Rank Distance Bicodes and their Generalization
W. B. Vasantha Kandasamy, Florentin Smarandache, N.Suresh Babu, R.S.Selvaraj
W. B. Vasantha Kandasamy Florentin Smarandache N. Suresh Babu R.S. Selvaraj
RANK DISTANCE BICODES AND THEIR GENERALIZATION RANK DISTANCE BICODES AND THEIR GENERALIZATION
W. B. Vasantha Kandasamy Florentin Smarandache N. Suresh Babu R.S. Selvaraj CONTENTS
Preface Chapter One
BASIC PROPERTIES OF RANK DISTANCE CODES Chapter Two
RANK DISTANCE BICODES AND THEIR PROPERTIES Chapter Three
RANK DISTANCE m-CODES Chapter Four
APPLICATIONS OF RANK DISTANCE m-CODES
FURTHER READING
INDEX
ABOUT THE AUTHORS 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. Our thanks are due to Dr. K. Kandasamy for proof-reading 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 Chapter One B ASIC P ROPERTIES OF R ANK D ISTANCE C ODES
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 V n denotes a linear space of dimension n over the Galois field GF(2 N ), N > 1. By fixing a basis for V n over GF(2 N ), we can represent any element x ∈ V n as an n-tuple (x , x , …, x n ) where x i ∈ GF(2 N ). Again, GF(2 N ) can be considered as a linear space of dimension ‘N’ over GF(2). Hence an element x i ∈ GF(2 N ) has a representation as a N-tuple ( α i1 , α i2 , …, α in ) over GF(2) with respect to some fixed basis. Hence associated with each x ∈ V n , (n ≤ N) there is a matrix, m(x) =
T11 1n21 2nN1 Nn α α⎡ ⎤⎢ ⎥α α⎢ ⎥⎢ ⎥⎢ ⎥α α⎣ ⎦ ……(cid:35) (cid:35)… where the i th column represents the i th coordinate ‘x i ’ of x over GF(2). If we assume x i ∈ GF(2 N ) has a representation as a N-tuple i 1i 1 2i 2 Ni N x u u ... u = α + α + + α where u , u , …, u N is some fixed basis of the field N q F regarded as a vector space over F q . If nN A denote the ensemble of all N × n matrices over F q and if n nN A : V A → is a bijection defined by the rule, for any vector x = (x , …, x n ) ∈ N nq F , the associated matrix denoted by, ( )
11 12 1n21 22 2nN1 N2 Nn a a aa a aA x a a a ⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦ ……(cid:35) (cid:35) (cid:35)… where the i th column represents the i th coordinate x i of x over F q . Now we proceed on to define the rank of an element n Vx ∈ . D EFINITION
The rank of a vector x ∈ N nq F is the rank of the associated matrix A(x). Let r(x) denote the rank of the vector x ∈ N nq F , over F q . By the usual properties of the rank of a matrix, it is easy to prove the following inequalities. r(x) ≥ ∈ V n . 2. r(x) = 0 if and only if x = 0. 3. r(x+y) ≤ r(x) + r(y) for every x, y ∈ V n . 4. r(ax) = | a | r(x) for every a ∈ GF(2) and x ∈ V n . Thus the function x → r(x) defines a norm on V n , (x → r(x) defines a norm on N nq F ) 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 N nq F . D EFINITION
The metric induced by the rank norm is defined as the rank metric on V n ( N nq F ) and it is denoted by d R . The rank distance between x, y ∈ V n is the rank of their difference : d R (x, y) = r(x – y). The vector space V n ( N nq F ) over N q F equipped with the rank metric d R is defined as a rank distance space. D EFINITION
A linear space V n over GF(2 N ), N > 1 of dimension n such that n ≤ 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. D EFINITION
A Rank Distance code (RD-code) of length n over GF(2 N ) is a subset of the rank space V n over GF(2 N ). A linear [n, k] RD code is a linear subspace of dimension k in the rank space V n . By C[n, k] we denote a linear [n, k] RD-code. D EFINITION
A generator matrix of a linear [n, k] RD code C is a k × n matrix over GF(2 N ) 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 = [I k , A k, n–k ] where I k is the identity matrix and A k, n–k is some matrix over GF(2 n ). This form is called the standard form. D EFINITION
Let G be a generator matrix of the linear RD code C[n, k], then a matrix H of order (n – k) × n over GF(2 N ) such that GH T = (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
11 12 1521 22 2531 32 35 3 5 ...G ...... × α α α⎡ ⎤⎢ ⎥= α α α⎢ ⎥⎢ ⎥α α α⎣ ⎦ be a generator matrix of the linear [5, 3] Rank Distance code C; over GF(2 ), here ( α , α , …, α ), ( α , α , …, α ), ( α , α , …, α ) forms a basis of C; α ij ∈ GF(2 ); 1 ≤ i ≤ ≤ j ≤
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. D EFINITION
Let C be a rank distance code, the minimum - rank distance d is defined by d = min{d R (x, y) | x, y ∈ C, x ≠ y}. In other words, d = min{r(x – y) | x, y ∈ C, x ≠ y}. i.e., d = min{r(x) | x ∈ C and x ≠ ∈ N nq e F with rank r(e) = −⎢ ⎥⎢ ⎥⎣ ⎦ d 12 . Let C denote an [n, k] RD – code over N q F . A generator matrix G of C is a k × n matrix with entries from N nq F whose rows form a basis for C. Then an (n – k) × n matrix H with entries from N nq F such that GH T = (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 ⊆ N nq F satisfies the following bound: d ≤ n – k + 1. Now based on this, the notion of Maximum Rank Distance; MRD codes were defined in [24, 27]. D EFINITION
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] = q s for any integer s. Let g , …, g n be any set of elements in N q F that are linearly independent over F q . A generator matrix G of an MRD code C is defined by [ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ] g g gg g gG g g gg g g − − − ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ………(cid:35) (cid:35) (cid:35)… . 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 g c = (g , g , …, g n ). Gabidulin has proved the following theorem. T HEOREM
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(2 N ) where N > 1. An element α ∈ GF(2 N ) can be denoted by a N-tuple (a , a , …, a N–1 ) as well as by a polynomial a + a x + … + a N–1 x N–1 over GF(2). D EFINITION
The circulant transpose (T c ) of a vector α = (a , a , …, a N–1 ) ∈ GF(2 N ) is defined as C T α = (a , a , …, a N–1 ). If α ∈ GF(2 N ) has polynomial representation a + a x + … + a N–1 x N–1 in [ ( )]( )( ) + N GF 2 xx 1 then by α i , we denote the vector corresponding to the polynomial [(a + a x + … + a N–1 x N–1 ).x i ] (mod x N + 1), for i = 0 to N – 1. (Note α = α ). D EFINITION
Let f: GF(2 N ) → [GF(2 N )] N be defined as ( ) ( , ,..., ) − = C C C
T T TN f α α α α ; we call f( α ) as the ‘word’ generated by α . MacWilliams F.J. and Sloane N.J.A., [61] defined circulant matrix associated with a vector in GF(2 N ) as follows. D EFINITION
A matrix of the form −− − ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ……(cid:35) (cid:35) (cid:35)…
NN N a a aa a aa a a is called the circulant matrix associated with the vector (a , a , …, a N–1 ) ∈ GF(2 N ). Thus with each α ∈ GF(2 N ) we can associate a circulant matrix whose i th column represents C Ti α , i = 0, 1, 2, …, N – 1. f is nothing but a mapping of GF(2 N ) on to the algebra of all N × N circulant matrices over GF(2). Denote the space f(GF(2 N )) by V N . We define norm of a word v ∈ V N as follows: D EFINITION
