and Solane define circulant m-matrix associated with a m-vector in GF(2N) ‰ GF(2N) ‰ … ‰ GF(2N). DEFINITION 3.14: A m-matrix of the form

ª a10 « 1 « aN 1 « # « 1 ¬« a1

a11 ! a1N 1 º ª a02 » « a01 ! a1N  2 » « aN2 1 ‰ # # » « # » « a12 ! a10 ¼» ¬« a12

86

a12 ! aN2 1 º » a02 ! aN2  2 » # # » » 2 a2 ! a02 ¼»

ª a0m « m a ‰... ‰ « N 1 « # « m ¬« a1

a1m ! aNm 1 º » a0m ! aNm  2 » # # » » m a2 ! a0m ¼»

is called the circulant m-matrix associated with the m-vector (a01 , a11 ,..., a1N 1 ) ‰ (a02 , a12 ,!, aN2 1 ) ‰ ! ‰ (a0m , a1m ,!, aNm 1 )  GF(2N) ‰ GF(2N) ‰ … ‰ GF(2N). Thus with each D = D1 ‰ D2 ‰ … ‰ Dm  GF(2N) ‰ GF(2N) ‰ … ‰ GF(2N) we can associate a circulant m-matrix whose ith mT1

T2

Tm

columns represents D1iC1 ‰ D 2iC2 ‰ ... ‰ D miCm ; i = 0, 1, 2, …, N–1. f = f1 ‰ f2 ‰ … ‰ fm is nothing but a m-mapping of GF(2N) ‰ GF(2N) ‰ … ‰ GF(2N) onto the pseudo false m-algebra of all N u N circulant m-matrices over GF(2). Denote the m-space of f(GF(2N)) = f1(GF(2N)) ‰ f2(GF(2N)) ‰ … ‰ fm(GF(2N)) by N N V ‰ VN  ‰ ... ‰ V

.  m  times

We define the m-norm of a m-word N N v v1 ‰ v 2 ‰ ... ‰ v m  V ‰ VN  ‰ ... ‰ V

 m  times

as follows : DEFINITION 3.15: The m-norm of a m-word v = v1 ‰ v2 ‰ … ‰ vm  VN ‰ VN ‰ … ‰ VN is defined as the m-rank of v = v1 ‰ v2 ‰ … ‰ vm over GF(2) [By considering it as a circulant mmatrix over GF(2)].

We denote the m-norm of v = v1 ‰ v2 ‰ … ‰ vm by r(v) = r1(v1) ‰ r2(v2) ‰ … ‰ rm(vm), we prove the following theorem: THEOREM 3.2: Suppose D D 1 ‰ D 2 ‰ ... ‰ D m  GF (2 N ) ‰ GF (2 N ) ‰ ... ‰ GF (2 N )  

m  times

87

has the m-polynomial representation g(x) = g1(x) ‰ g2(x) ‰ … ‰ gm(x) over GF(2) such that gcd(gi(x), xN +1) has degree N – ki for i = 1, 2, 3, …, m; 1 d k1, k2, …, km d N. Then the m-norm of the m-word generated by D = D1 ‰ D2 ‰ … ‰ Dm is k1 ‰ k2 ‰ … ‰ km. Proof: We know the m-norm of the m-word generated by D = D1 ‰ D2 ‰ … ‰ Dm is the m-rank of the circulant m-matrix T1

T1

T1

T2

T2

T2

(D10C1 , D11C1 ,..., D1NC11 ) ‰ (D 20C2 , D 21C2 ,..., D 2CN21 ) ‰ ! ‰ Tm

Tm

Tm

Cm (D m0Cm , D m1Cm ,..., D mN 1 )

where DiTC

T1

T2

Tm

D1iC1 ‰ D 2iC2 ‰ ! ‰ D miCm represents a m-polynomial

[x i g1 (x)]mod(x N  1) ‰ [x i g 2 (x)]mod(x N  1) ‰... ‰ [x i g m (x)]mod(x N  1)

over GF(2). Suppose the m-GCD {(g1(x), xN + 1) ‰ (g2(x), xN + 1) ‰ … ‰ (gm(x), xN + 1)} has m-degree N – k1 ‰ N – k2 ‰ … ‰ N – km. To prove that the m-word generated by D = D1 ‰ D2 ‰ … ‰ Dm has m-rank k1 ‰ k2 ‰ … ‰ km. It is enough to prove that the mspace generated by the N-polynomials {g1(x) mod(xN + 1), x˜g1(x) mod(xN + 1), …, xN–1˜g1(x) mod(xN + 1)} ‰ {g2(x) mod(xN + 1), x˜g2(x) mod(xN + 1), …, xN–1˜g2(x) mod(xN + 1)} ‰ … ‰ {gm(x) mod(xN + 1), x˜gm(x) mod(xN + 1), …, xN–1˜gm(x) mod(xN + 1)} has m-dimension k1 ‰ k2 ‰ … ‰ km. We will prove that the m-set of k1 ‰ k2 ‰ … ‰ km, m-polynomials {g1(x) mod xN + 1, x˜g1(x) mod xN + 1, …, xN–1˜g1(x) mod xN + 1} ‰ {g2(x) mod xN + 1, x˜g2(x) mod xN + 1, …, xN–1˜g2(x) mod xN + 1} ‰ … ‰ {gm(x) mod xN + 1, x˜gm(x) mod xN + 1, …, xN–1˜gm(x) mod xN + 1} forms a m-basis for the m-space. If possible let, a10 (g1 (x))  a11x(g1 (x))  ...  a1k1 1 (x k1 1g1 (x)) ‰ a 02 (g 2 (x))  a12 x(g 2 (x))  ...  a 2k 2 1 (x k 2 1g 2 (x)) ‰ ... ‰

a 0m (g m (x))  a1m x(g m (x))  ...  a mk m 1 (x k m 1g m (x))

88

= 0 ‰ 0 ‰ … ‰ 0 (mod xN + 1); where a iji  GF(2), 1 d ji d ki – 1 and i = 1, 2, …, m. This implies 1 N 1 N 1 xN ‰ x  ‰ ...‰x

m  times

m-divides (a10  a11 x  ...  a1k1 1x k1 1 )g1 (x) ‰

(a 02  a12 x  ...  a 2k 2 1 x k 2 1 )g 2 (x) ‰ ! ‰ (a 0m  a1m x  ...  a kmm 1x k m 1 )g m (x) . Now if g1(x) ‰ g2(x) ‰ … ‰ gm(x) = p1(x)a1(x) ‰ p2(x)a2(x) ‰ … ‰ pm(x)am(x) where pi(x) is the gcd(gi(x), xN + 1); i = 1, 2, …, m, then (ai(x), xN + 1) = 1. Thus xN + 1 m-divides

(a10  a11x  ...  a1k1 1x k1 1 )g1 (x) ‰ ... ‰ (a 0m  a1m x  ...  a kmm 1 x k m 1 )g m (x) implies the m-quotient (x N  1) (x N  1) ‰ ... ‰ p1 (x) p m (x) m-divides (a10  a11x  ...  a1k1 1x k1 1 )a1 (x) ‰ (a 02  a12 x  ...  a 2k 2 1 x k 2 1 )a 2 (x) ‰ ... ‰ (a 0m  a1m x  ...  a1k m 1x k m 1 )a m (x) . That is ª (x N  1) º ª (x N  1) º ª (x N  1) º « »‰« » ‰ ... ‰ « » ¬ p1 (x) ¼ ¬ p 2 (x) ¼ ¬ p m (x) ¼

m-divides

(a10  a11x  ...  a1k1 1x k1 1 ) ‰ (a 02  a12 x  ...  a 2k 2 1 x k 2 1 ) ‰ ... ‰ (a 0m  a1m x  ...  a1k m 1x k m 1 )

89

which is a contradiction, as

ª (x N  1) º ª (x N  1) º ª (x N  1) º ‰ ‰ ... ‰ « » « » « » ¬ p1 (x) ¼ ¬ p 2 (x) ¼ ¬ p m (x) ¼ has m-degree (k1, k2, . . ., km) where as the m-polynomial

(a10  a11x  ...  a1k1 1x k1 1 ) ‰ (a 02  a12 x  ...  a 2k 2 1 x k 2 1 ) ‰ ... ‰ (a 0m  a1m x  ...  a1k m 1x k m 1 ) has m-degree atmost ((k1 – 1), (k2 – 1), …, (km – 1)). Hence the m-polynomials {g1(x) mod(xN + 1), x˜g1(x) mod(xN + 1), …, x k1 1 ˜g1(x) mod(xN + 1)} ‰ {g2(x) mod(xN + 1), x˜g2(x) mod(xN + 1), …, x k 2 1 ˜g2(x) mod(xN + 1)} ‰ … ‰ {gm(x) mod(xN + 1), x˜gm(x) mod(xN + 1), …, x k m 1 ˜gm(x) mod(xN + 1)} are mlinearly independent over GF(2). We will prove, {g1(x) mod(xN + 1), x˜g1(x) mod(xN + 1), …, x k1 1 ˜g1(x) mod(xN + 1)} ‰ {g2(x) mod(xN + 1), x˜g2(x) mod(xN + 1), …, x k 2 1 ˜g2(x) mod(xN + 1)} ‰ … ‰ {gm(x) mod(xN + 1), x˜gm(x) mod(xN + 1), …, x k m 1 ˜gm(x) mod(xN + 1)} generate the m-space. For this it is enough to prove that xig1(x) ‰ xig2(x) ‰ … ‰ xigm(x) is a linear combination of these m-polynomials for kj d i d N – 1; j = 1, 2, 3, …, m. N x  1 ‰ x N   1‰! ‰ xN 

1 m  times

= p1(x)b1(x) ‰ p2(x)b2(x) ‰ … ‰ pm(x)bm(x) where bi(x)

bi0

biki

bi0  b1i x  ...  biki x ki ; i = 1, 2, 3, …, m. (Note that

1 since bi(x) divides xN + 1, i = 1, 2, 3, …, m). Also

we have gi(x) = pi(x)ai(x) for i = 1, 2, 3, …, m. Thus

90

§ g (x)bi (x) · xN 1 ¨ i ¸ © a i (x) ¹ for i = 1, 2, 3, …, m. That is g i (x)(bi0  b1i x  ...  bik i x k i ) 0 mod(x N  1) a i (x) (i = 1, 2, 3, …, m) that is g i (x)(bi0  b1i x  ...  biki 1x ki 1 ) a i (x)

ª g i (x) ˜ x ki « ¬ a i (x)

º N » mod(x  1) ¼

true for i = 1, 2, 3, ..., m. Hence x ki g i (x) (bi0 g i (x)  b1i x g i (x)  ...  biki 1x ki 1g i (x)) mod(x N  1) a linear combination of {gi(x) mod(xN + 1), [x˜gi(x)] mod(xN + 1), …, x k i 1 ˜gm(x) mod(xN + 1)} over GF(2). This is true for each i, for i = 1, 2, 3, …, m. Now it can be easily proved that xig1(x) ‰ xig2(x) ‰ … ‰ xigm(x) is a m-linear combination of {g1(x) mod(xN + 1), x˜g1(x) mod(xN + 1), …, x k1 1 ˜g1(x) mod(xN + 1)} ‰ {g2(x) mod(xN + 1), x˜g2(x) mod(xN + 1), …, x k 2 1 ˜g2(x) mod(xN + 1)} ‰ … ‰ {gm(x) mod(xN + 1), x˜gm(x) mod(xN + 1), …, x k m 1 ˜gm(x) mod(xN + 1)} for i > kj; j = 1, 2, …, m. Hence the m-space generated by the m-polynomials {g1(x) mod(xN + 1), x˜g1(x) mod(xN + 1), …, x k1 1 ˜g1(x) mod(xN + 1)} ‰ {g2(x) mod(xN + 1), x˜g2(x) mod(xN + 1), …, x k 2 1 ˜g2(x) mod(xN + 1)} ‰ … ‰ {gm(x) mod(xN + 1), x˜gm(x) mod(xN + 1), …, x k m 1 ˜gm(x) mod(xN + 1)} has m-dimension (k1, k2, …, km). That is the m-rank of a m-word generated by D = D1 ‰ D2 ‰ … ‰ Dm is (k1, k2, …, km). COROLLARY 3.1: If D = D1 ‰ D2 ‰ … ‰ Dm  GF (2 N ) ‰ ... ‰ GF (2 N ) then the m-norm of the m-word   

m  times

generated by D = D1 ‰ D2 ‰ … ‰ Dm is (N, N, …, N) and hence f(D) = f1(D1) ‰ f2(D2) ‰ … ‰ fm(Dm) is m-invertible. (We say

91

f(D) is m-invertible if each fi(D)is invertible for i = 1, 2, 3, …, m). Proof: The corollary follows immediately from the theorem since gcd(gi(x), xN+1) = 1 has degree 0 for i = 1, 2, 3, …, m; hence the m-rank of f(D) = f1(D1) ‰ f2(D2) ‰ … ‰ fm(Dm) is (N, N, …, N). DEFINITION 3.16: The m-distance between two m-words u, v  VN ‰ … ‰ VN is defined as, d(u, v) = d1(u1, v1) ‰ … ‰ dm(um, vm) = r1(u1 +v1) ‰ … ‰ rm(um + vm) where u = u1 ‰ u2 ‰ … ‰ um and v = v1 ‰ v2 ‰ … ‰ vm . DEFINITION 3.17: Let C = C1 ‰ C2 ‰ … ‰ Cm be a circulant rank m-code of m-length N1 ‰ N2 ‰ … ‰ Nm which is a msubspace of V N1 ‰ V N2 ‰ ... ‰ V Nm equipped with the m-distance m-function d1(u1, v1) ‰ d2(u2, v2) ‰ … ‰ dm(um, vm) = r1(u1 +v1) ‰ r2(u2 +v2) ‰… ‰ rm(um + vm) where V N1 ,V N2 ,...,V Nm are rank spaces defined over GF(2N) with Ni z Nj if i z j. C = C1 ‰ C2 ‰ … ‰ Cm is defined as the circulant m-code of m-length N1 ‰ N2 ‰ … ‰ Nm defined as a m-subspace of V N1 ‰ V N2 ‰ ... ‰ V Nm equipped with the m-distance m-function. DEFINITION 3.18: A circulant m-rank m-code of m-length N1 ‰ N2 ‰ … ‰ Nm is called m-cyclic if whenever (v11 ,..., v1N1 ) ‰

(v12 ,..., vN2 2 ) ‰ ... ‰ (v1m ,..., vNmm ) is a m-code word then it implies

(v12 , v31 ,..., v1N1 , v11 ) ‰ (v22 , v32 ,..., vN2 2 , v12 ) ‰ ... ‰ (v2m , v3m ,..., vNmm , v1m ) is also a m-code word.