The ‘norm’ of a word v ∈ V N 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. T HEOREM
Suppose α ∈
GF(2 N ) has the polynomial representation g(x) over GF(2) such that the gcd(g(x), x N +1) has degree N – k, where 0 ≤ k ≤ N. Then the ‘norm’ of the word generated by α is ‘k’. Proof: We know the norm of the word generated by α is the rank of the circulant matrix C C C
T T T0 1 N 1 ( , ,..., ) − α α α , where C Ti α represents the polynomial [x i g(x)] (mod x N +
1) over GF(2). Suppose the gcd(g(x), x N +1) is a polynomial of degree ‘N – k’, (0 ≤ k ≤ N – 1). To prove that the word generated by ‘ α ’ has rank ‘k’. It is enough to prove that the space generated by the N-polynomials g(x)(mod x N + ⋅ g(x)](mod x N + N–1 ⋅ g(x)] (mod x N +
1) has dimension ‘k’. We will prove that the set of k-polynomials g(x)(mod x N + ⋅ g(x)] (mod x N + N–1 ⋅ g(x)] (mod x N +
1) forms a basis for this space. If possible, let a (g(x)) + a (x ⋅ g(x)) + a (x ⋅ g(x)) + … + a k–1 (x k–1 ⋅ g(x)) ≡ N + 1)), where a i ∈ GF(2). This implies x N + 1 divides (a + a x + … + a k–1 x k–1 ) ⋅ g(x). Now if g(x) = h(x) a(x) where h(x) is the gcd(g(x), x N +1), then (a(x), x N + 1) = 1. Thus x N + 1 divides (a + a x + … + a k–1 x k–1 ) ⋅ g(x) implies that the quotient N (x 1)h(x) + divides (a + a x + … + a k–1 x k–1 ) ⋅ a(x). That is N (x 1)h(x) ⎡ ⎤+⎢ ⎥⎣ ⎦ divides (a + a x + … a k–1 x k–1 ) which is a contradiction as N (x 1)h(x) ⎡ ⎤+⎢ ⎥⎣ ⎦ has degree k where as the polynomial (a + a x + … + a k–1 x k–1 ) has degree atmost k – 1. Hence the polynomials g(x)mod(x N +1), [x ⋅ g(x)]mod(x N +1), …, [x k–1 ⋅ g(x)]mod(x N +1) are linearly independent over GF(2). We will prove that the polynomials g(x)mod(x N +1), [x ⋅ g(x)]mod(x N +1), …, [x k–1 ⋅ g(x)]mod(x N +1) generate the space. For this, it is enough to prove that x i ⋅ g(x) is a linear combination of these polynomials for k ≤ i ≤ N – 1. Let x N +1 = h(x)b(x), where b(x) = b + b x + … + b k x k (Note that here b = b k =1, since b(x) divides x N + 1). Also, we have g(x) = h(x). a(x). Thus N (g(x) b(x))x 1 a(x) ⋅+ = i.e., k0 1 k g(x)(b b x ... b x )a(x) + + + = 0 mod(x N + 1), that is k 10 1 k g(x) (b b x ... b x )a(x) − ⋅ + + + = k (g(x) x )a(x) ⎡ ⎤⋅⎢ ⎥⎣ ⎦ mod(x N + 1). (Since b k = 1). That is x k ⋅ g(x) = (b + b x + … + b k–1 x k–1 ⋅ g(x)) mod(x N + 1). Hence x k g(x) = b g(x) + b [x ⋅ g(x)] + … + b k–1 [x k–1 ⋅ g(x)] (mod(x N + N + 1), [x ⋅ g(x)]mod(x N + 1), …, [x k–1 ⋅ g(x)] mod(x N + 1) over GF(2). Now it can be easily proved that x i g(x) is a linear combination of g(x)mod(x N + 1), [x ⋅ g(x)]mod(x N + 1), …, [x k–1 ⋅ g(x)] mod(x N + 1) for i > k. Hence the space generated by the polynomial g(x)mod(x N + 1), [x ⋅ g(x)] mod(x N + 1), …, [x k–1 ⋅ g(x)] mod(x N + 1) has dimension k; i.e., the rank of the word generated by α is k. The following two corollaries are obvious. C OROLLARY If α ∈ GF(2 N ) is such that its polynomial representation g(x) is relatively prime to x N + 1, then the norm of the word generated by α is N and hence f( α ) is invertible. Proof: Follows from the theorem as gcd(g(x), x N +1) = 1 has degree 0 and hence rank of f( α ) is N. C OROLLARY
The norms of the vectors corresponding to the polynomials x +1 and x
N–1 + x
N–2 + … + x +1 are respectively N – 1 and 1.
Now we proceed on to define the distance function on V N . D EFINITION
The distance between two words v, u. in V N is defined as d(u, v) = r(u + v). Now we define circulant code of length N. D EFINITION
A circulant rank code of length N is defined as a subspace of V N equipped with the above defined distance function. D EFINITION
A circulant rank code of length N is called cyclic if, whenever (v , v , …, v N ) is a codeword, then it implies (v , v , …, v N , v ) 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). D EFINITION
A linear [n, k] RD code over GF(2 N ) 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. T HEOREM
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 C is given by (n k 1)2 − − . But (n k 1)2 − − is equal to the error 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 L r (n) error vectors where r = (n k 1)2 − − and L r (n) = n N N i 1i 1 n1 (2 1) ... (2 2 )i −= ⎡ ⎤+ − −⎢ ⎥⎣ ⎦ ∑ . Consider the same [n, k] code in Hamming metric. Let it be C , then the minimum distance of C is atmost (n – k +1). The error correcting capability of C is (n k 1 1)2 − + −⎢ ⎥ =⎢ ⎥⎣ ⎦ (n k 1)2 − − = r (since n – k is odd). Hence the number of error vectors corrected by a codeword is given by r N ii 0 n (2 1)i = ⎛ ⎞ −⎜ ⎟⎝ ⎠ ∑ , which is clearly less than L r (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]. T HEOREM
Let H = ( α ij ) be a 3 × n matrix of rank 3 over GF(2 N ), n ≤ N which satisfies the following condition. For any two distinct, non empty subsets P , P of {1, 2, …, n} there exists i , i ∈ {1, 2, 3} such that ∈ ∈ ∈ ∈ ⎛ ⎞ ⎛ ⎞⋅ ≠ ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ∑ ∑ ∑ ∑ i j i k i j i kj P k P j P k P α α α α then, H as a parity check matrix defines a [n, n – 3] single error correcting AMRD code over GF(2 N ). Proof:
Given H is a 3 × n matrix of rank 3 over GF(2 N ), so H as a parity check matrix defines a [n, n – 3] RD code C over GF(2 N ); where C = {x ∈ V n | xH T = 0}. 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) ≤
2; then, x can be written as x = y ⋅ M where y = (y , y ); y i ∈ GF(2 N ) and M = (m ij ) is a 2 × n matrix of rank 2 over GF(2). Thus (y ⋅ M)H T = 0 implies y(MH T ) = 0. Since y is non zero y(M ⋅ H T ) = 0 implies that the 2 × T has rank less than 2 over GF(2 N ). Now let P = {j such that m = 1} and P = {j such that m = 1}. Since M = (m ij ) is a 2 × n matrix of rank 2, P and P are disjoint nonempty subsets of {1, 2, …, n}, and
1j 2 j 3jj P j P j PT 1j 2 j 3jj P j P j P MH ∈ ∈ ∈∈ ∈ ∈ ⎛ ⎞α α α⎜ ⎟= ⎜ ⎟α α α⎜ ⎟⎝ ⎠ ∑ ∑ ∑∑ ∑ ∑ . But the selection of H is such that there exists i , i ∈ {1, 2, 3} such that i j i k i j i kj P k P j P k P ∈ ∈ ∈ ∈ ⎛ ⎞ ⎛ ⎞α ⋅ α ≠ α ⋅ α⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ∑ ∑ ∑ ∑ . Hence in MH T there exists a 2 × T ) = 2 over GF(2 N ), this contradicts the fact that rank (MH T ) < 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 F n 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 F n , distance atleast d apart from each other and constant Hamming weight ‘w’. 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. D EFINITION
A constant rank code of length n is a subset of a rank space V n with the property that every codeword has same rank. D EFINITION