Now we proceed on to define quasi MRD-m-codes. DEFINITION 3.19: Let C = C1 ‰ C2 ‰ … ‰ Cm be a RD rank mcode where each Ci z Cj if i z j. If some of the Ci’s are MRD codes and others are RD codes then we call C = C1 ‰ C2 ‰ … ‰ Cm to be a quasi MRD m-code.

92

Note: If r1 are MRD codes and r2 are RD codes r1 + r2 = m, (r1 t 1, r2 t 1). Then we call C to be a quasi (r1, r2) MRD m-code. We can say C1, …, Cr1 are Ci[ni, ki] RD-codes; i = 1, 2, 3, …, r1 and

Cj[nj, kj, dj] are MRD codes for j = 1, 2, 3, …, r2 with r1 + r2 = m. Thus C C1[n1 , k1 ] ‰ ... ‰ Cr1 [n r1 , k r1 ] ‰C1[n1 , k1 ,d1 ] ‰ ... ‰ Cr2 [n r2 , k r2 ,d r2 ] is a quasi (r1, r2) MRD m-code. Any m-code word in C would be of the form C = C1 ‰ C2 ‰ … ‰ Cm. The m-codes can be used in multi channel simultaneously when one needs both MRD codes and RD codes. This will be useful in applications in such type of channels. We proceed on to define the notion of quasi circulant m-codes of type I. DEFINITION 3.20: Let C1, C2, …, Cm be m distinct codes some circulant rank codes and others linear RD-codes defined over GF(2N). C = C1 ‰ C2 ‰ … ‰ Cm is defined as the quasi circulant m-code of type I. If in the quasi circulant m-code of type I some of the Ci’s are MRD codes i.e., C1, C2, …, Cm is a collection of RD codes, MRD codes and circulant rank codes then we define C = C1 ‰ C2 ‰ … ‰ Cm to be a quasi circulant m-code of type II.

We can define also mixed quasi circulant rank m-codes. DEFINITION 3.21: Let C1, C2, …, Cm be a collection of m-codes, all of them distinct Ci z Cj if i z j and Ci Œ Cj or Cj Œ Ci if i z j. If this collection of codes C1, C2, …, Cm are such that some of them are RD-codes, some MRD codes, some cyclic circulant rank codes and some only circulant codes then we define C = C1 ‰ C2 ‰ …‰ Cm to be a mixed quasi circulant rank m-code. DEFINITION 3.22: If C = C1 ‰ C2 ‰ … ‰ Cm be a collection of distinct circulant codes some Ci’s are circulant codes and some

93

of them are cyclic circulant codes then C is defined to be a mixed circulant m-code.

These codes will find applications in multi channels which have very high error probability and error correction. These multi channels (m-channels) are such that some of the channels have to work only with circulant codes not cyclic circulant codes and some only with cyclic circulant codes these mixed circulant mcodes will be appropriate. Now we proceed on to define the notion of almost maximum rank distance m-codes. DEFINITION 3.23: Let C1[n1, k1] ‰ C2[n2, k2] ‰ … ‰ Cm[nm, km] be a collection of m distinct almost maximum rank distance codes with minimum distances greater than equal to n1 – k1 ‰ n2 – k2 ‰ … ‰ nm – km defined over GF(2N). C is defined as the Almost Maximum Distance Rank-m-code or (AMRD-m-code) over GF(2N). An AMRD m-code is called a AMRD – tricode, if m = 3. An AMRD m-code whose minimum distance is greater than n1 – k1 ‰ n2 – k2 ‰ … ‰ nm – km is an MRD m-code hence the class of MRD m-codes is a subclass of the class of AMRD m-codes.

We have an interesting property about the AMRD m-codes. THEOREM 3.3: When (n1 – k1) ‰ (n2 – k2) ‰ … ‰ (nm – km) AMRD m-code C = C1[n1, k1] ‰ C2[n2, k2] ‰ … ‰ Cm[nm, km] is such that each (ni – ki) is odd for i = 1, 2, 3, …, m; then 1. The error correcting capability of the [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] AMRD m-code is equal to that of an [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] MRD m-code. 2. An [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] AMRD m-code is better than any [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] m-code in Hamming metric for error correction.

Proof: (1) Suppose C = C1 ‰ C2 ‰ … ‰ Cm is a [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] AMRD m-code such that (n1 – k1) ‰ (n2 –

94

k2) ‰ … ‰ (nm – km) is an odd m-integer (i.e., ni – ki z nj – kj if i z j are odd integer 1 d i, j d m). The maximum number of merrors corrected by C = C1 ‰ C2 ‰ … ‰ Cm is given by (n1  k1  1) (n 2  k 2  1) (n  k m  1) ‰ ‰ ... ‰ m . 2 2 2 But (n1  k1  1) (n 2  k 2  1) (n  k m  1) ‰ ‰ ... ‰ m 2 2 2 is equal to the error correcting capability of an [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km]; MRD m- code (since (n1 – k1), (n2 – k2), … (nm – km) are odd). That is, (n1 – k1) ‰ (n2 – k2) ‰ … ‰ (nm – km) is said to be m-odd if each ni – ki is odd for i = 1, 2, 3, …, m. Thus a [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] AMRD m-code is as good as an [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] MRD m-code.

Proof: (2) Suppose C = C1 ‰ C2 ‰ … ‰ Cm is a [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] AMRD m-code such that (n1 – k1) ‰ (n2 – k2) ‰ … ‰ (nm – km) are odd; then each m-code word of C can correct L r1 (n1 ) ‰ L r2 (n 2 ) ‰ ... ‰ L rm (n m ) L r (n) error m-vectors where r = r1 ‰ r2 ‰ … ‰ rm (n1  k1  1) (n 2  k 2  1) (n  k m  1) ‰ ‰ ... ‰ m 2 2 2 and L r (n) L r1 (n1 ) ‰ L r2 (n 2 ) ‰ ... ‰ L rm (n m ) n1 ªn º 1  ¦ « 1 » (2 N  1)...(2 N  2i 1 ) ‰ i 1 ¬ i ¼ n2 ªn º 1  ¦ « 2 » (2 N  1)...(2 N  2i 1 ) ‰ i 1 ¬ i ¼

nm ªn º ... ‰ 1  ¦ « m » (2 N  1)...(2 N  2i 1 ) . i 1 ¬ i ¼

Consider the same [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] m-code in Hamming metric. Let it be denoted by D = D1 ‰ D2 ‰ … ‰ Dm 95

then the minimum m-distance of D is atmost (n1 – k1 + 1) ‰ (n2 – k2 + 1) ‰…‰(nm – km + 1). The error correcting capability of D is « n1  k1  1  1 » « n 2  k 2  1  1 » « n  k m  1  1» ‰« ‰ ... ‰ « m « » » » 2 2 2 ¬ ¼ ¬ ¼ ¬ ¼ = r1 ‰ r2 ‰ … ‰ rm (since (n1 – k1) ‰ (n2 – k2) ‰ … ‰ (nm – km) are odd). Hence the number of error m-vectors corrected by the m-code word is given by r1

ª n1 º

i 0

¬ ¼

¦ « i » (2

N

r2 rm ªn º ªn º  1)i ‰ ¦ « 2 » (2 N  1)i ‰ ... ‰ ¦ « m » (2 N  1)i i 0 ¬ i ¼ i 0 ¬ i ¼

which is clearly less than L r1 (n1 ) ‰ L r2 (n 2 ) ‰ ... ‰ L rm (n m ) . Thus the number of error m-vectors that can be corrected by the [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] AMRD m-code is much greater than that of the same m- code considered in Hamming metric. For a given m-length n = n1 ‰ n2 ‰ … ‰ nm a single error correcting AMRD m-code is one having m-dimension (n1 – 3) ‰ (n2 – 3) ‰ … ‰ (nm – 3) and the minimum m-distance greater than or equal to 3 ‰ 3 ‰ … ‰ 3. We now proceed on to give a characterization of a single error correcting AMRD m-codes in terms of its parity check mmatrices. The characterization is based on the condition for the minimum distance proved by Gabidulin in [24, 27]. THEOREM 3.4: Let H = H1 ‰ H2 ‰…‰ Hm = (D ij1 ) ‰ (D ij2 ) ‰ …

‰ (D ijm ) be a (3 u n1) ‰ (3 u n2) ‰ … ‰ (3 u nm) m-matrix of mrank 3 over GF(2N); n1 d N and n2 d N which satisfies the following condition. For any two distinct, non empty m-subsets P1, P2, …, Pm where P1 P11 ‰ P21 ‰ ... ‰ Pm1 and P2 P12 ‰ P22 ‰ ... ‰ Pm2 of {1, 2, 3, …, n1} and {1, 2, 3, …, n2} respectively; there exists

96

i1 i11 ‰ i21 , i2 i12 ‰ i22 , …, im … ‰ {1, 2, 3} such that

i1m ‰ i2m  {1, 2, 3} ‰ {1, 2, 3} ‰

§ · § · 1 1 2 2 ¨¨ ¦ D i11 j11 ˜ ¦ D i12 k11 ¸¸ ‰ ¨¨ ¦ D i12 j12 ˜ ¦ D i22 k12 ¸¸ k11P21 k12 P22 © j11P11 ¹ © j12 P12 ¹ § · ‰... ‰ ¨ ¦ Dimm jm ˜ ¦ D imm k m ¸ ¨ jm Pm 1 1 k m Pm 2 1 ¸ 1 2 ©1 1 ¹ § · § · z ¨ ¦ D1i1 j1 ˜ ¦ D1i1k1 ¸ ‰ ¨ ¦ D i22 j2 ˜ ¦ D i22 k 2 ¸ ¨ j1P1 2 1 k1P1 1 1 ¸ ¨ j2 P2 2 1 k 2 P2 1 1 ¸ 1 2 1 2 ©1 1 ¹ ©1 1 ¹ § · ‰... ‰ ¨ ¦ D imm jm ˜ ¦ D imm k m ¸ . ¨ jm Pm 2 1 k m Pm 1 1 ¸ 1 2 ©1 1 ¹

Then H = H1 ‰ H2 ‰ … ‰ Hm as a parity check m-matrix defines a (n1, n1 – 3) ‰ (n2, n2 – 3) ‰ … ‰ (nm, nm – 3) single merror correcting AMRD m-code over GF(2). Proof: Given H = H1 ‰ H2 ‰ … ‰ Hm is a (3 u n1) ‰ (3 u n2) ‰ … ‰ (3 u nm) m-matrix of m-rank 3 ‰ 3 ‰ … ‰ 3 over GF(2N), so that H = H1 ‰ H2 ‰ … ‰ Hm as a parity check m-matrix defines a (n1, n1 – 3) ‰ (n2, n2 – 3) ‰ … ‰ (nm, nm – 3) RD mcode, where C1 {x  V n1 xH1T 0} ,

C2

{x  V n 2 xH T2

0} , …,

Cm {x  V xH 0} . It remains to prove that the minimum m-distance of C = C1 ‰ C2 ‰ … ‰ Cm is greater than or equal to 3 ‰ 3 ‰ … ‰ 3. We will prove that no non zero m-word of C = C1 ‰ C2 ‰ … ‰ Cm has m-rank less than 3 ‰ 3 ‰ … ‰ 3. nm

T m

The proof is by the method of contradiction. Suppose there exists a non zero m-code word x = x1 ‰ x2 ‰ … ‰ xm such that r1(x1) d 2, r2(x2) d 2, …, rm(xm) d 2, then x = x1

97

‰ x2 ‰ … ‰ xm can be written as x = x1 ‰ x2 ‰ … ‰ xm = y1 ‰ y2 ‰ … ‰ ym)(M1 ‰ M2 ‰ … ‰ Mm) where y1 (y11 , y12 ) , y 2 (y12 , y 22 ) , …, y m (y1m , y m2 ) ; y1i , yi2  GF(2 N ) ; 1 d i d m and M = M1 ‰ M2 ‰ … ‰ Mm

(m1ij ) ‰ (m ij2 ) ‰ ... ‰ (m ijm ) is a (2 u n1) ‰ (2 u n2) ‰ … ‰ (2 u nm) m-matrix of m-rank 2 ‰ 2 ‰ … ‰ 2 over GF(2). Thus (yM)HT = y1M1H1T ‰ y 2 M 2 H T2 ‰ ... ‰ y m M m H Tm =0‰0‰…‰0 implies that y(MHT) = y1 (M1H1T ) ‰ y 2 (M 2 H T2 ) ‰ ... ‰ y m (M m H Tm ) = 0 ‰ 0 ‰ … ‰ 0. Since y = y1 ‰ y2 ‰ … ‰ ym is non zero y(MHT) = (0 ‰ 0 ‰ … ‰ 0) implies yi (M i H iT ) 0 for i = 1, 2, 3, …, m; that is the 2 u 3 m-matrix M1H1T ‰ M 2 H T2 ‰ ... ‰ M m H Tm has m-rank less than 2 over GF(2N). Now let P11 ‰ P21 ‰ ... ‰ Pm1

P1 {j11 such that m11 j1 1

1} ‰ {j22 such that m12 j2 2

‰{jmm such that m1m jm

1} ‰ ...

1}

m

and P12 ‰ P22 ‰ ... ‰ Pm2

P2 {j11 such that m12 j1 1

1} ‰ {j22 such that m 22 j2 2

‰ {jmm such that m m2 jm

1} ‰ ...

1} .

m

Since M = M1 ‰ M2 ‰ … ‰ Mm

(m1ij ) ‰ (m ij2 ) ‰ ... ‰ (m ijm )

is a 2 u n1 ‰ 2 u n2 ‰ … ‰ 2 u nm m-matrix of m-rank 2 ‰ 2 ‰ … ‰ 2 and P1 and P2 are disjoint non empty m-subsets of {1, 2, …, n1} ‰ {1, 2, …, n2} ‰ … ‰ {1, 2, …, nm} respectively and MH T

M1H1T ‰ M 2 H T2 ‰ ... ‰ M m H Tm

98

§ ¦ D11j1 ¨ j11P11 1 ¨ ¨ ¨ D11j1 ¨ j¦ 1 © 11P21 § ¦ D1j2 2 ¨ j22 P12 2 ¨ ¨ ¨ D1j2 2 ¨ j2¦ 2 2 © 2 P2

¦D

¦D

· ¸ ¸ ¸‰ 1 ¸ ¦ D3 j11 ¸ j11P21 ¹

1 2 j11

j11P11

¦D

1 2 j11

j11P21

¦D

2 2 j22

¦D

2 2 j22

j22 P12

j22 P22

§ ¦ D1jm1 ¨ j11P11 1 ¨ ¨ ¨ D1jm2 ¨ j2¦ 2 2 © 2 P2

1 3 j11

j11P11

¦D

· ¸ ¸ ¸ ‰ ... ‰ 2 ¸ ¦ D3 j22 ¸ j22 P22 ¹ j22 P12

¦D

m 2 j11

¦D

m 2 j22

j11P11

j22 P22

2 3 j22

¦D

· ¸ ¸ ¸. D 3mj2 ¸ ¦ 2 ¸ 2 2 j2 P2 ¹ j11P11

m 3 j11

But the selection of H = H1 ‰ H2 ‰ … ‰ Hm is such that there exists i1p ,i p2  {1, 2, 3}; p = 1, 2, …, m such that

¦D

j11P1

i11 j11

˜

¦D

k1P2

i12 k1

‰

¦D

j22 P1

z

¦D

j11P1

i12 j11

˜

¦D

k1P2

i11k1

¦D

j11P1

¦D

j22 P1 i1m j11

‰

1 im 2 j1

˜

i12 j11

k 2 P2

¦D

k m P2

¦D

j22 P1

˜

i 22 j11

¦D

k m P2

¦D

˜

i 22 k 2

‰ ... ‰

i 2m k m

˜

¦D

k 2 P2

i1m k m

i12 k 2

‰ ... ‰

.

Hence in MHT there exists a 2 u 2 m-submatrices whose determinant is non zero; i.e., r(MH T ) r1 (M1H1T ) ‰ r2 (M 2 H T2 ) ‰ ... ‰ rm (M m H Tm ) over GF(2N). But this is a contradiction to fact that,

99

rank(MH T ) rank(M1H1T ) ‰ rank(M 2 H T2 ) ‰ ... ‰ rank(M m H Tm ) < 2 ‰ 2 ‰ … ‰ 2. Hence the proof. Now using constant rank code we proceed on to define the notion of constant rank m-codes of m-length n1 ‰ n2 ‰ … ‰ nm. DEFINITION 3.24: Let C1 ‰ C2 ‰ … ‰ Cn be a RD-m-code where Ci is a constant rank code of length ni, i = 1, 2, …, m (Each Ci is a subset of the rank space V ni ; i = 1, 2, …, m) then C = C1 ‰ C2 ‰ … ‰ Cm is a constant m-rank code of m-length n1 ‰ n2 ‰ … ‰ nm; that is every m-code word has same m-rank. DEFINITION 3.25: A(n1, r1, d1) ‰ A(n2, r2, d2) ‰ … ‰ A(nm, rm, dm) is defined as the maximum number of m-vectors in V n1 ‰ V n2 ‰ ... ‰ V nm of constant m-rank, r1 ‰ r2 ‰ … ‰ rm and m-distance between any two m-vectors is at least d1 ‰ d2 ‰ … ‰ dm [By (n1, r1, d1) ‰ (n2, r2, d2) ‰ … ‰ (nm, rm, dm) m-set we mean a m-subset of m-vectors of V n1 ‰ V n2 ‰ ... ‰ V nm having constant m-rank r1 ‰ r2 ‰ … ‰ rm and m-distance between any two m-vectors is atleast d1 ‰ d2 ‰ … ‰ dm].

We analyze the m-function A(n1, r1, d1) ‰ A(n2, r2, d2) ‰ … ‰ A(nm, rm, dm) by the following theorem: THEOREM 3.5: 1. A(n1, r1, 1) ‰ A(n2, r2, 1) ‰ … ‰ A(nm, rm, 1) = Lr1 ( n1 ) ‰ Lr2 (n2 ) ‰ ... ‰ Lrm (nm ) , the number of m-vectors of m-

rank r1 ‰ r2 ‰ … ‰ rm in V n1 ‰ V n2 ‰ ... ‰ V nm . 2. A(n1, r1, d1) ‰ A(n2, r2, d2) ‰ … ‰ A(nm, rm, dm) = 0 ‰ 0 ‰ … ‰ 0 if ri > 0 or di > ni and di > 2ri (i = 1, 2, 3, … , m), Proof: (1) Follows from the fact that L r1 (n1 ) ‰ L r2 (n 2 ) ‰ ... ‰ L rm (n m )

is the number of m-vectors of m-length n1 ‰ n2 ‰ … ‰ nm, constant m-rank r1 ‰ r2 ‰ … ‰ rm and m-distance between any

100

two distinct m-vectors in a m-rank space V n1 ‰ V n 2 ‰ ... ‰ V n m is always greater than or equal to 1 ‰ 1 ‰ … ‰ 1. (2) Follows immediately from the definition of A(n1, r1, d1) ‰ A(n2, r2, d2) ‰ … ‰ A(nm, rm, dm). THEOREM 3.6: A(n1, 1, 2) ‰ A(n2, 1, 2) ‰ … ‰ A(nm, 1, 2) = 2 n1  1 ‰ 2 n2  1 ‰.... ‰ 2 nm  1 over any Galois field GF (2N).

Proof: Denote by V1 ‰ V2 ‰ … ‰ Vm the set of m-vectors of m-rank 1 ‰ 1 ‰ … ‰ 1 in V n1 ‰ V n 2 ‰ ... ‰ V n m ; we know each non zero element D1 ‰ D2 ‰ … ‰ Dm  GF(2N), there exists (2n1  1) ‰ (2n 2  1) ‰ ... ‰ (2n m  1) m-vectors of m-rank one having D1 ‰ D2 ‰ … ‰ Dm as a coordinate. Thus the mcardinality of V1 ‰ V2 ‰ … ‰ Vm is (2 N  1)(2n1  1) ‰ (2 N  1)

(2n 2  1) ‰ ... ‰ (2 N  1)(2n m  1) . Now m-divide V1 ‰ V2 ‰ … ‰ Vm into (2n1  1) ‰ (2n 2  1) ‰ ... ‰ (2n m  1) blocks of (2N –1) ‰ (2N –1) ‰ … ‰ (2N –1) m-vectors such that each block consists of the same pattern of all nonzero m-elements of GF(2N) ‰ GF(2N) ‰ … ‰ GF(2N). Then from each m-block element almost one m-vector can be choosen such that the selected m-vectors are atleast rank 2 apart from each other. Such a m-set we call as (n1, 1, 2) ‰ (n2, 1, 2) ‰ … ‰ (nm, 1, 2) m-set. Also it is always possible to construct such a m-set. Thus A(n1, 1, 2) ‰ A(n2, 1, 2) ‰ … ‰ A(nm, 1, 2) = (2n1  1) ‰ (2n 2  1) ‰... ‰ (2n m  1) . THEOREM 3.7: A(n1, n1, n1) ‰ A(n2, n2, n2) ‰ … ‰ A(nm, nm, nm) = (2N –1) ‰ (2N –1) ‰ … ‰ (2N –1) (i.e., A(ni, ni, ni) = 2N – 1; i = 1, 2, 3, …, m) over GF(2N).

Proof: Denote by Vn1 ‰ Vn 2 ‰ ... ‰ Vn m the m-set of all m-

vectors of m-rank n1 ‰ n2 ‰…‰ nm in the m-space V n1 ‰ V n 2 ‰… ‰V n m . We know the m-cardinality of Vn1 ‰ Vn 2 ‰ ... ‰ Vn m is

101

(2N – 1)(2N – 2) … (2 N  2n1 1 ) ‰ (2N – 1) (2N – 2) … (2 N  2n 2 1 ) ‰ … ‰ (2N – 1) (2N – 2) … (2 N  2n m 1 )

and by the definition in a (n1, n1, n1) ‰ (n2, n2, n2) ‰ … ‰ (nm, nm, nm) m-set, the m-distance between any two m-vector should be n1 ‰ n2 ‰ … ‰ nm. Thus no two m-vectors can have a common symbol at a co-ordinate place i1 ‰ i2 ‰ … ‰ im; (1 d i1 d n1, 1 d i2 d n2, …, 1 d im d nm). This implies that A(n1, n1, n1) ‰ A(n2, n2, n2) ‰ … ‰ A(nm, nm, nm) d (2N –1) ‰ (2N –1) ‰ … ‰ (2N –1). Now we construct a (n1, n1, n1) ‰ (n2, n2, n2) ‰ … ‰ (nm, nm, nm) m-set as follows: Select N m-vectors from Vn1 ‰ Vn 2 ‰ ... ‰ Vn m such that i.

Each m-basis m-elements of GF(2n1 ) ‰ GF(2n 2 ) ‰ … ‰

GF(2n m ) should occur (can be as a m-combination) atleast once in each m-vector. ii. If the (i1th ,i 2th ,...,i mth ) m-vector is choosen ((i1  1) th , (i 2  1) th ,...,(i m  1) th ) m-vector should be selected such that its m-rank m-distance from any m-linear combination of the previous (i1i2… im) m-vectors is n1 ‰ n2 ‰ … ‰ nm. Now the set of all m-linear combinations of these N ‰ N ‰ … ‰ N, mvectors over GF(2) ‰ GF(2) ‰ … ‰ GF(2) will be such that the m-distance between any two m-vectors is n1 ‰ n2 ‰ … ‰ nm. Hence it is (n1, n1, n1) ‰ (n2, n2, n2) ‰ … ‰ (nm, nm, nm) m-set. Also the m-cardinally of this (n1, n1, n1) ‰ (n2, n2, n2) ‰ … ‰ (nm, nm, nm) m-sets is (2N –1) ‰ (2N –1) ‰ … ‰ (2N –1) (we do not count all zero m-linear combinations). Thus A(n1, n1, n1) ‰ A(n2, n2, n2) ‰ … ‰ A(nm, nm, nm) (2N –1) ‰ (2N –1) ‰ … ‰ (2N –1). Recall a [n, 1] repetition RD code is a code generated by the matrix G = (1, 1, …, 1) over F2N . Any non zero code word has rank 1.

102

DEFINITION 3.26: A [n1, 1] ‰ [n2, 1] ‰ … ‰ [nm, 1] repetition RD m-code is a m-code generated by the m-matrix G = G1 ‰ G2 ‰ … ‰ Gm = (1 1 … 1) ‰ (1 1 … 1) ‰ … ‰ (1 1 … 1) (Gi z Gj if i z j; 1 d i, j d m) over F2 N . Any non zero m-code word has mrank 1 ‰ 1 ‰ … ‰ 1. DEFINITION 3.27: Let C = C1 ‰ C2 ‰ … ‰ Cm be a linear [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] RD m-code defined over F2 N . The

covering m-radius of C = C1 ‰ C2 ‰ … ‰ Cm is defined as the smallest m-tuple of integers (r1, r2, …, rm) such that all mvectors in the rank m-space F2nN1 ‰ F2nN2 ‰ ... ‰ F2nNm are with in the rank m-distance r1 ‰ r2 ‰ … ‰ rm of some m-code word. The covering m-radius of C = C1 ‰ C2 ‰ … ‰ Cm is denoted by t(C1) ‰ t(C2) ‰ … ‰ t(Cm) = t(C) ­ min( r1 ( x1  C1 ))½ ­ min( r2 ( x2  C2 ))½ = maxn ® ¾ ‰ maxn2 ® ¾ 1 x1 F N c  C 1 ¿ x2 F2N ¯c2  C2 ¿ 2 ¯ 1 min( r ( x  C )) ­ ½ m m m ‰... ‰ maxn ® ¾. xm F Nm c  C m ¯ m ¿ 2

THEOREM 3.8: The linear [n1, k1] ‰ [n2, k2] ‰ … ‰ [nm, km] RD-m-code C = C1 ‰ C2 ‰ … ‰ Cm satisfies t(C) = t(C1) ‰ t(C2) ‰ … ‰ t(Cm) d (n1 – k1) ‰ (n2 – k2) ‰ … ‰ (nm – km).

Proof: Let C = C1 ‰ C2 ‰ … ‰ Cn be a (n1, k1) ‰ … ‰ (nm, km) RD-m-code. Consider the m-generator m-matrix G = G1 ‰ G2 ‰ … ‰ Gm = (I k1 , A k1 ,n1  k1 ) ‰ (I k 2 , A k 2 ,n 2  k 2 ) ‰ ! ‰ (I k m , A k m ,n m  k m ) .

Suppose x = x1 ‰ x2 ‰ … ‰ xm = (x , x12 , !, x1k1 , x1k1 1 ,!, x1n1 ) ‰ … 1 1

‰ (x1m , x 2m , !, x kmm , x kmm 1 ,!, x nmm )

103

be any m vector in V n1 ‰ ! ‰ V n m . Let C = (C1 ‰ C2 ‰ … ‰ Cn)(G1 ‰ G2 ‰ … ‰ Gm) = C1G1 ‰ C2G2 ‰ … ‰ CnGm 1 = (x1 ! x1k1 ) G1 ‰ … ‰ (x1m ! x kmm ) Gm.