A(n, r, d) is defined as the maximum number of vectors in V n , 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 V n having constant rank r and distance between any two vectors is at least d). We analyze the function A(n, r, d). T HEOREM A(n, r, 1) = L r (n), the number of vectors of rank r in V n . 2. A(n, r, d) = 0 if r > 0 or d > n or d > 2r. Proof. (1) is obvious from the fact that L r (n) is the number of vectors of length n, constant rank r and the distance between any two distinct vectors in the rank space V n is always greater than or equal to one. (2) Follows immediately from the definition of A(n, r, d). T HEOREM
A(n, 1, 2) = 2 n – 1 over any Galois field GF(2 N ). Proof:
Let V denote the set of vectors of rank 1 in V n . We know for each non zero element α ∈ GF(2 N ) there exists (2 n – 1) vectors of rank one having α as a coordinate. Thus the cardinality of V is (2 N – 1) (2 n – 1). Now divide V into (2 n – 1) blocks of (2 N – 1) vectors such that each block consists of the same pattern of all non-zero elements of GF(2 N ). 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) = 2 n – 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(2 ). Let 0, 1, 2, 3 be the basic symbols. Then GF(2 )={0, 1, 2, 3, (12), (13), (23), (123)} (Note that by (ijk), we denote the linear combination of i + j + k over GF(2)). Suppose n = 3, divide F into 2 – 1 blocks of 2 – 1 vectors as follows: 001 010 100 110 002 020 200 220 … … … … (00) (123) 0(123)0 (123)00 (123)(123)0 101 011 111 202 022 222 … … … (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 = 2 – 1. Now we prove another interesting theorem. T HEOREM
A(n, n, n) = 2 N – 1 over any GF(2 N ). Proof:
Denote by V n the set of vectors of rank n in the space V n . We know that the cardinality of V n is (2 N – 1) (2 N – 2) … (2 N – 2 n–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 ≤ i ≤ n). This implies that A(n, n, n) ≤ N – 1. Now we construct a (n, n, n) set as follows: Select N vectors from V n such that, 1. Each basis element of GF(2 n ) should occur (can be as a combination) atleast once in each vector. 2. If the i th 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 2 N – 1. (We do not count the all zero linear combination). Thus A(n, n, n) = 2 N – 1. We illustrate this by the following example. Example 1.3:
Consider GF(2 N ) for any N > 1. As in example 1.2, we represent GF(2 N ) 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). 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. T HEOREM
A(4, 2, 4) = 5 over any Galois field GF(2 N ). Proof:
Consider V , the 4-dimensional space over GF(2 N ). We denote the elements of V as 4-tuples (a b c d) where a, b, c, d ∈ GF(2 N ). Denote by V , the set of vectors of rank 2 in V . The cardinality of V is 35 × (2 N –
1) (2 N – (2 1)(2 2)(2 1)(2 2)V L (4) (2 1)(2 2) ⎞− − − −= = ⎟− − ⎠ . Thus for each distinct non-zero pair of elements of GF(2 N ) there are 35 vectors in V . Let a, b ∈ GF(2 N ) be such that a ≠
0, b ≠ ≠ b. 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 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). T HEOREM
A(n, n, d) ≤ (2 N – 1) (2 N – 2) … (2 N – 2 n–d ) over any field GF(2 N ). Proof:
Let V n be the set of vectors of rank n in the space V n . The cardinality of V n is given by (2 N – 1) (2 N – 2) … (2 N – 2 n–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 (2 N – 1) (2 N – 2) … (2 N – 2 n–d ). This proves, A(n, n d) ≤ (2 N – 1) (2 N – 2) … (2 N – 2 n–d ). These AMRD codes are useful for error correction in data storage systems. Chapter Two R ANK D ISTANCE B ICODES
AND THEIR P ROPERTIES
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. Let V n and V m be n-dimensional and m-dimensional vector spaces over the field N q F ; n ≤ N and m ≤ N (m ≠ n). That is V n = N nq F and V m = N mq F . We know V = V n ∪ V m is a (m, n) dimensional vector bispace (or bivector space) over the field N q F . By fixing a bibasis of V = V n ∪ V n over N q F we can represent any element x ∪ y ∈ V = (V n ∪ V m ) as a (n, m)-tuple; (x , …, x n ) ∪ (y , …, y m ) where x i, y j ∈ N q F ; 1 ≤ i ≤ n and 1 ≤ j ≤ m. Again N q F can be considered as a pseudo false linear bispace of dimension (N, N) over F q (we say a linear vector bispace V = V n ∪ V n to be a pseudo false linear bispace if m = n = N). Hence the elements x i, y j ∈ N q F has a representation as N-bituple ( α , …, α Ni ) ∪ ( β , …, β Nj ) over F q with respect to some fixed bibasis. Hence associated with each x ∪ y ∈ V n ∪ V m (n ≠ m) there is a bimatrix m(x) ∪ m(y) =
11 1n 11 1m21 2n 21 2mN1 Nn N1 Nm a a b ba a b ba a b b ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥∪⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ … …… …(cid:35) (cid:35) (cid:35) (cid:35)… … where the i th ∪ j th column represents the i th ∪ j th coordinate of x i ∪ y j of x ∪ y over F q . 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 Z Z = ∪ is a false pseudo bivector space over Z . Likewise Z Z ∪ is a pseudo false bivector space over Z . Also ZZV ∪= is a pseudo false bivector space over Z . However throughout this book we will be using only vector bispaces over Z or N Z unless otherwise specified. Now we see to every x ∪ y in the bivector space V n ∪ V m 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 F q or GF(2). D EFINITION
The birank of an element x ∪ y ∈ (V n ∪ V m ) is defined as the birank of the bimatrix m(x) ∪ m(y) over GF(2) or F q . (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 r (x) ∪ r (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 ∈ (V n ∪ V m ) (x ∈ V n and y ∈ V m ) we have r(x ∪ y) = r (x) ∪ r (y) ≥ ∪ (x) ≥ (y) ≥ ∈ V n and y ∈ V m ). (ii) r(x ∪ y) = r (x) ∪ r (y) = 0 ∪ ∪ y = 0 ∪ r((x + x ) ∪ (y + y )) ≤ {r (x ) + r (x )} ∪ r (y ) + r (y ) for every x , x ∈ V n and y , y ∈ V m . That is we have r((x + x ) ∪ (y + y )) = r (x + x ) ∪ r (y + y ) ≤ r (x ) + r (x ) ∪ r (y ) + r (y ); (as we have for every x , x ∈ V n , r(x + x ) ≤ r (x ) + r (x ) and for every y , y ∈ V m ; r (y + y ) ≤ r (y ) + (y )). (iv) r (a x) ∪ r (a y) = |a |r (x) ∪ |a |r (y) for every a , a , ∈ F q or GF(2) and x ∈ V n and y ∈ V m . Thus the bifunction x ∪ y → r (x) ∪ r (y) defines a binorm on V n ∪ V m . D EFINITION
The bimetric induced by the birank binorm is defined as the birank bimetric on V n ∪ V m and is denoted by ∪ R R d d . If x ∪ y , x ∪ y ∈ V n ∪ V m then the birank bidistance between x ∪ y and x ∪ y is ( , ) ( , ) ( ) ( ) ∪ = − ∪ − R R d x x d y y r x x r y y (here R d (x , x ) = r (x – x ) for every x , x in V n , the rank distance between x and x likewise for y , y ∈ V m ). D EFINITION
A linear bispace V n ∪ V m over GF(2 N ), N > 1 of bidimension n ∪ m such that n ≤ N and m ≤ N equipped with the birank bimetric is defined as the birank bispace. D EFINITION
A birank bidistance RD bicode of bilength n ∪ m over GF(2 N ) is a bisubset of the birank bispace V n ∪ V m over GF(2 N ). D EFINITION
A linear [n , k ] ∪ [n , k ] RD bicode is a linear bisubspace of bidimension k ∪ k in the birank bispace V n ∪ V m . By C [n , k ] ∪ C [n , k ], we denote a linear [n , k ] ∪ [n , k ] RD bicode. We can equivalently define a RD bicode as follows: D EFINITION
Let V n and V m , m ≠ n be rank spaces over GF(2 N ), N > 1. Suppose P ⊂ V n and Q ⊂ V m be subsets of the rank spaces over GF(2 N ). Then P ∪ Q ⊆ V n ∪ V m is a rank distance bicode of bilength (n, m) over GF(2 N ). D EFINITION
Let C [n , k ] be [n , k ] RD code (i.e., a linear subspace of dimension k , in the rank space V n ) and C [n , k ] be [n , k ] RD code (i.e., a linear subspace of dimension k in the rank space V m ) (m ≠ n); then C [n , k ] ∪ C [n , k ] is defined as the linear RD bicode of the linear bisubspace of dimension (k , k ) in the rank bispace V n ∪ V m . Now we proceed onto define the notion of the generator bimatrix of a linear [n , k ] ∪ [n , k ] RD bicode. D EFINITION
A generator bimatrix of a linear [n , k ] ∪ [n , k ] RD-bicode C ∪ C is a k × n ∪ k × n bimatrix over GF(2 N ) whose birows form a bibasis for C ∪ C . A generator bimatrix G = G ∪ G of a linear RD bicode C [n , k ] ∪ C [n , k ] can be brought into the form G = G ∪ G = , [ , ] − k k n k I A ∪ [ , ] × − k k n k I A where k I , k I is the identity matrix and × − i i i k n k
A , i = 1, 2 is some matrix over GF(2 N ). This form of G = G ∪ G is called the standard form. D EFINITION
If G = G ∪ G is a generator bimatrix of C [n , k ] ∪ C [n , k ] then a bimatrix H = H ∪ H of order (n – k × n , n – k × n ) over GF(2 N ) such that GH T = (G ∪ G ) (H ∪ H ) T = (G ∪ G ) ( ∪ T T
H H ) = ∪ T T
G H G H = 0 ∪
0 is called a parity check bimatrix of C [n , k ] ∪ C [n , k ]. Suppose C = C ∪ C is a linear [n , k ] ∪ [n , k ] RD code with G = G ∪ G and H = H ∪ H as its generator and parity check bimatrices respectively, then C = C ∪ C has two representation, (i) C = C ∪ C is a row bispace of G = G ∪ G (i.e., C is the row space of G and C is the row space of G ) (ii) C = C ∪ C is the solution bispace of H = H ∪ H ; i.e., C is the solution space of H and C is the solution space of H . Now we proceed on to define the notion of minimum rank bidistance of a rank distance bicode C = C ∪ C . D EFINITION
Let C = C ∪ C be a rank distance bicode, the minimum-rank bidistance is defined by d = d ∪ d where, d = min{d R (x, y) | x, y ∈ C , x ≠ y} and d = min{d R (x, y) | x, y ∈ C , x ≠ y} i.e., d = d ∪ d = min {r (x) | x ∈ C and x ≠ ∪ min {r (x) | x ∈ C and x ≠
0} . If an RD bicode C = C ∪ C has the minimum rank bidistance d = d ∪ d then it can correct all bierrors d = d ∪ d ∈ ∪ N N n mq q
F F with birank r(e) = (r ∪ r )(e ∪ e ) = r (e ) ∪ r (e ) − −⎢ ⎥ ⎢ ⎥≤ ∪⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ d d . Let C = C ∪ C denote an [n , k ] ∪ [n , k ] RD-bicode over N q F . A generator bimatrix G = G ∪ G of C = C ∪ C is a k × n ∪ k × n bimatrix with entries from N q F whose rows form a bibasis for C = C ∪ C . Then an (n – k ) × n ∪ (n – k ) × n bimatrix H = H ∪ H with entries from N q F such that GH T = (G ∪ G ) (H ∪ H ) T = (G ∪ G ) ( ∪ T T
H H ) = ∪ T T
G H G H = 0 ∪
0 is called the parity check bimatrix of C = C ∪ C . 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 = d ∪ d of any linear [n , k ] ∪ [n , k ] RD bicode C = C ∪ C ⊆ ∪ N N n mq q
F F satisfies the following bounds. d = d ∪ d ≤ n – k + 1 ∪ n – k +1. Based on this notion we now proceed on to define the new notion of Maximum Rank Distance (MRD) bicodes. D EFINITION
An [n , k , d ] ∪ [n , k , d ] RD bicode C = C ∪ C is called a Maximum Rank Distance (MRD) bicode if the singleton-style bound is reached, that is d = d ∪ d = n – k + 1 ∪ n – k +1. Now we proceed on to briefly give the construction of MRD bicode. Let [s] = [s ] ∪ [s ] = s s q q ∪ for any two integers s and s . Let {g , …, g n } ∪ {h , h , …, h m } be any set of elements in N q F that are linearly independent over over F q . A generator bimatrix G = G ∪ G of an MRD bicode C = C ∪ C is defined by G = G ∪ G [ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ] g g g h h hg g g h h hg g g h h hg g g h h h − − − − − − ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= ∪⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ … …… …… …(cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35)… … It can be easily proved that the bicode C = C ∪ C given by the generator bimatrix G = G ∪ G , has the rank bidistance d = d ∪ d = (n – k + 1) ∪ (n – k + 1). Any bimatrix of the above from is called a Frobenius bimatrix with generating bivector g C = C C g h ∪ = (g , …, g n ) ∪ (h , h , …, h m ). One can prove the following theorem: T HEOREM
Let C[n, k] = C (n , k ) ∪ C (n , k ) be the linear (n , k , d ) ∪ (n , k , d ) MRD bicode with d = 2t + 1 and d = 2t + 1. Then C[n, k] = C (n , k ) ∪ C (n , k ), bicode corrects all bierrors of birank atmost t = t ∪ t and detects all bierrors of birank greater than t = t ∪ t . Consider the Galois field GF(2 N ), N > 1. An element α ∪ β ∈ GF(2 N ) ∪ GF(2 N ) can be denoted by a biN-tuple (a , …, a N–1 ) ∪ (b , b , …, b N–1 ) as well as by the bipolynomial a + a x + … + a N – 1 x N – 1 ∪ b + b x+ …+ b N – 1 x N – 1 over GF(2). We now proceed on to define the new notion of circulant bitranspose. D EFINITION
The circulant bitranspose = ∪
C C C
T T T of a bivector α = α ∪ β = (a , …, a N–1 ) ∪ (b , b , …, b N–1 ) ∈ GF(2 N ) is defined as = ∪ C CC
T TT α α β = (a , a , …, a N–1 ) ∪ (b , b , …, b N–1 ). If α = α ∪ β ∈ GF(2 N ) ∪ GF(2 N ) has the bipolynomial representation a + a x + … + a N – 1 x N – 1 ∪ b + b x+ …+ b N – 1 x N – 1 in (2)( ) (2)( )1 1 N N
GF x GF xx x ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥∪+ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ then by; α i = α ∪ β we denote the bivector corresponding to the bipolynomial [(a + a x + … + a N – 1 x N – 1 ) ⋅ x i ] mod(x n +1) ∪ [(b + b x+ …+ b N – 1 x N – 1 ) ⋅ x i ] mod(x n +1) for i = 0, 1, 2, . . . , N – 1. (Note α = α ∪ β = α ). Now we proceed on to define the biword generated by α = α ∪ β . D EFINITION
Let f = f ∪ f : GF(2 N ) ∪ GF(2 N ) → [GF(2 N )] N ∪ [GF(2 N )] N be defined as, f( α ) = f ( α ) ∪ f ( β ) ( , , , ) ( , , , ) C C C C C C
T T T T T TN N α α α β β β − − = ∪ … … . We call f( α ) = f ( α ) ∪ f ( β ) as the biword generated by α = α ∪ β . We analogous to the definition given in MacWilliams and Sloane [61] define circulant bimatrix associated with a bivector in GF(2 N ) ∪ GF(2 N ). D EFINITION
A bimatrix of the from − −− − − − ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= ∪⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ … …… …(cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35)… …
N NN N N N a a a b b ba a a b b ba a a b b b is called the circulant bimatrix associated with the bivector (a , a , …, a N–1 ) ∪ (b , b , …, b N–1 ) ∈ GF(2 N ) ∪ GF(2 N ). Thus with each α = α ∪ β ∈ GF(2 N ) ∪ GF(2 N ), we can associate a circulant bimatrix whose i th bicolumn represents C C