Then C is a m-code word of C and r(x +c) = r1(x1 + c1) ‰ … ‰ rm(xm + cm) d n1 – k1 ‰ … ‰ nm – km. Hence the proof. For any [n1, k1] ‰ [n2, k1] ‰ … ‰ [nm, k1] repetition RD m-code generated by the m-matrix G = G1 ‰ G2 ‰ … ‰ Gm = (1 1 … 1) ‰ (1 1 … 1) ‰ … ‰ (1 1 … 1) (Gi z Gj if i z j; 1 d i, j d m) over F2N . A non zero m-code word of it has m-rank 1 ‰ 1 ‰ … ‰ 1. We proceed on to define the notion of covering m-radius. THEOREM 3.9: The covering m-radius of a [n1, 1] ‰ [n2, 1] ‰ … ‰ [nm, 1] repetition RD m-code over F2 N is (n1 – 1) ‰ (n2 –

1) ‰ … ‰ (nm – 1). Proof: The Cartesian m-product of 2 linear RD m-codes C C1[n11 , k11 ] ‰ C 2 [n12 , k12 ] ‰ ... ‰ C m [n1m , k1m ] and D D1[n12 , k12 ] ‰ D 2 [n 22 , k 22 ] ‰ ... ‰ D m [n m2 , k 2m ] over F2N is given by C u D = C1 u D1 ‰ C2 u D2 ‰ … ‰ Cm u Dm {(a11 , b11 ) a11  C1 and b11  D1} ‰ {(a12 , b12 ) a12  C2 and b12  D 2 } ‰ ! ‰ {(a1m , b1m ) a1m  Cm and b1m  D m } . C u D is a {(n11  n12 ) ‰ (n12  n 22 ) ‰ ... ‰ (n1m  n 2m ), (k11  k12 ) ‰ (k12  k 22 ) ‰ ... ‰ (k1m  k 2m )} linear RD m-code.

(We assume n1i  n i2 d N for i = 1, 2, … , m).

104

Now the reader is expected to prove the following theorem: THEOREM 3.10: If C = C1 ‰ C2 ‰ … ‰ Cm and D = D1 ‰ D2 ‰ … ‰ Dm be two linear RD m-codes then t(C u D) d (t(C1)  t(D1)) ‰ (t(C2)  t(D2)) ‰ … ‰ (t(Cm)  t(Dm)).

Hint: If C = C1 ‰ C2 ‰ … ‰ Cm and D = D1 ‰ D2 ‰ … ‰ Dm then C u D = {C1 u D1} ‰ {C2 u D2} ‰ … ‰ {Cm u Dm} and t(C u D) = t(C1 u D1) ‰ t(C2 u D2) ‰ … ‰ t(Cm u Dm) d {t(C1)  t(D1)} ‰ {t(C2)  t(D2)} ‰ … ‰ {t(Cm)  t(Dm)}. Next we proceed on to define the notion of m-divisible linear RD m-codes we have earlier defined the notion of bidivisible linear RD bicodes DEFINITION 3.28: C = C1[n1, k1, d1] ‰ C2[n2, k2, d2] ‰ … ‰ Cm[nm, km, dm] be a linear RD m-code over Fq N , ni d N, 1 d i d

m and N > 1. If there exists (m1, m2, …, mm) (mi > 1; i = 1, 2, …, m) such that mi ; ri ( ci ; q ) 1 d i d ni; i = 1, 2, …, m for all ci  Ci; then we say the m-code C is m-divisible. THEOREM 3.11: Let C = C1[n1, 1, n1] ‰ C2[n2, 1, n2] ‰ … ‰ Cm[nm, 1, nm] (ni z nj, i z j; 1 d i, j d m) be a MRD m-code for all ni d N, 1 d i d m. Then C is a m-divisible m-code.

Proof: Since there cannot exists m-code words of m-rank greater than (n1, n2, …, nm) in an [n1, 1, n1] ‰ [n2, 1, n2] ‰ … ‰ [nm, 1, nm] MRD m-code. C is a m-divisible m-code. DEFINITION 3.29: Let Ci = [ni, ki] be a linear RD-code, i = 1, 2, 3, …, m1 and Cj = [nj, kj, dj] linear divisible RD codes, j = 1, 2, 3, …, m2 defined over GF(2N). Let m = m1 + m2, then the RD linear m-code C = C1 ‰ C2 ‰ … ‰ Cm is defined as quasi divisible RD m-code, ni d N, 1 d i d m.

105

DEFINITION 3.30: Let Ci = Ci[ni, ki, di] be a MRD code which is not divisible and Cj = Cj[nj, kj, dj] be a divisible MRD code defined over GF (2N); i = 1, 2, 3, …, m1 and j = 1, 2, 3, …, m2 such that m = m1 + m2. C = C1 ‰ C2 ‰ … ‰ Cm is defined to be a quasi divisible MRD m-code. DEFINITION 3.31: Let C1, C2, …, Cm1 be circulant rank codes

and Cj[nj, kj, dj] a divisible RD-code defined over GF(2N); j = 1, 2, …, m2 such that m1 + m2 = m. Then C = C1 ‰ C2 ‰ … ‰ Cm is defined to be a quasi divisible circulant rank m-code. DEFINITION 3.32: Let C1, C2, …, Cm1 be AMRD codes and Cj[nj,

kj, dj] be a divisible RD code defined over GF(2N), j = 1, 2, …, m2 such that m1 + m2 = m. Then C = C1 ‰ C2 ‰ … ‰ Cm is defined to be the quasi divisible AMRD m-code. We see non divisible MRD m-codes exists as there exists non divisible MRD bicodes. DEFINITION 3.33: Let Ci = Ci[ni, ki, di], i = 1, 2, …, m be MRD codes defined over Fq N , ni d N; i = 1, 2, …, m with ni z nj if i z

j, 1 d i, j d m. As1 [ n1 ,d1 ] ‰ As2 [ n2 ,d 2 ] ‰ ... ‰ Asm [ nm ,d m ] be the number of m-code words with rank m-norms si in the linear [ni, ki, di] MRD-code 1 d i d m. Then m-spectrum of the MRD m-code C = C1 ‰ C2 ‰ … ‰ Cm is described by the formulae A0(n1, d1) ‰ A0(n2, d2) ‰ … ‰ A0(nm, dm) = 1 ‰ 1 ‰ … ‰ 1 Ad1  m1 ( n1 ,d1 ) ‰ Ad2  m2 ( n2 ,d 2 ) ‰ ... ‰ Adm  mm ( nm ,d m ) d  m1 º ( m1  j1 )( m1  2j1 1 )( Q ª n1 º m1 j1  m1 ª 1 « d  m » ¦ ( 1 ) «d  j » q 1 ¼ j1 0 ¬ 1 ¬ 1 1¼

106

j1 1 1 )

‰

ª n2 º m2 j2  m2 « d  m » ¦ ( 1 ) ¬ 2 2 ¼ j2 0

ª d 2  m2 º ( m2  j2 )( m2  2j2 1 )( Q « »q ¬ d 2  j2 ¼

ª nm º mm jm  mm « » ¦ ( 1 ) ¬ d m  mm ¼ jm 0

j2 1 1 )

ª d m  mm º ( mm  jm )( mm  2jm 1 )( Q « »q ¬ d m  jm ¼

‰ ... ‰ jm 1 1 )

where Q = qN; ª ni º « » ¬ mi ¼

( q ni  1 )( q ni  2 )...( q ni  q mi 1 ) ; i = 1, 2, …, m. ( q mi  1 )( q mi  2 )...( q mi  q mi 1 )

Using the m-spectrum of a MRD m-code we prove the following theorem: THEOREM 3.12: All MRD m-codes C1[n1, k1, d1] ‰ C2[n2, k2, d2] ‰ … ‰ Cm[nm, km, dm] with di < ni (i.e., with ki t 2) 1 d i d m are non m-divisible.

Proof: This is proved by making use of the m-spectrum of the MRD m-code. Clearly A d1 (n1 ,d1 ) ‰ A d 2 (n 2 ,d 2 ) ‰ ... ‰ A dm (n m ,d m ) z 0 ‰ 0 ‰ ... ‰ 0. If the existence of a m-code word with m-rank (d1+1) ‰ (d2+1) ‰ … ‰ (dm+1) is established then the proof is complete as the m-gcd {(d1, d1+1) ‰ (d2, d2+1) ‰ … ‰ (dm, dm+1)} = 1 ‰ 1 ‰ … ‰ 1. So the proof is to show that, A d1 1 (n1 ,d1 ) ‰ A d2 1 (n 2 ,d 2 ) ‰ ... ‰ A d m 1 (n m ,d m )

is non zero (i.e., A di 1 (n i ,d i ) z 0 ; i = 1, 2, …, m). Now A d1 1 (n1 ,d1 ) ‰ A d2 1 (n 2 ,d 2 ) ‰ ... ‰ A d m 1 (n m ,d m )

· ª n1 º § ªd1  1º 2 « d  1» ¨¨  « d » ª¬ (Q  1)  (Q  1) º¼ ¸¸ ‰ ¬ 1 ¼© ¬ 1 ¼ ¹

107

· ª n 2 º § ªd 2  1º 2 « d  1» ¨¨  « d » ª¬(Q  1)  (Q  1) º¼ ¸¸ ‰ ... ‰ ¬ 2 ¼© ¬ 2 ¼ ¹ · ª n m º § ª d m  1º 2 « d  1» ¨¨  « d » ª¬ (Q  1)  (Q  1) º¼ ¸¸ ¬ m ¼© ¬ m ¼ ¹ § ª n1 º ª d1  1º · « d  1» (Q  1) ¨¨ Q  1  « d » ¸¸ ‰ ¬ 1 ¼ ¬ 1 ¼¹ © § ª n2 º ª d 2  1º · « d  1» (Q  1) ¨¨ Q  1  « d » ¸¸ ‰ ... ‰ ¬ 2 ¼ ¬ 2 ¼¹ © § ª nm º ªd m  1º · « d  1» (Q  1) ¨¨ Q  1  « d » ¸¸ . ¬ m ¼ ¬ m ¼¹ © Suppose ªd1  1º ªd 2  1º ªd m  1º Q 1 « ‰ Q 1 « ‰ ... ‰ Q  1  « » » » ¬ d1 ¼ ¬ d2 ¼ ¬ dm ¼

= 0 ‰ 0 ‰ … ‰ 0; i.e., qN  1

q d 1  1 q 1

q 1

qd  1 . q N 1

i.e.,

Clearly, qd  1 1 q N 1 For, if

qd  1 t 1 then q N 1  q d  1 q N 1

108

which is impossible as d < n d N. Thus q – 1 < 1 which implies q < 2 a contradiction. Hence, A d 1 (n 1 , d 1 ) ‰ A d  1 (n 2 , d 2 ) ‰ ... ‰ A d  1 (n m , d m ) is non 1

2

m

zero. Thus expect C1(n1, 1, n1) ‰ C2(n2, 1, n2) ‰ … ‰ Cm(nm, 1, nm) MRD m-codes all C1[n1, k1, d1] ‰ C2[n2, k2, d2] ‰ … ‰ Cm[nm, km, dm] MRD m-codes with di d ni; i = 1, 2, …, m are non divisible. Now finally we define the fuzzy rank distance m-codes (m t 3). Recall Von Kaenel introduced the idea of fuzzy codes with hamming metric and we have defined fuzzy RD codes with Rank metric, we have in the earlier chapter defined the notion of fuzzy RD bicodes. We now proceed onto define the new notion of fuzzy RD m-codes (m t 3) when (m 3), we call the fuzzy RD m-code to be a fuzzy RD tricode. In the chapter two we have recalled the notion of three types of errors namely asymmetric, symmetric and unidirectional. We proceed onto define the notion of fuzzy RD m-codes m t 3. DEFINITION 3.34: Let V n1 ‰ V n2 ‰ ... ‰ V nm denote the (n1, n2, …, nm) dimensional vector m-space of (n1, n2, …, nm)-tuples over F2 N ; ni d N and N >1; 1 d i d m.

Let ui, vi  V ni ; i = 1, 2, …, m, where, ui ( u1i ,u2i ,...,uni i ) and vi

( v1i ,v2i ,...,vni i )

with

u ij , vij  F2 N ; 1 d j d ni; 1 d i d m. A fuzzy RD m-code word fu1 ‰u2 ‰...‰um

f u11 ‰ fu22 ‰ ... ‰ f umm is a

fuzzy m-subset of V n1 ‰ V n2 ‰ ... ‰ V nm defined by, fu1 ‰u2 ‰...‰um

f u11 ‰ fu22 ‰ ... ‰ f umm

{( v1 , f u11 ( v1 )) v1 V n1 } ‰ {( v2 , f u22 ( v2 )) v2 V n2 } ‰ ... ‰

109

{ ( vm , fumm ( vm )) vm V nm } where fu11 ( v1 ) ‰ f u22 ( v2 ) ‰ ... ‰ fumm ( vm ) is the membership mfunction. DEFINITION 3.35: For the symmetric error m-model assume p1 ‰ p2 ‰ … ‰ pm to represent the m-probability that no transition (i.e., error) is made and q1 ‰ q2 ‰ … ‰ qm to represent the mprobability that a m-rank error occurs so that, p1 + q1 ‰ p2 + q2 ‰…‰ pm + qm = 1 ‰ 1 ‰ … ‰ 1, then fu11 ( v1 ) ‰ f u22 ( v2 ) ‰ ! ‰ f umm ( vm )

p1n1  r1 q1r1 ‰ p2n2  r2 q2r2 ‰ ! ‰ pmnm  rm qmrm ri ( ui  vi ,2 )

where ri

ui  vi , i = 1, 2, …, m.