T Ti i α β ∪ ; i = 0, 1, 2, …, N – 1. f = f ∪ f is nothing but a bimapping of GF(2 N ) ∪ GF(2 N ) on to the pseudo false bialgebra of all N × N circulant bimatrices over GF(2). Denote the bispace f(GF(2 N )) = f (GF(2 N )) ∪ f (GF(2 N )) by V N ∪ V N . We define binorm of a biword v = v ∪ v ∈ V N ∪ V N as follows. D EFINITION
The binorm of a biword v = v ∪ v ∈ V N ∪ V N is defined as the birank of v = v ∪ v over GF(2 N ) (by considering it as a circulant bimatrix over GF(2)). We denote the binorm of v = v ∪ v by r(v) = r (v ) ∪ r (v ), we prove the following theorem: T HEOREM
Suppose α = α ∪ β ∈ GF(2 N ) ∪ GF(2 N ) has the bipolynomial representation g (x) ∪ h (x) over GF(2) such that gcd (g (x), x N + 1) has degree N – k and gcd (h (x), x N + 1) has degree N – k where 1 ≤ k , k ≤ N; then the binorm of the biword generated by α = α ∪ β is k ∪ k . Proof:
We know the binorm of the biword generated by α = α ∪ β is the birank of the circulant bimatrix T T T T T T0 1 N 1 0 1 N 1 ( , , ... , ) ( , , ... , ) − − = α α α ∪ β β β where
T TTi 1i 2i α = α ∪ β represents the bipolynomial [x i g (x)] [mod x N +1] ∪ [x i h (x)] [mod x N +1] over GF(2). Suppose the bigcd (g (x), x N +1) ∪ (h (x), x N +1) has bidegree N – k ∪ N – k , (0 ≤ k , k ≤ N – 1). To prove that the biword generated by α = α ∪ β has birank k ∪ k . It is enough to prove that the bispace generated by the N- bipolynomials {g (x) mod (x N + 1), (xg (x)) [mod x N +1], …, [x N-1 g (x)] mod[x N + 1]} ∪ { h (x) mod (x N + 1), (xh (x)) [mod x N +1], …, [x N-1 h (x)] mod (x N + 1)} has bidimension k ∪ k . We will prove that the biset of k ∪ k bipolynomials {g (x) mod (x N + 1), (xg (x)) mod (x N + 1), …, (x N-1 g (x)) mod(x N + 1)} ∪ {h (x) mod (x N + 1), (xh (x)) (mod x N + 1), …, (x N-1 h (x)) (mod x N + 1)} forms a bibasis for this bispace. If possible let a (g (x)) + a (xg (x)) + … + k 1 a − ( k 1 x − g (x)) ∪ b (h (x)) + b (xh (x)) + … + k 1k 1 2 b (x h (x)) −− ≡ ∪ N + 1) where a i , b i ∈ GF(2). This implies x
N+1 ∪ x N+1 bidivides (a + a x + … + k 1k 1 a x −− )g (x) ∪ (b + b x + … + k 1k 2 b x −− )h (x) Now if g (x) ∪ h (x) = p (x) a (x) ∪ p (x) b (x) where, p (x) ∪ p (x) is the bigcd {(g (x), x N + 1) ∪ (h (x), x N + 1)} then (a (x), x N + 1) ∪ (b (x), x N + 1) = 1 ∪
1. Thus x N + 1 bidivides (a + a x + … + k 1k 1 a x −− )g (x) ∪ (b + b x + … + k 1k 2 b x −− )h (x) which inturn implies that the biquotient ( ) ( ) N N1 2 x 1 x 1p (x) p (x) + +∪ bidivides (a + a x + … + k 1k 1 a x −− )a (x) ∪ (b + b x + … + k 1k 2 b x −− )b (x) That is ( ) ( ) N N1 2 x 1 x 1p (x) p (x) + +∪ bidivides (a + a x + … + k 1k 1 a x −− ) ∪ (b + b x + … + k 1k 2 b x −− ) which is a contradiction as ( ) ( ) N N1 2 x 1 x 1p (x) p (x) + +∪ has bidegree k ∪ k where as the bipolynomial (a + a x + … + k 1k 1 a x −− ) ∪ (b + b x + … + k 1k 2 b x −− ) has bidegree atmost k –1 ∪ k –1. Hence the bipolynomials {g (x) mod (x N + 1), xg (x) mod (x N + 1), …, k 1 x − g (x) mod(x N + 1)} ∪ {h (x) mod (x N + 1), xh (x)mod (x N + 1), …, k 1 x − h (x) mod (x N + 1)} are bilinearly independent over GF(2) ∪ GF(2), i.e.; {g (x) mod (x N + 1), xg (x) mod (x N + 1), …, k 1 x − g (x) mod(x N + 1)} ∪ {h (x) mod (x N + 1), xh (x)mod (x N + 1), …, k 1 x − h (x) mod (x N + 1)} bigenerate the bispace. For this it is enough to prove that x i g (x) ∪ x i h (x) is a linear bicombination of these bipolynomials for k ≤ i ≤ N –1 and k ≤ i ≤ N – 1. Let x N + 1 ∪ x N + 1 = p (x)q (x) ∪ p (x)q (x) where q (x) ∪ q (x) = kk10kk10 xd...xddxc...xcc +++∪+++ (Note: c c 1 = = and d d 1 = = , since q (x) ∪ q (x) bidivides x N + 1 ∪ x N + 1, i.e., g (x) divides x N + 1 and g (x) divides x N + 1). Also we have g (x) = p (x) a (x) and h (x) = p (x) b (x) i.e., g (x) ∪ h (x) = p (x) a (x) ∪ p (x) b (x). Thus x N + 1 ∪ x N + 1 = p (x) p (x)g (x) h (x)a (x) b (x) ⎛ ⎞ ⎛ ⎞⋅ ∪ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ that is N11 1 p (x)g (x) 0 mod(x 1)a (x) ⋅ ≡ + and
N22 2 p (x)h (x) 0 mod(x 1)b (x) ⋅ ≡ + x N + 1 ∪ x N + 1 = p (x) p (x)g (x) h (x)a (x) b (x) ⎛ ⎞ ⎛ ⎞⋅ ∪ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ . Now k 11 0 1 k 11 g (x)(a a x ... a x )a (x) −− + + + = k N1 1 g (x)x mod(x 1)a (x) ⎡ ⎤ +⎢ ⎥⎣ ⎦ (since a k =1). That is k k 1 N1 0 1 k 1 1 x g (x) (a a x ... a x g (x)) mod(x 1) −− = + + + + . Hence, k 1 0 1 1 1 x g (x) (a g (x) a (xg (x)) ... = + + + k 1 Nk 1 1 a [x g (x)]) mod(x 1) −− + ; a linear combination of g (x) mod(x N + 1), [xg (x)] mod (x N + 1), …, [ k 1 x − g (x)] mod(x N + 1) over GF(2). Now it can be easily proved that x i g (x) is a linear combination of g (x) mod (x N +1), xg (x) mod (x N +1), …, k 1 N1 x g (x) mod(x 1) − + for i > k . Similar argument holds good for x i h (x) where i > k . Hence the bispace generated by the bipolynomial {g (x) mod (x N + 1), xg (x) mod (x N + 1), …, k 1 x − g (x) mod(x N + 1)} ∪ ∪ {h (x) mod (x N + 1), xh (x)mod (x N + 1), …, k 1 x − h (x) mod (x N + 1)} has bidimension k ∪ k ; i.e., the birank of the biword generated by α = α ∪ β is k ∪ k . Now we have the two corollaries to be true. C OROLLARY If α = α ∪ β ∈ GF(2 N ) ∪ GF(2 N ) is such that its bipolynomial representation g (x) ∪ h (x) is relatively prime to x N +1 ∪ x N +1 (i.e. g (x) is relatively prime to x N +1 and h (x) is relatively prime to x N + 1) then the binorm of the biword generated by α = α ∪ β is (N, N) and f( α ) = f ( α ) ∪ f ( β ) is invertible. Proof:
Follows from the theorem bigcd(g (x) ∪ h (x), x N +1 ∪ x N + 1) = bigcd(g (x), x N + 1) ∪ (h (x), x N + 1) = 1 ∪
1 has bidegree 0 ∪ α ) = f ( α ) ∪ f ( β ) is (N, N). C OROLLARY
The binorms of the bivectors corresponding to the bipolynomials x + 1 ∪ x + 1 and x N – 1 + x
N – 2 + … + x + 1 ∪ x N – 1 + x
N – 2 + … + x + 1 are respectively (N – 1, N – 1) and (1, 1).
Now we proceed on to define the bidistance bifunction on V N ∪ V N . D EFINITION
The bidistance between two biwords u = u ∪ u and v = v ∪ v in V N ∪ V N is defined as d(u, v) = d (u , v ) ∪ d (u , v ) = r(u + v) = r (u + v ) ∪ r (u + v ). Now we proceed on to define the new notion of circulant rank bicodes of bilength N = N ∪ N . D EFINITION
Let C be a circulant rank code of length N which is a subspace of N V equipped with the distance function d (u , v ) = r (u , v ) and C be a circulant rank code of length N which is the subspace of N V equipped with distance function d (u , v ) = r (u , v ) where N V and N V are spaces defined over GF(2 N ) with N ≠ N . C = C ∪ C is defined as the circulant birank bicode of bilength N = N ∪ N defined as a bisubsapce of ∪ N N
V V equipped with the bidistance bifunction d (u , v ) ∪ d (u , v ) = r (u + v ) ∪ r (u + v ). D EFINITION
A circulant birank bicode of bilength N = N ∪ N is called bicyclic if whenever ( ... ) ( ... ) ∪ N N v v u u is a bicodeword then it implies ( ) ( ) ∪ … … N N v v v v u u u u is also a bicodeword.
Now we proceed on to define semi MRD bicode. D EFINITION
Let C [n , k ] be a [n , k ] RD code and C [n , k , d ] be a MRD code. C [n , k ] ∪ C [n , k , d ] is defined as the semi MRD bicode if n ≠ n and k ≠ k and C [n , k ] 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. D EFINITION
Let C = C [n , k ] be a RD-code and C be a circulant rank code both are subspaces of rank spaces defined over the same field GF(2 N ). Then C ∪ C 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. D EFINITION
Let [n , k , d ] = C be a MRD code and C be a circulant rank code both are subspaces or rank spaces defined over the same field GF(2 N ) or N q F . C ∪ C is defined to be semi circulant bicode of type II. 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. D EFINITION
Let C [n , k ] be a RD-code which is not a MRD code and C be a cyclic circulant rank code, both take entries from the same field GF(2 N ). C ∪ C 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. D EFINITION
Let [n , k , d ] = C be a MRD-code which is a subspace of V N , V N defined over GF(2 N ). C be a cyclic circulant rank code with entries from GF(2 N ). C ∪ C 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. D EFINITION
Let C be a circulant rank code and C be a cyclic circulant rank code, C ∪ C is defined as the semicyclic circulant rank bicode, both C and C 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. D EFINITION
Let C [n , k ] and C [n , k ] be any two distinct Almost Maximum Rank Distance (AMRD) codes with the minimum distances greater than or equal to n – k and n – k respectively defined over GF(2 N ). Then C [n , k ] ∪ C [n , k ] is defined as the Almost Maximum Rank Distance bicode (AMRD-bicode) over GF(2 N ). An AMRD bicode whose minimum bidistance is greater than n – k ∪ n – k 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. T HEOREM
When n – k ∪ n – k is an odd pair of biintegers (i.e., n – k and n – k are odd integers); (i) The error correcting capability of the [n , k ] ∪ [n , k ] AMRD bicode is equal to that of an [n , k ] ∪ [n , k ] MRD bicode. (ii) An [n , k ] ∪ [n , k ] AMRD bicode is better than any [n , k ] ∪ [n , k ] bicode in Hamming metric for error correction. Proof: (i) Suppose C = C ∪ C is a [n , k ] ∪ [n , k ] AMRD bicode such that n – k ∪ n – k is an odd biinteger (i.e., n – k ≠ n – k are odd integers). The maximum number of bierrors corrected by C = C ∪ C is given by (n k 1) (n k 1)2 2 − − − −∪ . But (n k 1) (n k 1)2 2 − − − −∪ is equal to the error correcting capability of an [n , k ] ∪ [n , k ] MRD bicode (Since n – k and n – k are odd). Thus when n – k ∪ n – k is biodd (i.e., both n – k and n – k are odd) the [n , k ] ∪ [n , k ] AMRD bicode is as good as an [n , k ] ∪ [n , k ] MRD bicode. (ii) Suppose C = C ∪ C is a [n , k ] ∪ [n , k ] AMRD bicode such that n – k and n – k are odd. Then, each bicodeword of C can correct r 1 r 2 r L (n ) L (n ) L (n) ∪ = error bivectors where (n k 1) (n k 1)r r r 2 2 − − − −= ∪ = ∪ and r r 1 r 2 L (n) L (n ) L (n ) = ∪ n 1 N N i 1i 1 n1 (2 1) (2 2 )i −= ⎡ ⎤= + − − ∪⎢ ⎥⎣ ⎦ ∑ … n 2 N N i 1i 1 n1 (2 1) (2 2 )i −= ⎡ ⎤+ − −⎢ ⎥⎣ ⎦ ∑ … . Consider the same [n , k ] ∪ [n , k ] bicode in Hamming metric. Let it be D ∪ D = D, then the minimum bidistance of D is atmost (n – k +1) ∪ (n – k + 1). The error correcting capability of D is (n k 1 1) (n k 1 1) r r2 2 − + − − + −∪ = ∪ ([n – k ] and [n – k ] are odd). Hence the number of error bivectors corrected by a codeword is given by r r1 2N i N ii 0 i 0 n n(2 1) (2 1)i i = = ⎡ ⎤ ⎡ ⎤− ∪ −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ∑ ∑ which is clearly less than r 1 r 2 L (n ) L (n ) ∪ . Thus the number of error bivectors that can be corrected by the [n , k ] ∪ [n , k ] AMRD bicode is much greater than that of the same bicode considered in Hamming metric. For a given bilength n = n ∪ n , a single error correcting AMRD bicode is one having bidimension n – 3 ∪ n –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]. T HEOREM
Let H = H ∪ H = ( ) ( ) ∪ ij ij α α be a 3 × n ∪ × n bimatrix of birank 3 over GF(2 N ); n ≤ N and n ≤ N which satisfies the following condition. For any two distinct, non empty bisubsets P , P where = ∪ P P P and = ∪
P P P of {1, 2, …, n } and {1, 2, 3, …, n } respectively there exists = ∪ i i i , = ∪ i i i ∈ {1, 2, 3} ∪ {1, 2, 3} such that, ∈ ∈ ∈ ∈ ⎛ ⎞ ⎛ ⎞⋅ ∪ ⋅ ≠⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ∑ ∑ ∑ ∑ j P k P j P k P i j i k i j i k α α α α ∈ ∈ ∈ ∈ ⎛ ⎞ ⎛ ⎞⋅ ∪ ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ∑ ∑ ∑ ∑ j P k P j P k P i j i k i j i k α α α α . then, H = H ∪ H as a parity check bimatrix defines a [n , n –3] ∪ [n , n –3] single bierror correcting AMRD bicode over GF(2 N ). Proof: Given H = H ∪ H is a 3 × n ∪ × n bimatrix of birank 3 ∪
3 over
GF(2 N ), so H = H ∪ H as a parity check bimatrix defines a [n , n – 3] ∪ [n , n – 3] RD bicode C = C ∪ C over GF(2 N ) where, n T1 1 C {x V | xH 0} = ∈ = and n T2 2 C {x V | xH 0} = ∈ = . It remains to prove that the minimum bidistance of C = C ∪ C is greater than or equal to 3 ∪
3. We will prove that no non zero bicodeword of C = C ∪ C has birank less than 3 ∪
3. The proof is by method of contradiction. Suppose there exists a non zero bicodeword x = x ∪ x such that r (x ) ≤
2 and r (x ) ≤
2, then x = x ∪ x can be written as x = x ∪ x = (y ∪ y ) (M ∪ M ) where y (y , y ) = and y (y , y ) = ; y , y , y , y GF(2 ) ∈ and M = M ∪ M = (m ) (m ) ∪ is a 2 × n ∪ × n bimatrix of birank 2 ∪ (yM)H T = T T1 1 1 2 2 2 y M H y M H ∪ = 0 ∪ T T T1 1 1 2 2 2 y(MH ) y (M H ) y (M H ) = ∪ = 0 ∪
0. Since y = y ∪ y is non zero; y(MH T ) = 0 ∪
0; implies y (M T1 H ) = 0 and y (M T2 H ) = 0 that is the 2 ×
3 bimatrix
T T1 1 2 2
M H M H ∪ has birank less than 2 over GF(2 N ). Now let P P P = ∪ {j such that m 1} {j such that m 1} = = ∪ = and
P P P = ∪ {j such that m 1} {j such that m 1} = = ∪ = . Since M = M ∪ M = ij ij (m ) (m ) ∪ is a 2 ×
2 bimatrix of birank 2 ∪
2. P and P are disjoint non empty bisubsets of {1, 2, …, n } ∪ {1, 2, …, n } respectively and T T T1 1 1 2 2
M H M H M H = ∪ ∈ ∈ ∈∈ ∈ ∈ ⎛ ⎞α α α⎜ ⎟⎜ ⎟⎜ ⎟= ∪⎜ ⎟⎜ ⎟α α α⎜ ⎟⎝ ⎠ ∑ ∑ ∑∑ ∑ ∑ ∈ ∈ ∈∈ ∈ ∈ ⎛ ⎞α α α⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟α α α⎜ ⎟⎝ ⎠ ∑ ∑ ∑∑ ∑ ∑ . But the selection of H = H ∪ H is such that their exists i ,i ,i ,i ∈ {1, 2, 3} such that ∈ ∈ ∈ ∈ α ⋅ α ∪ α ⋅ α ∑ ∑ ∑ ∑ ∈ ∈ ∈ ∈ ≠ α ⋅ α ∪ α ⋅ α ∑ ∑ ∑ ∑ . Hence in MH T = T T1 1 2 2
M H M H ∪ ,there exists a 2 × ( ) ( ) ( ) T T T1 1 1 2 2 2 r MH r M H r M H = ∪ = 2 ∪
2 over GF(2); this contradicts the fact that birank of
T T T1 1 2 2
MH M H M H 2 2 = ∪ < ∪ . Hence the result. Now using constant rank code, we proceed on to define the notion of constant rank bicodes of bilength n ∪ n . D EFINITION
Let C = C ∪ C be a rank bicode, where C is a constant rank code of length n (a subset of rank space n V ) and C is a constant rank code of length n (a subset of rank space n V ) then C is a constant rank bicode of bilength n ∪ n ; that is every bicodeword has same birank. D EFINITION