DEFINITION 3.36: For unidirectional and asymmetric error mmodels assume q1 ‰ q2 ‰ … ‰ qm to represent the probability that (1 o 0) ‰ (1 o 0) ‰ … ‰ (1 o 0) m-transition or (0 o 1) ‰ (0 o 1) ‰ … ‰ (0 o 1) m-transition occurs. Then

fu11 ( v1 ) ‰ f u22 ( v2 ) ‰ ... ‰ fumm ( vm ) n1

nm

n2

– fu11 ( vi1 ) ‰ – fu22 ( vi2 ) ‰ ... ‰ – fumm ( vim ) i 1

i

i 1

i

i 1

i

where fu11 ( vi1 ) ‰ f u22 ( vi2 ) ‰ ! ‰ fumm ( vim ) inherits its definition i

i

i

from the unidirectional and asymmetric m-models respectively, since each ui1 ‰ ui2 ‰ ! ‰ uim or vi1 ‰ vi2 ‰ ! ‰ vim itself is an N-m-tuple over F2. That is since uij , vij  F2 N ; j = 1, 2, …, m; each ui1 ‰ ui2 ‰ ! ‰ uim or vi1 ‰ vi2 ‰ ! ‰ vim itself is an N, mtuple from F2. uij ( ui1j ‰ uij2 ‰ ! ‰ uiNj ) , vkj

( vkj1 ‰ vkj2 ‰ ! ‰ vkNj )

where, uipj vklj  F2 , 1 d p, l d N and 1 d i d m. Then for unidirectional error m-model

110

fu11 ( vi1 ) ‰ f u22 ( vi2 ) ‰ ! ‰ f umm ( vim ) i

i

i

­0 ‰ 0 ‰ ! ‰ 0 ° min( ki11 ,ki12 ) ‰ ... ‰ min( ki1m ,kim2 ) z 0 ‰ ... ‰ 0 ® 2 2 2 m m m ° mi1  di1 di1 q1 ‰ p2mi  di q2di ‰ ... ‰ pmmi  di qmdi otherwise ¯ p1 N

where kitj

¦ max( 0,u

j is

 visj ) where j = 1, 2, …, m and i = 1,

s 1

2, …, m. ­°ki1j if ki j2 ® j j °¯ki 2 if ki1

di j

0 0

j = 1, 2, …, m.

mij

­N j j °¦ uis if ki 2 0 °s 1 N ° N uisj if ki1j 0  ® ¦ s 1 ° °max  u j ,N  u j if k j ¦ is ¦ is i1 ° ¯

kid2

0

j = 1, 2, …, m. For the asymmetric error m-model fu11 ( vi1 ) ‰ f u22 ( vi2 ) ‰ ... ‰ f umm ( vim ) i

i

i

­0 ‰ ! ‰ 0 ° if min( ki11 ,ki12 ) ‰ ! ‰ min( ki1m ,kim2 ) z 0 ‰ ! ‰ 0 ® 2 2 2 m m m ° mi1  di1 di1 q1 ‰ p2mi  di q2di ‰ ! ‰ pmmi  di qmdi otherwise ¯ p1 where di j

ki1j and j = 1, 2, …, m. N

mij

¦u

j is

, j = 1, 2, …, m

s 1

for asymmetric (1 o 0) ‰ (1 o 0) error m-model and di j j = 1, 2, …, m and

111

ki j2 ,

N

mij

N  ¦ uisj , s 1

j = 1, 2, …, m for the asymmetric 0 o 1 ‰ 0 o 1 error mmodel. The results for minimum m-distance of a fuzzy RD-m-code can be derived as in case of minimum bidistance of a fuzzy RD bicode. The notions related to m-covering radius of RD bicodes can be analogously transformed to RD-p-codes (p t 3). PROPOSITION 3.1: If C11 ‰ C21 ‰ ! ‰ Cm1 and C12 ‰ C22 ‰ ! ‰ Cm2 are RD m-codes with C11 ‰ C21 ‰ ! ‰ Cm1 Ž C12 ‰ C22 ‰ ! ‰ Cm2

(i.e., C 1j Ž C 2j ; j = 1, 2, …, m) then

tm1 ( C11 ) ‰ tm2 ( C21 ) ‰ ... ‰ tmm ( Cm1 ) t tm1 ( C12 ) ‰ tm2 ( C22 ) ‰ ... ‰ tmm ( Cm2 ) (i.e., t mi (C1i ) t t mi (Ci2 ) ; i = 1, 2, …, m). Proof: Let Si Ž V ni with 1 d j d m; |Si| = mi; i = 1, 2, …, m cov(C12 ,S1 ) ‰ cov(C 22 ,S2 ) ‰ ... ‰ cov(C2m ,Sm ) = min{cov(x1 ,S1 ); x1  C12 } ‰ min{cov(x 2 ,S2 ); x 2  C 22 } ‰ ... ‰ min{cov(x m ,Sm ); x m  C 2m }

d min{cov(x1 ,S1 ) x1  C11} ‰ min{cov(x 2 ,S2 ) x 2  C12 } ‰ ... ‰ min{cov(x m ,Sm ) x m  C1m } cov(C11 ,S1 ) ‰ cov(C12 ,S2 ) ‰ ... ‰ cov(C1m ,Sm ) .

Thus

t m1 (C12 ) ‰ t m2 (C22 ) ‰ ... ‰ t mm (C2m ) d t m1 (C11 ) ‰ t m2 (C12 ) ‰ ... ‰ t mm (C1m ) .

112

PROPOSITION 3.2: For any RD m-code C = C1 ‰ C2 ‰ … ‰ Cm and a m-tuple of positive integers (m1, m2, …, mm) tm1 ( C1 ) ‰ tm2 ( C2 ) ‰ ! ‰ tmm ( Cm ) d

tm1 1 ( C1 ) ‰ tm2 1 ( C2 ) ‰ ! ‰ tmm 1 ( Cm ). Proof: Since Si Ž V ni ; i = 1, 2, …, m; S1 ‰ S2 ‰ … ‰ Sm is a m-subset of V n1 ‰ V n 2 ‰ ! ‰ V n m . Now t m1 (C1 ) ‰ t m2 (C 2 ) ‰ ! ‰ t mm (C m ) max{cov(C1 ,S1 ) S1 Ž V n1 , S1 max{cov(C2 ,S2 ) S2 Ž V , S2

m1} ‰ m 2 } ‰ ... ‰

n2

max{cov(C m ,Sm ) Sm Ž V , Sm nm

d max{cov(C1 ,S1 ) S1 Ž V n1 , S1 max{cov(C 2 ,S2 ) S2 Ž V , S2 n2

mm } m1  1} ‰

m 2  1} ‰ ... ‰

max{cov(C m ,Sm ) Sm Ž V , Sm

m m  1}

nm

t m1 1 (C1 ) ‰ t m2 1 (C2 ) ‰ ... ‰ t mm 1 (Cm ) . PROPOSITION 3.3: For any m-set of positive integers {n1, m1, k1, K1} ‰ {n2, m2, k2, K2} ‰ … ‰ {nm, mm, km, Km}; tm1 [ n1 ,k1 ] ‰ tm2 [ n2 ,k2 ] ‰ ! ‰ tmm [ nm ,km ]

d tm1 1 [ n1 ,k1 ] ‰ tm2 1 [ n2 ,k2 ] ‰ ! ‰ tmm 1 [ nm ,km ] . Proof: Given C1[n1, k1] ‰ C2[n2, k2] ‰ … ‰ Cm[nm, km] to be a RD m-code with Ci Ž V ni ; i = 1, 2, …, m. Now t m1 [n1 , k1 ] ‰ t m2 [n 2 , k 2 ] ‰ ... ‰ t mm [n m , k m ] min{t m1 (C1 ) C1 Ž V n1 , dim C1 min{t m 2 (C 2 ) C 2 Ž V n 2 , dim C 2

113

k 1} ‰

k 2 } ‰ ... ‰

min{t mm (Cm ) C m Ž V n m ,dim C m

km}

d min{t m1 1 (C1 ) C1 Ž V n1 ,dim C1

k1 } ‰

min{t m2 1 (C 2 ) C2 Ž V ,dim C 2

k 2 } ‰ ... ‰

n2

min{t mm 1 (Cm ) Cm Ž V ,dim Cm nm

km}

t m1 1[n1 , k1 ] ‰ t m2 1[n 2 , k 2 ] ‰ ... ‰ t mm 1[n m , k m ] . Similarly we have t m1 [n1 , K1 ] ‰ t m2 [n 2 , K 2 ] ‰ ... ‰ t mm [n m , K m ]

d t m1 1[n1 , K1 ] ‰ t m2 1[n 2 , K 2 ] ‰ ... ‰ t mm 1[n m , K m ] . That is t m1 [n1 , K1 ] ‰ t m2 [n 2 , K 2 ] ‰ ... ‰ t mm [n m , K m ]

min{t m1 (C1 ) C1 Ž V n1 , C1 min{t m2 (C2 ) C2 Ž V n 2 , C2

K1} ‰ K 2 } ‰ ... ‰

min{t mm (C m ) C m Ž V , Cm

Km}

d min{t m1 1 (C1 ) C1 Ž V n1 , C1

K1 } ‰

nm

min{t m2 1 (C 2 ) C2 Ž V , C2 n2

K 2 } ‰ ... ‰

min{t mm 1 (Cm ) Cm Ž V n m , Cm

Km}

d t m1 1[n1 , K1 ] ‰ t m2 1[n 2 , K 2 ] ‰ ... ‰ t mm 1[n m , K m ] . PROPOSITION 3.4: For any m-set of positive integers {n1, m1, k1, K1} ‰ {n2, m2, k2, K2} ‰ … ‰ {nm, mm, km, Km}; tm1 [ n1 ,k1 ] ‰ tm2 [ n2 ,k2 ] ‰ ... ‰ tmm [ nm ,km ]

t tm1 [ n1 ,k1  1] ‰ tm2 [ n2 ,k2  1] ‰ ... ‰ tmm [ nm ,km  1] .

114

Proof: Given C = C1 ‰ C2 ‰ … ‰ Cm is a RD m-code, hence a m-subspace of V n1 ‰ V n 2 ‰ ! ‰ V n m . Consider t m1 [n 1 , k 1  1] ‰ t m 2 [n 2 , k 2  1] ‰ ... ‰ t m m [n m , k m  1]

min{t m1 (C1 ) C1 Ž V n1 ,dim C1 min{t m 2 (C 2 ) C 2 Ž V , dim C 2 n2

k1  1} ‰ k 2  1} ‰ ... ‰

min{t mm (Cm ) Cm Ž V n m ,dim Cm

k m  1}

d min{t m1 (C1 ) C1 Ž V n1 ,dim C1 min{t m2 (C 2 ) C 2 Ž V ,dim C 2 n2

k 2 } ‰ ... ‰

min{t mm (Cm ) Cm Ž V ,dim Cm nm

k1} ‰

km}

since for each C1 ‰ C2 ‰ … ‰ Cm Ž C12 ‰ C22 ‰ … ‰ Cm2 t m1 (C12 ) ‰ t m2 (C 22 ) ‰ ... ‰ t mm (Cm2 )

d t m1 (C1 ) ‰ t m2 (C2 ) ‰ ... ‰ t mm (Cm ) t m1 [n1 , k1 ] ‰ t m2 [n 2 , k 2 ] ‰ … ‰ t mm [n m , k m ]. . Similarly t m1 (n1 , K1  1) ‰ t m 2 (n 2 , K 2  1) ‰ ... ‰ t m m (n m , K m  1) d t m1 (n 1 , K 1 ) ‰ t m 2 (n 2 , K 2 ) ‰ ... ‰ t m m (n m , K m ) .

Using these results and the fact k imi [n i , t i ] denotes the smallest dimension of a linear RD code of length ni and mi covering radius ti and K imi [n i , t i ] denotes the least cardinality of the RD codes of length ni and mi-covering radius ti the following results can be easily proved.

Result 1: For any m-set of positive integers {n1, m1, t1} ‰ {n2, m2, t2} ‰ … ‰ {nm, mm, tm}; and

115

k m1 [n1 , t1 ] ‰ k m2 [n 2 , t 2 ] ‰ ... ‰ k mm [n m , t m ]

d k m1 1[n1 , t1 ] ‰ k m2 1[n 2 , t 2 ] ‰ ... ‰ k mm 1[n m , t m ] and K m1 (n1 , t1 ) ‰ K m2 (n 2 , t 2 ) ‰ ... ‰ K mm (n m , t m )

d K m1 1 (n1 , t1 ) ‰ K m2 1 (n 2 , t 2 ) ‰ ... ‰ K mm 1 (n m , t m ) . Result 2: For any m-set of positive integers {n1, m1, t1} ‰ {n2, m2, t2} ‰ … ‰ {nm, mm, tm}. we have, k m1 [n1 , t1 ] ‰ k m2 [n 2 , t 2 ] ‰ ... ‰ k mm [n m , t m ] t k m1 [n1 , t1  1] ‰ k m2 [n 2 , t 2  1] ‰ ... ‰ k mm [n m , t m  1] and K m1 (n1 , t1 ) ‰ K m2 (n 2 , t 2 ) ‰ ... ‰ K mm (n m , t m )

t K m1 (n1 , t1  1) ‰ K m2 (n 2 , t 2  1) ‰ ... ‰ K mm (n m , t m  1) . We say a m-function f1 ‰ f2 ‰ … ‰ fm is a non decreasing mfunction in some m-variable say x1 ‰ x2 ‰ … ‰ xm if each fi happen to be a non-decreasing function in the variable xi; i = 1, 2, …, m. With this understanding we have for (m1, m2, … , mm)covering m-radius of a fixed RD m-code C1 ‰ C2 ‰ … ‰ Cm, t m1 [n1 , k1 ] ‰ t m2 [n 2 , k 2 ] ‰ ... ‰ t mm [n m , k m ] , k m1 [n1 , t1 ] ‰ k m2 [n 2 , t 2 ] ‰ ... ‰ k mm [n m , t m ] , t m1 (n1 , K1 ) ‰ t m2 (n 2 , K 2 ) ‰ ... ‰ t mm (n m , K m ) and K m1 (n1 , t1 ) ‰ K m2 (n 2 , t 2 ) ‰ ... ‰ K mm (n m , t m ) are non decreasing m-functions of (m1, m2, … , mm). The relationships between the multi covering m-radii of two RD m-codes that are built using them are described. Let Ci C1i ‰ Ci2 ‰ .... ‰ Cim for i = 1, 2 be a