A(n , r , d ) ∪ A(n , r , d ) is defined as the maximum number of bivectors in ∪ n n V V , constant birank r ∪ r and bidistance between any two bivectors is atleast d ∪ d . (By a (n , r , d ) ∪ (n , r , d ) biset we mean a bisubset of bivectors of ∪ n n V V having constant birank r ∪ r and bidistance between any two bivectors is atleast d ∪ d ). We analyze the bifunction A(n , r , d ) ∪ A(n , r , d ) by the following theorem. T HEOREM (i)
A(n , r , d ) ∪ A(n , r , d ) ( ) ( ) = ∪ r r L n L n , the number of bivectors of birank r ∪ r in ∪ n n V V . (ii)
A(n , r , d ) ∪ A(n , r , d ) = ∪ > 0 and r > 0 or d > n and d > n or d > 2r and d > 2r Proof: (i) Follows from the fact that r 1 r 2
L (n ) L (n ) ∪ is the number of bivectors of bilength n ∪ n , constant birank r ∪ r and bidistance between any two distinct bivectors in a rank bispace n n V V ∪ , is always greater than or equal to 1 ∪
1. (ii) Follows immediately from the definition of A(n , r , d ) ∪ A(n , r , d ). T HEOREM
A(n , 1, 2) ∪ A(n , 1, 2) = − ∪ − n n N ). Proof: Denote by V ∪ V the set of bivectors of birank 1 ∪ n n V V ∪ . We know for each non zero element α ∪ α ∈ GF(2 N ) there exists (2 1) (2 1) n n − ∪ − bivectors of birank 1 ∪
1 having α ∪ α as a coordinate. Thus the cardinality of V ∪ V is n nN N (2 1)(2 1) (2 1)(2 1) − − ∪ − − .Now bidivide V ∪ V into n n (2 1) (2 1) − ∪ − blocks of N N (2 1) (2 1) − ∪ − bivectors such that each block consists of the same pattern of all nonzero bielements of GF(2 N ) ∪ GF(2 N ). 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 (n , 1, 2) ∪ (n , 1, 2) biset. Also it is always possible to construct such a biset. Thus A(n , 1, 2) ∪ A(n , 1, 2) n n = − ∪ − . T HEOREM
A(n , n , n ) ∪ A(n , n , n ) = 2 N – 1 ∪ N – 1 (i.e. A(n i , n i , n i ) = 2 N – 1; i = 1, 2); over any GF(2 N ). Proof: Denote by n n
V V ∪ ; the biset of all bivectors of birank n ∪ n in the bispace n n V V ∪ . We know the bicardinality of n n V V ∪ is (2 N – 1) (2 N – 2) … n 1N (2 2 ) − − ∪ (2 N – 1) (2 N – 2) … n 1N (2 2 ) − − , by the definition of a (n , n , n ) ∪ (n , n , n ) biset the bidistance between any two bivectors should be n ∪ n . Thus no two bivectors can have a common symbol at a coordinate place i ∪ i ; (1 ≤ i ≤ n , 1 ≤ i ≤ n ). This implies that A(n , n , n ) ∪ A(n , n , n ) ≤ N – 1 ∪ N – 1. Now we construct a (n , n , n ) ∪ (n , n , n ) biset as follows. Select N bivectors from n n V V ∪ such that 1. Each bibasis bielement of
N N
GF(2 ) GF(2 ) ∪ should occur (can be as a bicombination) atleast once in each bivector. 2. If the th th1 2 (i ,i ) bivector is chosen th th1 2 [(i 1) ,(i 1) ] + + bivector should be selected such that its birank bidistance from any bilinear combination of the previous (i , i ) bivectors is n ∪ n . 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 n ∪ n . Hence it is a (n , n , n ) ∪ (n , n , n ) biset. Also the bicardinality of this (n , n , n ) ∪ (n , n , n ) biset is 2 N – 1 ∪ N – 1 (we do not count all zero bilinear combination); thus A(n , n , n ) ∪ A(n , n , n ) = 2 N – 1 ∪ N – 1. Recall a [n, 1] repetition RD code is code generated by the matrix G = (1 1 1 … 1) over N q F . Any non zero codeword has rank 1. D EFINITION
A [n , 1] ∪ [n , 1] repetition RD bicode is a bicode generated by the bimatrix G = G ∪ G = (11 … 1) ∪ (1 1 … 1) (G ≠ G ) over N F . Any non zero bicodeword has birank 1 ∪ We proceed onto define the notion of covering biradius. D EFINITION
Let C = C ∪ C be a linear [n , k ] ∪ [n , k ] RD bicode defined over N F . The covering biradius of C = C ∪ C is defined as the smallest pair of integers (r , r ) such that all bivectors in the rank bispace ∪ N N n n
F F are within the rank bidistance r ∪ r of some bicodeword. The covering biradius of C = C ∪ C is denoted by t(C ) ∪ t(C ). In notation, t(C) = t(C ) ∪ t(C ) { } N n max min r ( x c )x F c C ⎧ ⎫= +⎨ ⎬∈ ∈⎩ ⎭ { } N n max min r ( x c )x F c C ⎧ ⎫∪ +⎨ ⎬∈ ∈⎩ ⎭ . T HEOREM
The linear [n , k ] ∪ [n , k ] RD bicode C = C ∪ C satisfies t(C) = t(C ) ∪ t(C ) ≤ n – k ∪ n – k . Proof is direct. T HEOREM
The covering biradius of a [n , 1] ∪ [n , 1] repetition RD-bicode over N F is [n – 1] ∪ [n – 1]. Direct from theorem 2.8 as k = k = 1. Next we proceed on to define the Cartesian biproduct of two linear RD-bicodes. The Cartesian biproduct of two linear RD-bicodes C C [n , k ] C [n , k ] = ∪ and
D D [n , k ] D [n , k ] = ∪ over N F is given by C × D = C × D ∪ C × D . { } (a , b ) | a C , b D = ∈ ∈ ∪ { } (a , b ) | a C , b D ∈ ∈ . C × D is a { } (n n ) (n n ), (k k ) (k k ) + ∪ + + ∪ + linear RD bicode (We assume = (n n ) N + ≤ and (n n ) N + ≤ ). Now the reader is left with the task of proving the following theorem: T HEOREM
If C = C ∪ C and D = D ∪ D be two linear RD bicodes then t(C × D) ≤ (t(C ) + t(D ) ∪ t(C ) + t(D )). Hint for the proof. If C = C ∪ C and D = D ∪ D then C × D = {C × D } ∪ {C × D } and t(C × D ) = (C × D ) ∪ (C × D ) ≤ {t(C ) + t(D )} ∪ {t(C ) + t(D )}. . 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 N q F , n ≤ N and N > 1 with length n, dimension k and minimum distance d. If there exists m > 1 an integer such that m r(c;q) for all 0 ≠ c ∈ C then the code C is defined to be divisible. (r(x; q) denotes the rank norm of x over the field F q ). D EFINITION
Let C = C (n , k , d ) ∪ C (n , k , d ) be a linear RD bicode over N q F , n ≤ N, n ≤ N and N > 1. If there exists (m , m ) (m > 1 and m > 1) such that ( ) ; m r c q and ( ) ; m r c q for all c ∈ C and for all c ∈ C then we say the bicode C is bidivisible. T HEOREM
Let C = [n , 1 , n ] ∪ [n , 1 , n ] (n ≠ n ) be a MRD-bicode for all n ≤ N and n ≤ N; C is a bidivisible bicode. Proof:
Since there cannot exist bicodewords of birank greater than (n , n ) in an [n , 1, n ] ∪ [n , 1, n ] MRD-bicode. D EFINITION
Let C = [n , k ] be a linear RD-code and C = (n , k , d ) linear divisible RD-code defined over GF(2 N ). Then the RD-bicode C = C ∪ C is defined to be a semidivisible RD-bicode. D EFINITION
Let C = (n , k , d ) be a MRD code which is not divisible and C = (n , k , d ) a divisible RD code defined over GF(2 N ); then the bicode C = C ∪ C is defined to be a semidivisible MRD bicode. D EFINITION
Let C be a circulant rank code and C = (n , k , d ) a divisible RD code defined over GF(2 N ) then C = C ∪ C is defined to be semidivisible circulant bicode. D EFINITION
Let C be a AMRD code and C = (n , k , d ) be a divisible RD code defined over GF(2 N ) then C = C ∪ C 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. D EFINITION
Let C be a (n , k , d ) MRD code and C = (n , k , d ) MRD code over N q F ; n ≤ N and n ≤ N; (n ≠ n ). ( , ) ( , ) s s A n d A n d ∪ be the number of bicodewords with rank norms s and s in the linear (n , k , d ) MRD code and (n , k , d ) MRD code respectively. Then the bispectrum of the MRD bicode C ∪ C is described by the formulae; A (n , d ) ∪ A (n , d ) = 1 ∪ ( , ) ( , ) d m d m A n d A n d + + ∪ = ( ) ( )( ) ( ) m j m 1 1 1 1 jj n d m m -j m -j -1q Qd m d j + += +⎡ ⎤ ⎡ ⎤− −⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦ ∑ ( ) ( )( ) ( ) m j m 2 2 2 2 jj n d m m -j m -j -1q Qd m d j + += +⎡ ⎤ ⎡ ⎤∪ − −⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦ ∑ m = 0, 1, …, n – d , m = 0, 1, …, n – d , where ( )( ) ( )( )( ) ( )
11 11 n n n mm m m m q q q q qnm q q q q q −− − − −⎡ ⎤ =⎢ ⎥ − − −⎣ ⎦ and ( )( ) ( )( )( ) ( )