116

[n11 , k11 ,d11] ‰ [n12 , k12 ,d12 ] ‰ ... ‰ [n1m , k1m ,d1m ] and [n21 , k21 ,d21 ] ‰ [n 22 , k 22 ,d 22 ] ‰ ... ‰ [n m2 , k 2m ,d 2m ] RD m-codes over F2N with n1i , n i2 , n1i  n i2 d N for i = 1, 2, …, m. PROPOSITION 3.5: Let C 1 C11 ‰ C21 ‰ ... ‰ Cm1 and C 2 C12 ‰ C22 ‰ ... ‰ Cm2 be RD m-codes described above C C 1 u C 2 ( C11 u C12 ) ‰ ( C21 u C22 ) ‰ ... ‰ ( Cm1 u Cm2 ) { ( x1 y1 ) x1 C11 , y1  C12 } ‰

{ ( x2 y2 ) x2 C21 , y2  C22 } ‰ ... ‰ { ( xm ym ) xm Cm1 , ym  Cm2 } . Then C1 u C2 is a [ n11  n21 ‰ n12  n22 ‰ ... ‰ n1m  n2m , k11  k21 ‰ k12  k22 ‰ ... ‰ k1m  k2m , min{ d 11 ,d 21 } ‰ min{ d 12 ,d 22 } ‰ ... ‰ min{ d 1m ,d 2m }] rank distance m-code over F2 N and tm1 ( C11 u C12 ) ‰ tm2 ( C21 u C22 ) ‰ ... ‰ tmm ( Cm1 u Cm2 ) d tm1 ( C11 )  tm1 ( C12 ) ‰ tm2 ( C21 )

tm2 ( C22 ) ‰ ! ‰ tmm ( Cm1 )  tmm ( Cm2 ) . Proof: Let Si Ž V n1  n 2 ; for i = 1, 2, …, m. and Si i

i

{s1i ,...,simi } j

for i = 1, 2, …, m with sij (x ji y ji ) for i = 1, 2, …, m; x ji  V n1 j

and y ji  V n 2 , 1 d i d mi; 1 d i d m. Let

S12 1 m

S

S11 {x11 ,..., x1m1 }, S12

{y11 ,..., y1m1 } ,

{x 21 ,..., x 2m2 }, S22

{y 21 ,..., y 2m2 } , …,

{x m1 ,..., x mmm } and S2m

117

{y m1 ,..., y mmm } .

Now t mi (C1i ) being the mi covering radius of C1i ; i = 1, 2, …, m, there exists c1i  C1i such that S1i Ž Bit m (C1i ) ; i = 1, 2, …, m. i

This implies as in case of RD bicodes rj (s ji  C j ) rj ((x ji y ji )  (C1j C2j )) rj (x ji  C1j y ji  C2j ) d rj (x ji  C1j )  rj (y ji  C 2j ) d t m j (C1j )  t m j (C 2j ) ;

j = 1, 2, …, m. Thus t m (C) t m1 (C11 u C12 ) ‰ t m2 (C12 u C22 ) ‰ ... ‰ t mm (C1m u C2m ) d t m1 (C11 )  t m1 (C12 ) ‰ t m2 (C12 )  t m2 (C22 ) ‰ ... ‰ t mm (C1m )  t mm (C2m ). . For any (r, r,..., r) (r a positive integer (m t3) the (r, r, …,   

m  times

r) fold repetition RD m-code C1 ‰ C2 ‰ … ‰ Cm is the m-code

C {(c1 | c1 | ...| c1 ) c1  C1} ‰ {(c2 | c 2 | ... | c 2 ) c 2  C 2 } ‰ ... ‰ {(cm | cm | ...| c m ) c m  Cm } where the m-code word C1 ‰ C2 ‰ … ‰ Cm is a concatenation of (r, r, …, r) times, this is a [rn1, k1, d1] ‰ [rn2, k2, d2] ‰ … ‰ [rnm, km, dm] rank distance m-code with ni d N and rni d N; i = 1, 2, …, m. Thus any m-code word in C1 ‰ C2 ‰ … ‰ Cm would be of the form {(c1 | c1 | ...| c1 )} ‰ {(c 2 | c 2 | ...| c 2 )} ‰ ... ‰ {(c m | c m | ...| c m )} such that xi  Ci for i = 1, 2, …, m. We can also define (r1, r2, …, rm) fold repetition m-code (ri z rj if i z j; 1 d i, j d m). DEFINITION 3.37: Let C1 ‰ C2 ‰ … ‰ Cm be a [n1, k1, d1] ‰ [n2, k2, d2] ‰ … ‰ [nm, km, dm] RD m-code.

118

^

`

( ci | ci | ...| ci ) ci  Ci  

Let ci =  be a ri-fold repetition RD-code ri  times

Ci, i = 1, 2, …, m. Then C ‰ C2 ‰ … ‰ Cm is defined as the (r1, r2, …, rm)-fold repetition m-code each rini  N for i = 1, 2, …, m. 1

We prove the following interesting theorem. THEOREM 3.13: For an (r, r, …, r) fold repetition m-code C1 ‰ C2 ‰ … ‰ Cm tm1 ( C1 ) ‰ tm2 ( C2 ) ‰ ! ‰ tmm ( Cm )

tm1 ( C 1 ) ‰ tm2 ( C 2 ) ‰ ! ‰ tmm ( C m ) . Proof: Let Si

{vi1 , vi2 ,..., vimi } Ž V ni for i = 1, 2, …, m;

such that cov(Ci ,Si ) t mi (Ci ) ; i = 1, 2, …, m. Let

vci1 (vi1 | vi1 | ... | vi1 ) vci2 (vi2 | vi2 | ...| vi2 ) and so on vcim1 Let Si

(vim1 | vim1 | ... | vim1 ) ; 1 d i d m.

{vci1 , vci2 ,..., vcimi } be the set of mi-vectors of length mi

for, i = 1, 2, …, m. An r fold repetition of any RD code word retains the same rank weight. Hence (Ci ,Sci ) t mi (Ci ) true for i = 1, 2, …, m. Since t mi (Ci ) cov(Ci ,Sci ) it follows that

t mi (Ci ) t t mi (Ci ) for i = 1, 2, …, m; i.e.,

t m1 (C1 ) ‰ t m2 (C2 ) ‰ ... ‰ t mm (Cm ) t t m1 (C1 ) ‰ t m2 (C 2 ) ‰ ... ‰ t mm (Cm )

119

-----

I

Conversely let Si

{vi1 , vi2 ,..., vimi } be a set of mi vectors i = 1,

2, …, m of length rni with vij

(vcij | ...| vcij ) ; j = 1, 2, …, mi, i =

1, 2, …, m. vci  V n i , 1 d i d mi. Then there exists ci  Ci such that d R i (ci , vci ) d t mi (Ci ) ; i = 1, 2, …, mi; 1 d i d m. This implies d R i ((ci | ci | ... | ci ), vij ) d t mi (Ci ) for every i(1 d i d m). Thus t mi (Ci ) d t mi (Ci ) , i = 1, 2, …, m; i.e.,

t m1 (C1 ) ‰ t m2 (C2 ) ‰ ... ‰ t mm (Cm ) d t m1 (C1 ) ‰ t m2 (C 2 ) ‰ ... ‰ t mm (C m )

------- II

From I and II

t m1 (C1 ) ‰ t m2 (C2 ) ‰ ... ‰ t mm (Cm ) t m1 (C1 ) ‰ t m2 (C 2 ) ‰ ... ‰ t mm (Cm ) . Now we proceed on to analyse the notion of multi covering mbounds for RD m-codes. The (m1, m2, … , mm) covering mradius t m1 (C1 ) ‰ t m2 (C 2 ) ‰ ... ‰ t mm (Cm ) of a RD m-code C = C1 ‰ C2 ‰ … ‰ Cm is a non-decreasing m-function of m1 ‰ m2 ‰ … ‰ mm. Thus a lower m-bound for t m1 (C1 ) ‰ t m2 (C2 ) ‰ ... ‰ t mm (Cm ) implies a m-bound for t m1 1 (C1 ) ‰ t m2 1 (C 2 ) ‰ ... ‰ t mm 1 (Cm ) . First m-bound exhibits that for m1 ‰ m2 ‰ … ‰ mm t 2 ‰ 2 ‰ … ‰ 2 the situation of (m1, m2, …, mm) covering m-radii is quite different for ordinary covering radii. PREPOSITION 3.6: If m1 ‰ m2 ‰ … ‰ mm ! 2 ‰ 2 ‰ … ‰ 2 then the (m1, m2, …, mm) covering m-radii of a RD m-code C = C1 ‰ C2 ‰ … ‰ Cm of m-length (n1, n2, …, nm) is at least ª nm º ª n1 º ª n2 º « 2 » ‰ « 2 » ‰ ... ‰ « 2 » . « » « » « »

120

Proof: Let C = C1 ‰ C2 ‰ … ‰ Cm be a RD m-code of m-length (n1, n2, … , nm) over GF(2N). Let m1 ‰ m2 ‰ … ‰ mm 2 ‰ 2 ‰ … ‰ 2. Let t1, t2, …, tm be the 2-covering m-radii of the RD m-code C = C1 ‰ C2 ‰ … ‰ Cm. Let x = x1 ‰ x2 ‰ … ‰ xm  V n1 ‰ V n 2 ‰ ... ‰ V n m . Choose y = y1 ‰ y2 ‰ … ‰ ym  V n1 ‰ V n 2 ‰ ... ‰ V n m such that all the (n1, n2, …, nm) coordinate of x  y (x1  y1 ) ‰ (x 2  y 2 ) ‰ ... ‰ (x m  y m ) are linearly independent that is d R (x, y) d R (x1 ‰ x 2 ‰ ... ‰ x m , y1 ‰ y 2 ‰ ... ‰ y m ) d R1 (x1 , y1 ) ‰ d R 2 (x 2 , y 2 ) ‰ ... ‰ d R m (x m , y m )

(R = R1 ‰ R2 ‰ … ‰ Rm and d R

d R1 ‰ d R 2 ‰ ... ‰ d R m ) = n1 ‰

n2 ‰ … ‰ nm. Then for any c = c1 ‰ c2 ‰ … ‰ cm  C1 ‰ C2 ‰ … ‰ Cm. d R (x  c)  d R (c  y) d R1 (x1 ,c1 )  d R1 (c1 , y1 ) ‰ d R 2 (x 2 ,c2 )  d R 2 (c2 , y 2 ) ‰ ... ‰ d R m (x m ,c m )  d R m (c m , y m ) t d R1 (x1 , y1 ) ‰ d R 2 (x 2 , y 2 ) ‰ ... ‰ d R m (x m , y m )

= n1 ‰ n2 ‰ … ‰ nm; this implies that one of d R1 (x1 ,c1 ) ‰ d R 2 (x 2 ,c2 ) ‰ ... ‰ d R m (x m ,c m ) and d R1 (c1 , y1 ) ‰ d R 2 (c2 , y 2 ) ‰ ... ‰ d R m (c m , y m ) is at least n1 n 2 n ‰ ‰ ... ‰ m . 2 2 2

(That is one of d R i (x i ,ci ) and d R i (ci , yi ) is atleast

ni ; i = 1, 2, 2

…, m) and hence ªn º ªn º ªn º t = t1 ‰ t2 ‰ … ‰ tm t « 1 » ‰ « 2 » ‰ ... ‰ « m » . «2» «2» « 2 »

121

Since t is a non-decreasing m-function of m1 ‰ m2 ‰ … ‰ mm it follows that t m (C)

t m1 (C1 ) ‰ t m2 (C2 ) ‰ ... ‰ t mm (C m ) ªn º ªn º ªn º t « 1 » ‰ « 2 » ‰ ... ‰ « m » «2» «2» « 2 »

for m1 ‰ m2 ‰ … ‰ mm t 2 ‰ 2 ‰ … ‰ 2. m-bounds of the multi covering m-radius of V n1 ‰ V n 2 ‰ ... ‰ V n m can be used to obtain m-bounds on the multi covering m-radii of arbitrary mcodes. Thus a relationship between (m1, m2, …, mm) covering m-radii of an RD m-code and that of its ambient m-space V n1 ‰ V n 2 ‰ ... ‰ V n m is established. THEOREM 3.14: Let C = C1 ‰ C2 ‰ … ‰ Cm be RD m-code of m-length n1 ‰ n2 ‰ … ‰ nm over F2 N ‰ F2 N ‰ ! ‰ F2 N . Then

for any positive m-integer tuple (m1, m2, …, mm) tm1 1 ( C1 ) ‰ tm2 2 ( C2 ) ‰ ... ‰ tmmm ( Cm ) d t11 ( C1 )  tm1 1 (V n1 ) ‰ t12 ( C2 )  tm2 2 (V n2 ) ‰ ... ‰

t1m ( Cm )  tmmm (V nm ) . Proof: Let S = S1 ‰ S2 ‰ … ‰ Sm Ž V n1 ‰ V n 2 ‰ ! ‰ V n m (i.e., Si Ž V ni ; i = 1, 2, …, m) with |S| = |S1| ‰ |S2| ‰ … ‰ |Sm| = m1 ‰ m2 ‰ … ‰ mm. Then there exists u = u1 ‰ u2 ‰ … ‰ um  V n1 ‰ V n 2 ‰ ... ‰ V n m such that cov(u,S)

cov(u 1 ,S1 ) ‰ cov(u 2 ,S 2 ) ‰ ... ‰ cov(u m ,S m )

d t (V n1 ) ‰ t 2m2 (V n 2 ) ‰ ... ‰ t mmm (V n m ) . 1 m1

Also there is a c = c1 ‰ c2 ‰ … ‰ cm  C1 ‰ C2 ‰ … ‰ Cm such that d R (c, u) d R1 (c1 , u1 ) ‰ d R 2 (c 2 , u 2 ) ‰ ... ‰ d R m (c m , u m ) d t11 (C1 ) ‰ t12 (C2 ) ‰ ... ‰ t1m (C m ) . Now, 122

cov(c,S) cov(c1 ,S1 ) ‰ cov(c2 ,S2 ) ‰ ... ‰ cov(c m ,Sm ) max{d R1 (c1 , y1 ) y1  S1} ‰ max{d R 2 (c 2 , y 2 ) y 2  S2 } ‰ ... ‰ max{d R m (c m , y m ) y m  Sm } d max{d R1 (c1 , u1 )  d R1 (u1 , y1 ) y1  S1} ‰ max{d R 2 (c 2 , u 2 )  d R 2 (u 2 , y 2 ) y 2  S2 } ‰ ... ‰ max{d R m (c m , u m )  d R m (u m , y m ) y m  Sm } d R1 (c1 , u1 )  cov(u1 ,S1 ) ‰ d R 2 (c 2 , u 2 )  cov(u 2 ,S2 ) ‰ ... ‰ d R m (c m , u m )  cov(u m ,Sm ) d t11 (C1 )  t1m1 (V n1 ) ‰ t12 (C2 )  t 2m2 (V n 2 ) ‰ ... ‰

t1m (Cm )  t mmm (V n m ) . Thus for every S = S1 ‰ S2 ‰ … ‰ Sm Ž V n1 ‰ V n 2 ‰ ... ‰ V n m with |S| = |S1| ‰ |S2| ‰ … ‰ |Sm| = m = m1 ‰ m2 ‰ … ‰ mm one can find c = c1 ‰ c2 ‰ … ‰ cm  C1 ‰ C2 ‰ … ‰ Cm such that cov(c,S)

cov(c1 ,S1 ) ‰ cov(c 2 ,S 2 ) ‰ ... ‰ cov(c m ,Sm )

d t (C1 )  t1m1 (V n1 ) ‰ t12 (C2 )  t 2m2 (V n 2 ) ‰ ... ‰ 1 1

t1m (C m )  t mmm (V n m ) . Since

cov(c,S) cov(c1 ,S1 ) ‰ cov(c2 ,S2 ) ‰ ... ‰ cov(c m ,Sm ) min{cov(a1 ,S1 ) a1  C1} ‰ min{cov(a 2 ,S2 ) a 2  C 2 } ‰ ... ‰ min{cov(a m ,Sm ) a m  C m }

123

d {t11 (C1 )  t1m1 (V n1 )} ‰ {t12 (C 2 )  t 2m2 (V n 2 )} ‰ ... ‰

{t1m (Cm )  t mmm (V n m )} for all S = S1 ‰ S2 ‰ … ‰ Sm Ž V n1 ‰ V n 2 ‰ ... ‰ V n m with |S| = |S1| ‰ |S2| ‰ … ‰ |Sm| = m1 ‰ m2 ‰ … ‰ mm, it follows that t1m1 (C1 ) ‰ t 2m2 (C2 ) ‰ ... ‰ t mmm (Cm ) max{cov(C1 ,S1 ) S1 Ž V n1 , S1 max{cov(C 2 ,S2 ) S2 Ž V , S2 n2

m1} ‰ m 2 } ‰ ... ‰

max{cov(Cm ,Sm ) Sm Ž V n m , Sm

mm }

d t11 (C1 )  t1m1 (V n1 ) ‰ t12 (C2 )  t 2m2 (V n 2 ) ‰ ... ‰ t1m (C m )  t mmm (V n m ) . PROPOSITION 3.7: For any m-tuple of integer (n1, n2, …, nm); n1 ‰ n2 ‰ … ‰ nm t 2 ‰ 2 ‰ … ‰ 2; t21 (V n1 ) ‰ t22 (V n2 ) ‰ ! ‰ t2m (V nm ) d n1  1 ‰ n2  1 ‰ ! ‰ nm  1

where V ni

F2nNi ; i = 1, 2, …, ni; ni d N, i = 1, 2, …, m.

Proof: Let x i (x1i , x i2 ,..., x in i ) and yi

(y1i , yi2 ,..., yini )  V ni ;

i = 1, 2, …, m. Let u = u1 ‰ u2 ‰ … ‰ um  V n1 ‰ V n 2 ‰ ... ‰ V n m where u i (x1i , u i2 , u 3i ,..., u ini 1 , yini ) ; i = 1, 2, …, m. Thus u = u1 ‰ u2 ‰ … ‰ um m-covers x1 ‰ x2 ‰ … ‰ xm and y1 ‰ y2 ‰ … ‰ ym  V n1 ‰ V n 2 ‰ ... ‰ V n m within a m-radius n1–1 ‰ n2–1 ‰ … ‰ nm–1 as d R i (u i , x i ) d n i  1 ; i = 1, 2, …, m. Thus for any pair of m-vectors x1 ‰ x2 ‰ … ‰ xm, y1 ‰ y2 ‰ … ‰ ym in V n1 ‰ V n 2 ‰ ... ‰ V n m there always exists a m-vector namely u = u1 ‰ u2 ‰ … ‰ um which m-covers x1 ‰ x2 ‰ … ‰ xm and y1 ‰ y2 ‰ … ‰ ym within a m-radius n1–1 ‰ n2–1 ‰ … ‰ nm–1.

124

Hence t12 (V n1 ) ‰ t 22 (V n 2 ) ‰ ... ‰ t 2m (V n m ) d n1  1 ‰ n 2  1 ‰ ... ‰ n m  1 . Now we proceed on to describe the notion of generalized sphere m-covering m-bounds for RD – m-codes. A natural question is for a given t1 ‰ t2 ‰ … ‰ tm, m1 ‰ m2 ‰ … ‰ mm and n1 ‰ n2 ‰ … ‰ nm what is the smallest RD m-code whose m1 ‰ m2 ‰ … ‰ mm, m-covering m-radius is atmost t1 ‰ t2 ‰ … ‰ tm. As it turns out even for m1 ‰ m2 ‰ … ‰ mm t 2 ‰ 2 ‰ … ‰ 2, it is necessary that t1 ‰ t2 ‰ … ‰ tm be atleast n1 n 2 n ‰ ‰ ... ‰ m . 2 2 2 Infact the minimal t1 ‰ t2 ‰ … ‰ tm for which such a m-code exists is the (m1, m2, …, mm), m-covering m-radius of C1 ‰ C2 ‰ … ‰ Cm = F2nN1 ‰ F2nN2 ‰ ... ‰ F2nNm . Various external values associated with this notion are n1 n2 nm 1 2 m t m1 (V ) ‰ t m2 (V ) ‰ ... ‰ t mm (V ) , the smallest (m1, m2, …, mm) covering m-radius among m-length n1 ‰ n2 ‰ … ‰ nm RDm-codes t1m1 (n1 , K1 ) ‰ t 2m2 (n 2 , K 2 ) ‰ ... ‰ t mmm (n m , K m ) ; the smallest (m1, m2, …, mm) covering m-radius among all (n1, K1) ‰ (n2, K2) ‰ … ‰ (nm, Km) RD-m-codes. K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) ‰ ... ‰ K mmm (n m , t m ) is the smallest m-cardinality of a m-length n1 ‰ n2 ‰ … ‰ nm. RD-m-code with m1 ‰ m2 ‰ … ‰ mm covering m-radius t1 ‰ t2 ‰ … ‰ tm and so on. It is the latter quantity that is studied in the book for deriving new lower m-bounds. From the earlier results K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) ‰ ... ‰ K mmm (n m , t m ) is undefined if

125

t1 ‰ t2 ‰ … ‰ tm 

n1 n 2 n ‰ ‰ ... ‰ m . 2 2 2

When this is the case it is accepted to say K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) ‰ ... ‰ K mmm (n m , t m ) = f ‰ f ‰ … ‰ f. There are other circumstances when K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) ‰ ... ‰ K mmm (n m , t m ) is undefined. For instance K1 Nn1 (n1 , n1  1) ‰ K 2Nn2 (n 2 , n 2  1) ‰ ... ‰ K mNnm (n m , n m  1) 2

2

2

= f ‰ f ‰ … ‰ f 1 m1 ! V(n1 , t ), m 2 ! V(n 2 , t 2 ),..., m m ! V(n m , t m ) ; since in this case no m-ball of m-radius t1 ‰ t2 ‰ … ‰ tm mcovers any m-set of m1 ‰ m2 ‰ … ‰ mm distinct m-vectors. More generally one has the fundamental issue of whether K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) ‰ ... ‰ K mmm (n m , t m ) is m-finite for a given n1 , m1 , t1 , n 2 , m 2 , t 2 ,..., n m , m m , t m . This is the case if and only if t1m1 (V n1 ) d t1 , t 2m2 (V n 2 ) d t 2 ,..., t mmm (V n m ) d t m since t1m1 (V n1 ) ‰ t 2m2 (V n 2 ) ‰ ... ‰ t mmm (V n m ) lower m-bounds the (m1, m2, …, mm) covering m-radius of all other m-codes of m-dimension n1 ‰ n2 ‰ … ‰ nm when t1 ‰ t2 ‰ … ‰ tm = n1 ‰ n2 ‰ … ‰ nm every m-code word m-covers every m-vector, so a m-code of size 1 ‰ 1 ‰ … ‰ 1 will (m1, m2, …, mm) m-cover V n1 ‰ V n 2 ‰ ... ‰ V n m for every m1 ‰ m2 ‰ … ‰ mm. Thus K1m1 (n1 , n1 ) ‰ K 2m2 (n 2 , n 2 ) ‰ ... ‰ K mmm (n m , n m ) =1‰1‰…‰1

126

for every m1 ‰ m2 ‰ … ‰ mm. If t1 ‰ t2 ‰ … ‰ tm = n1 – 1 ‰ n2 – 1 ‰ … ‰ nm – 1 what happens to K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) ‰ ... ‰ K mmm (n m , t m ) ? When m1 = m2 = … = mm, K12 (n1 , n1  1) ‰ K 22 (n 2 , n 2  1) ‰ ... ‰ K m2 (n m , n m  1) d 1  L n1 (n1 ) ‰ 1  L n 2 (n 2 ) ‰ ... ‰ 1  L n m (n m ) .

For 0 ‰ 0 ‰ ... ‰ 0 = (0, 0, …, 0) ‰ (0, 0, …, 0) ‰ (0, 0, …, 0) will m-cover m-norm less than or equal to n1 – 1 ‰ n2 – 1 ‰ … ‰ nm – 1 within m-radius n1 – 1 ‰ n2 – 1 ‰ … ‰ nm – 1. That is 0 ‰ 0 ‰ ... ‰ 0 = (0, 0, …, 0) ‰ (0, 0, …, 0) ‰ (0, 0, …, 0) will m-cover all m-norm n1 – 1 ‰ n2 – 1 ‰ … ‰ nm – 1 mvectors within the m-radius n1 – 1 ‰ n2 – 1 ‰ … ‰ nm – 1. Hence remaining m-vectors are m-rank n1 ‰ n2 ‰ … ‰ nm mvectors. Thus 0 ‰ 0 ‰ ... ‰ 0 = (0, 0, …, 0) ‰ (0, 0, …, 0) ‰ (0, 0, …, 0) and these m-rank (n1 ‰ n2 ‰ … ‰ nm) m-vector can m-cover the ambient m-space within the m-radius n1 – 1 ‰ n2 – 1 ‰ … ‰ nm – 1. Therefore, K12 (n1 , n1  1) ‰ K 22 (n 2 , n 2  1) ‰ ... ‰ K m2 (n m , n m  1) d 1  L n1 (n1 ) ‰ 1  L n 2 (n 2 ) ‰ ... ‰ 1  L n m (n m ) . PROPOSITION 3.8: For any RD m-code of m-length n1 ‰ n2 ‰ … ‰ nm over F2 N ‰ F2 N ‰ ! ‰ F2 N ,

K 21 ( n1 ,n1  1 ) ‰ K 22 ( n2 ,n2  1 ) ‰ ! ‰ K 2m ( nm ,nm  1 ) d m1 Ln1 ( n1 )  1 ‰ m2 Ln2 ( n2 )  1 ‰ ! ‰ mm Lnm ( nm )  1 provided m1 ‰ m2 ‰ … ‰ mm is such that m1 Ln1 ( n1 )  1 ‰ m2 Ln2 ( n2 )  1 ‰ ... ‰ mm Lnm ( nm )  1 d V n1 ‰ V n2 ‰ ... ‰ V nm .

Proof: Consider a RD m-code C = C1 ‰ C2 ‰ … ‰ Cm such that

127

C

C1 ‰ C 2 ‰ ... ‰ C m

= m1L n1 (n1 )  1 ‰ m 2 L n 2 (n 2 )  1 ‰ ! ‰ m m L n m (n m )  1 . Each m-vector in V n1 ‰ V n 2 ‰ ... ‰ V n m has L n1 (n1 ) ‰ L n 2 (n 2 ) ‰ ... ‰ L n m (n m ) rank complements, that is from each v1 ‰ v 2 ‰ ... ‰ v m  V n1 ‰ V n 2 ‰ ... ‰ V n m there are L n1 (n1 ) ‰ L n 2 (n 2 ) ‰ ... ‰ L n m (n m )

m-vector

m-vectors at rank m-distance n1 ‰ n2 ‰ … ‰ nm. This means for any set S1 ‰ S2 ‰ ... ‰ Sm Ž V n1 ‰ V n 2 ‰ ... ‰ V n m of (m1, m2, …, mm) m-vectors there always exists a c1 ‰ c2 ‰ … ‰ cm  C1 ‰ C2 ‰ … ‰ Cm which m-covers S1 ‰ S2 ‰ … ‰ Sm; m-rank distance n1 – 1 ‰ n2 – 1 ‰ … ‰ nm – 1. Thus

cov(u1 ,S1 ) ‰ cov(u 2 ,S2 ) ‰ ... ‰ cov(u m ,Sm ) d n1 – 1 ‰ n2 – 1 ‰ … ‰ nm – 1 which implies cov(u1 ,S1 ) ‰ cov(u 2 ,S2 ) d n1  1 ‰ n 2  1 . Hence K1m1 (n1 , n1  1) ‰ K 2m2 (n 2 , n 2  1) ‰ ... ‰ K mmm (n m , n m  1) d m1L n1 (n1 )  1 ‰ m 2 L n 2 (n 2 )  1 ‰ ... ‰ m m L n m (n m )  1 .

By bounding the number of (m1, m2, …, mm) m-sets that can be covered by a given m-code word, one obtains a straight forward generalization of the classical sphere m-bound. THEOREM (Generalized sphere bound for RD-m-codes): For any ( n1 ,K1 ) ‰ ( n2 ,K 2 ) ‰ ! ‰ ( nm ,K m ) RD m-code C = C1 ‰ C2 ‰ … ‰ Cm, § V ( n1 ,tm1 ( C1 )) · § V ( n2 ,tm2 ( C2 )) · 2 K1 ¨ ¸‰K ¨ ¸ m1 m2 © ¹ © ¹ V ( n ,t ( C )) § · m mm m ‰! ‰ K m ¨¨ ¸¸ mm © ¹

128

§ 2 Nn1 t¨ ¨ m © 1

· § 2 Nn2 ¸¸ ‰ ¨¨ ¹ © m2

§ 2 Nnm · ‰ ‰ ! ¨¨ ¸¸ ¹ © mm

· ¸¸ . ¹

Hence for any ni ,ti ,mi triple i = 1, 2, …, m K m1 1 ( n1 ,t1 ) ‰ K m2 2 ( n2 ,t2 ) ‰ ... ‰ K mmm ( nm ,tm ) § 2 Nnm · § 2 Nn1 · § 2 Nn2 · ¨¨ ¸¸ ¨¨ ¸¸ ¨¨ m ¸¸ m1 ¹ m2 ¹ © © t ‰ ‰ ... ‰ © m ¹ § V ( n1 ,t1 ) · § V ( n2 ,t2 ) · § V ( nm ,tm ) · ¨ ¸ ¨ ¸ ¨ ¸ © m1 ¹ © m2 ¹ © mm ¹ where V ( n1 ,t1 ) ‰ V ( n2 ,t2 ) ‰ ... ‰ V ( nm ,tm ) t1

t2

tm

i1 0

i2 0

im 0

¦ L1i1 ( n1 ) ‰ ¦ L2i2 ( n2 ) ‰ ... ‰ ¦ Lmim ( nm )

number of m-vectors in a sphere m-radius t1 ‰ t2 ‰ … ‰ tm and L1i1 ( n1 ) ‰ L2i2 ( n2 ) ‰ ! ‰ Lmim ( nm ) is the number of m-vectors in V n1 ‰ V n2 ‰ ! ‰ V nm whose rank m-norm is i1 ‰ i2 ‰ … ‰ im. Proof: Each set of (m1, m2, …, mm) m-vectors in V n1 ‰ V n 2 ‰ ... ‰ V n m must occur in a sphere of m-radius t m1 (C1 ) ‰ t m2 (C 2 ) ‰ ... ‰ t mm (Cm ) around at least one code mword. Total number of such m-sets is V n1 ‰ V n 2 ‰ ... ‰ V n m choose m1 ‰ m2 ‰ … ‰ mm where V n1 ‰ V n 2 ‰ ... ‰ V n m 2 Nn1 ‰ 2 Nn 2 ‰ ... ‰ 2 Nn m . The number of m-sets of (m1, m2, …, mm) m-vectors in a neighborhood of m-radius t m1 (C1 ) ‰ t m2 (C2 ) ‰ ... ‰ t mm (Cm ) is V(n1 , t m1 (C1 )) ‰ V(n 2 , t m2 (C 2 )) ‰ ... ‰ V(n m , t mm (C m ))

129

choose m1 ‰ m2 ‰ … ‰ mm. There are K-code m-words. Hence § V(n1 , t m1 (C1 )) · § V(n 2 , t m2 (C2 )) · 2 K1 ¨ ¸‰K ¨ ¸ m1 m2 © ¹ © ¹ § V(n m , t mm (C m )) · ‰! ‰ K m ¨ ¸ mm © ¹

§ 2 Nn1 · § 2 Nn2 t¨ ‰ ¨ m ¸¸ ¨¨ m © 1¹ © 2

· § 2 Nnm ¸¸ ‰ ... ‰ ¨¨ ¹ © mm

· ¸¸ . ¹

Thus for any n1 ‰ n2 ‰ … ‰ nm, t1 ‰ t2 ‰ … ‰ tm and m1 ‰ m2 ‰ … ‰ mm K1m1 (n1 , t1 ) ‰ K 2m2 (n 2 , t 2 ) ‰ ... ‰ K mmm (n m , t m )

§ 2 Nn1 · § 2 Nn2 · § 2 Nnm · ¨¨ ¸ ¨¨ ¸ ¨¨ ¸ m1 ¸¹ m 2 ¸¹ m m ¸¹ © © © . t ‰ ‰ ... ‰ § V  n1 , t 1 · § V  n 2 , t 2 · § V nm , tm · ¨¨ ¸¸ ¨¨ ¸¸ ¨¨ ¸¸ © m1 ¹ © m 2 ¹ © mm ¹ COROLLARY 3.2: If § 2 Nnm · § 2 Nn1 · § 2 Nn2 · ‰ ‰ ‰ ... ¨¨ ¸¸ ¨¨ ¸¸ ¨¨ ¸¸ © m1 ¹ © m2 ¹ © mm ¹ N nm N § V ( n1 ,t1 ) · N n2 § V ( n2 ,t2 ) · ! 2 n1 ¨ ¸‰2 ¨ ¸ ‰ ... ‰ 2 © m1 ¹ © m2 ¹

§ V ( nm ,tm ) · ¨ ¸ © mm ¹

then K m1 1 ( n1 ,t1 ) ‰ K m2 2 ( n2 ,t2 ) ‰ ... ‰ K mmm ( nm ,tm ) f ‰ f ‰ ... ‰ f .

130

Chapter Four

APPLICATIONS OF RANK DISTANCE m-CODES

In this chapter we proceed onto give some applications of Rank Distance bicodes and those of the new classes of rank distance bicodes and their generalizations. Rank distance m-codes can be used in multi disk storage systems by constructing or building m- Redundant Array of Inexpensive Disks (m-RAID). These linear MRD m-codes can also be used in m-public key m-cryptosystems. Circulant rank m-codes can be used in multi communication channels or m-channels having very high m-error m-probability for m-error correction. The AMRD m-codes is useful for m-error correction in data multi(m-) storage systems. These m-codes will equally be as good as the MRD m-codes and are better than the corresponding m-codes in the Hamming metric. In data multi storage systems these MRD and AMRD mcodes can be used simultaneously in criss cross error corrections

131

by suitably programming; appropriately the functioning of the implementation. Using these m-codes one can save time, space and economy. With computerization in every walk of life these mcodes can perform simultaneously bulk m-error correction in bulk data transmission if appropriately programmed. Interested researcher can develop m-algorithms (algorithms that can work in m-channels simultaneously) of these rank distance m-codes.

132

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INDEX

(m, n) dimensional vector bispace, 28 (m1, m2) bicovering biradius, 69 (m1, m2) covering biradii, 69 (m1, m2, m3, …, mm) covering m-radii, 120-2 (r, r) repetition RD bicode, 67 (r1, r2) fold repetition bicode, 67 A Almost Maximum Rank Distance (AMRD) codes, 15-6 AMRD bicode, 40-1 AMRD m-codes, 94-5 Associated matrix, 8-9 Asymmetric error bimodel, 59-60 Asymmetric error m-models, 110-1 B Bibasis, 28-29 Bicirculant transpose, 32

144

Bicyclic, 39 Bidegree, 34-5 Bidimension of the bispace, 28 Bidistance between biwords, 38 Bidistance bifunction, 38-9 Bidivisible bicode, 49 Bimatrix, 26 Bimetric induced by the birank binorm, 27-8 Binorm, 27 Bipolynomial representation, 31-2 Biprobability, 58 Birank bidistance, 26-9 Birank bimetric, 28 Birank bispace, 28-9 Birank, 28 Birows, 28-9 Bispectrum of a MRD bicode, 50-1 Bivector bispace, 25-28 C Circulant bimatrix, 33 Circulant birank bicode, 38-9 Circulant bitranspose, 32 Circulant code of length N, 12-5 Circulant m-matrix, 86-8 Circulant matrix, 12-3 Circulant m-rank m-code, 93 Circulant m-transpose, 85-6 Circulant rank bicodes, 38 Constant birank, 45 Constant m-rank code of m-length, 100-1 Constant rank bicode, 45 Constant rank code of length n, 19 Constant rank code, 14-9 Constant weight codes, 18-9 Covering biradius, 48 Cyclic circulant code, 93-4

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F Frobenius bimatrix, 31 Frobenius m-matrix, 84 Fuzzy RD bicode, 60-1 Fuzzy RD code, 54-5 Fuzzy RD m-code, 109 Fuzzy RD tricode, 109 G Generating bivector, 31 Generator bimatrix, 28-9, 33 Generator matrix of a linear RD code, 9-11 Generator matrix of a MRD code, 11 Generator m-matrix, 81 H Hamming metric, 10 L Linear bisubspace, 28 M Maximum rank bidistance, 29-30 Maximum Rank Distance (MRD) code, 11 Maximum Rank Distance (MRD)-m-code, 83 Maximum Rank Distance bicode (MRD bicode), 30-31 Maximum rank distance, 11 m-basis, 77-78, 82-3 m-bounds, 125-6 m-cardinality, 101-2 m-column, 78 m-degree of m-polynomial, 86-9 m-distance m-function, 92

146

m-divisible m-code, 104-5 Membership bifunction, 57-8 Membership m-function, 109-10 m-error, 82-3 m-function, 100-1 m-GCD, 88 Minimum rank m-distance, 83 m-invertible, 91-3 Mixed circulant m-code, 93-4 Mixed quasi circulant m-code of type II, 93 Mixed quasi circulant rank m-code, 93-4 m-linear combination, 91, 102 m-matrix, 79-80 m-polynomial, 85-6 m-probability, 110 m-rank m-distance, 80 m-rank m-metric, 80 m-rank m-space, 79-80 m-rank, 79-80, 101 MRD m-code, 93-4 Multi covering m-radii, 122 m-word generated, 91-3 N N-m-tuple representation, 78 Non bidivisible bicode, 50 Non divisible AMRD m-code, 105-9 Non divisible MRD m-code, 106-7 P Parity check bimatrix, 30 Parity check matrix, 10-11 Parity check m-matrix, 81 Pseudo false linear bispace, 25-6 Pseudo false m-space, 77-8

147

Q Quasi (r1, r2) MRD m-code, 92-3 Quasi circulant m-code of type I, 93 Quasi divisible AMRD m-code, 106-7 Quasi divisible circulant rank m-code, 106-7 Quasi divisible MRD m-code, 105-7 Quasi MRD m-code, 92 R Rank Distance (RD) bicodes, 26-8 Rank Distance (RD) codes, 7-9 Rank distance m-code of m-length, 80-1 Rank distance m-code, 77-8 Rank distance space, 9-11 Rank distance tricode, 77-8 Rank distance, 9 Rank metric, 9-10 Rank norm, 9-10 Rank of a matrix, 8-10 RD bicode of bilength, 28 Repetition bicode, 67 ri- fold repetition RD code, 118-9 S Semi circulant rank bicode of type I, 39 Semi circulant rank bicode of type II, 39 Semi cyclic circulant rank bicode of type I, 40 Semi cyclic circulant rank bicode, 40 Semi divisible AMRD bicode, 50 Semi divisible circulant bicode, 50 Semi divisible MRD bicode, 50 Semi divisible RD bicode, 50 Semi MRD bicode, 39 Single error correcting AMRD m-code, 96-8 Single Style bound theorem, 10-11 Singleton Style bound, 30

148

Solution bispace, 29 Solution m- space, 80-2 Symmetric error bimodel, 59-60 Symmetric error m-model, 110-1

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ABOUT THE AUTHORS Dr.W.B.Vasantha Kandasamy is an Associate Professor in the Department of Mathematics, Indian Institute of Technology Madras, Chennai. In the past decade she has guided 13 Ph.D. scholars in the different fields of non-associative algebras, algebraic coding theory, transportation theory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. She has to her credit 646 research papers. She has guided over 68 M.Sc. and M.Tech. projects. She has worked in collaboration projects with the Indian Space Research Organization and with the Tamil Nadu State AIDS Control Society. She is presently working on a research project funded by the Government of India’s Department of Atomic Energy. This is her 46th book. On India's 60th Independence Day, Dr.Vasantha was conferred the Kalpana Chawla Award for Courage and Daring Enterprise by the State Government of Tamil Nadu in recognition of her sustained fight for social justice in the Indian Institute of Technology (IIT) Madras and for her contribution to mathematics. The award, instituted in the memory of Indian-American astronaut Kalpana Chawla who died aboard Space Shuttle Columbia, carried a cash prize of five lakh rupees (the highest prizemoney for any Indian award) and a gold medal. She can be contacted at [email protected] Web Site: http://mat.iitm.ac.in/home/wbv/public_html/

Dr. Florentin Smarandache is a Professor of Mathematics at the University of New Mexico in USA. He published over 75 books and 150 articles and notes in mathematics, physics, philosophy, psychology, rebus, literature. In mathematics his research is in number theory, non-Euclidean geometry, synthetic geometry, algebraic structures, statistics, neutrosophic logic and set (generalizations of fuzzy logic and set respectively), neutrosophic probability (generalization of classical and imprecise probability). Also, small contributions to nuclear and particle physics, information fusion, neutrosophy (a generalization of dialectics), law of sensations and stimuli, etc. He can be contacted at [email protected]

Dr. N. Suresh Babu is a faculty in the College of Engineering, Trivandrum. He has obtained his doctorate degree in coding/ communication theory from department of mathematics, Indian Institute of Technology, Madras. He has worked as a post doctoral research fellow, under Prof. Zimmermann, K. H., a renowned coding theorist, working in the University of Bayreuth, Germany.

Dr. R. S. Selvaraj is presently working as a faculty in NIIT Warangal. He has obtained his doctor of philosophy in coding/ communication theory from department of mathematics, Indian Institute of Technology, Madras.

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