12 12 n n n mm m m m q q q q qnm q q q q q −− − − −⎡ ⎤ =⎢ ⎥ − − −⎣ ⎦ with Q = q N . Using the bispectrum of a MRD bicode we prove the following theorem: T HEOREM
All C [n , k , d ] ∪ C [n , k , d ] MRD bicodes with d < n and d < n (i.e., with k ≥
2 and k ≥
2) are non bidivisible. Proof:
This is proved by making use of the bispectrum of the MRD bicodes. Clearly d 1 1 d 2 2
A (n ,d ) A (n , d ) ∪ ≠ ∪
0. If the existence of a bicodeword with birank d + 1 ∪ d + 1 is established then the proof is complete as bigcd{(d , d + 1) ∪ (d , d + 1)} = 1 ∪
1. So the proof is to show that d 1 1 1 d 1 2 2
A (n ,d ) A (n ,d ) + + ∪ is non zero (i.e., d 1 1 1 A (n ,d ) 0 + ≠ and d 1 2 2 A (n ,d ) 0 + ≠ ). Now d 1 1 1 d 1 2 2 A (n ,d ) A (n ,d ) + + ∪ = n d 1(d 1 d +⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥+⎣ ⎦ ⎣ ⎦ [(Q –1) + (Q – 1)] ∪ n d 1(d 1 d +⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥+⎣ ⎦ ⎣ ⎦ [(Q –1) + (Q – 1)] = ( ) n d 1Q 1 Q 1d 1 d ⎛ ⎞+⎡ ⎤ ⎡ ⎤− + + −⎜ ⎟⎢ ⎥ ⎢ ⎥+⎣ ⎦ ⎣ ⎦⎝ ⎠ ( ) n d 1Q 1 Q 1d 1 d ⎛ ⎞+⎡ ⎤ ⎡ ⎤∪ − + + −⎜ ⎟⎢ ⎥ ⎢ ⎥+⎣ ⎦ ⎣ ⎦⎝ ⎠ . Suppose that ( ) ( ) d 1 d 1Q 1 Q 1 0 0d d + +⎡ ⎤ ⎡ ⎤+ − ∪ + − = ∪⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ . i.e., i d 1N q 1q 1 q 1 + −+ = − i.e., i dN 1 q 1q 1 q − −− = Clearly i dN 1 q 1 1q − − < . For if i dN 1 q 1 1q − − ≥ then q N–1 < i d q – 1, which is not possible as d i < n i ≤ N; i = 1, 2. Thus q – 1 < 1 which implies q < 2 a contradiction. Hence d 1 1 1 d 1 2 2
A (n ,d ) A (n ,d ) + + ∪ is non zero. Thus except C (n , 1, n ) ∪ C (n , 1, n ) MRD bicodes all C [n , k , d ] ∪ C [n , k , d ] MRD bicodes with d < n and d < n are non divisible. 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 → → →
0 and 0 →
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. D EFINITION [41, : Let n F denote the n-dimensional vector space of n-tuples over F . Let u, v ∈ n F where u = (u , …, u n ) and v = (v , …, v n ). 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 f u is the fuzzy subset of n F defined by f u = {(v, f u (v)) | v ∈ n F } where f u (v) is the membership function. (i) For the symmetric error model with the Hamming distance, , ( ) n d di i ui d u v f v p q −= = − = ∑ . (ii) For unidirectional error model, ( ) ( ) u m d d if min k kf v p q otherwise − ⎧ ≠= ⎨⎩ where, ( ) max 0, n i ii k u v = = − ∑ and ( ) max 0, n i ii k v u = = − ∑ k if kd k if k =⎧= ⎨ =⎩ ( )
21 11 1 2 n ii n ii i i u if km n u if ku n u if k k = = ⎧ =⎪⎪⎪= ⎨ − =⎪⎪ Σ − Σ = =⎪⎩ ∑ ∑
For the asymmetric error model ( ) u m d d if k kf v p q otherwise − ⎧ ≠= ⎨⎩ where d = k and m = Σ u i for asymmetric 1 →
0 error model and d = k and m = n – Σ u i for asymmetric 0 → D EFINITION
Asymmetric distance d a between u and v is defined as d a (u, v) = max(k , k ), for u, v ∈ n F . D EFINITION
The generalized Hamming distance between fuzzy sets is a metric in the set f n = {f u : u ∈ n F } that is ( , ) ( ) ( ) n u v u vz F d f f f z f z ∈ = − ∑ for f u , f v ∈ f n . Note that if u ∈ n2 F represents a received word and C is a codeword then f c (u) is the probability that c was transmitted. T HEOREM
Let u, v ∈ n F be such that d H (u, v) = d. If p ≠ q and p ≠
0, 1 then ( , ) d i d i d i iu v i dd f f p q p qi − −= ⎛ ⎞= −⎜ ⎟⎝ ⎠ ∑ for a symmetric error model. T HEOREM
Let u, v ∈ n F be such that d a (u, v) = d a . If p ≠ q and p ≠
0, 1 then ( , ) ( ) a du v d f f 2 1 q = − 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]. D EFINITION
Let V n denote the n-dimensional vector space of n-tuples over N F , n ≤ N and N > 1. Let u, v ∈ V n where u = (u , u , …, u n ) and v = (v , v , …, v n ) with each u i , v i ∈ n F . A fuzzy RD word f u is the fuzzy subset of V n defined by f u = {(v, f u (v)) | v ∈ V n } where f u (v) is the membership function. D EFINITION
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 f u (v) = p n–r q r where r = r(u – v; 2) = ||u – v||. D EFINITION
For the unidirectional and asymmetric error models assume q to represent the probability that 1 → → ( ) ( ) i nu u ii f v f v = = ∏ where ( ) i u i f v inherits its definition from the above equation for unidirectional and asymmetric error models respectively, since each u i or v i itself in an N-tuple over F . That is since u i , v i ∈ N F each u i or v i itself is an N-tuple from F . Let u i = (u i1 , u i2 , …, u iN ) and v j = (v j1 , v j2 , …, v jN ) where u is , v jt ∈ F , 1 ≤ s, t ≤ N. Then for unidirectional error model, ( ) ( ) i i i i i iu i m d d if min k ,kf v p q otherwise − ⎧ ≠= ⎨⎩ where, ( ) ni is iss k max ,u v = = − ∑ and ( ) ni is iss k max ,v u = = − ∑ i ii i i k if kd k if k =⎧= ⎨ =⎩ ( )
21 11 1 2 n is is ni is is is is i i u if km n u if kmax u ,n u if k k . = = ⎧ =⎪⎪⎪= ⎨ − =⎪⎪ Σ − Σ = =⎪⎩ ∑ ∑ . For the asymmetric error model ( ) i i i i i iu i m d d if min k ,kf ( v ) p q otherwise − ⎧ ≠= ⎨⎩ where d i = k i1 and Ni iss m u = = ∑ for asymmetric 0 → i = k i2 and ni iss m n u = = − ∑ for asymmetric 0 → D EFINITION
Let f n = {f u : u ∈ V n }. Let ψ : V n → f n be defined as ψ (u) = f u . Clearly ψ is a bijection. Let C[n, k, d] be an RD code which is a subspace of V n of dimension k and minimum rank distance d. Then ψ (C) ⊆ f n is a fuzzy RD code. If c ∈ C then f c is a fuzzy RD codeword of ψ (C). For any RD code ψ (C) its minimum distance is defined as, { } ( ) a b min a b a bf , f C d ( ( C )) min d( f , f ) : f f ψ ψ ∈ = ≠ where ( ) ( ) ( ) , n a b a bz V d f f f z f z ∈ = − ∑ is a metric in f n . D EFINITION
If u ∈ V n represents a received codeword and c ∈ C then f c (u) gives the probability that c was transmitted. Let θ (u) = {f a | a ∈ C, f a (u) ≥ f b (u), b ∈ C}. A code for which | θ (u)| = 1 for all u ∈ V n is said to be uniquely decodable. In such a case u is decoded as ψ –1 ( θ (u)). Now we proceed on to define the notion of fuzzy RD bicodes. D EFINITION
Let n n
V V ∪ denote (n , n ) dimensional vector bispace of (n , n )-tuples over N F ; n ≤ N and n ≤ N, N > 1. Let , , , n n u v V u v V ∈ ∈ where ( ,..., ), ( ,..., ) n n u u u v v v = = , ( ,..., ) n u u u = and ( ,..., ) n v v v = with , , , N i i i i u u v v F ∈ . A fuzzy RD bicodeword u u u u f f f ∪ = ∪ is a fuzzy bisubset of n n V V ∪ defined by, u u u u f f f ∪ = ∪ = {( , ( )) | } {( , ( )) | } n nu u v f v v V v f v v V ∈ ∪ ∈ where ( ) ( ) u u f v f v ∪ is the membership bifunction. D EFINITION
For the symmetric error bimodel assume p ∪ p to represent the biprobability that no transition (i.e., error) is made and q ∪ q to represent the biprobability that a birank error occurs so that p + q ∪ p + q = ∪
1. Then ( ) ( ) n r r n r ru u f v f v p q p q − − ∪ = ∪ where r = r (u – v , 2) = ||u – v || and r = r (u – v , 2) = ||u – v ||. D EFINITION
For unidirectional and asymmetric error bimodels assume q ∪ q to represent the probability that (1 → ∪ (1 →
0) bitransition or (0 → ∪ (0 →
1) bitransition occurs. Then ( ) ( ) ( ) ( ) i i n nu u i iu ui i f v f v f v f v = = ∪ = ∪ ∏ ∏ where ( ) ( ) u u f v f v ∪ inherits its definition from the unidirectional and asymmetric bimodels respectively since each i i u u ∪ or i i v v ∪ itself is an N-bituple over F . That is since , , , N i i i i u u v v F ∈ each i i u u ∪ or i i v v ∪ itself is an N-tuple from F . ( , ,..., ) i i i iN u u u u = , ( , ,..., ) j j j jN v v v v = , ( , ,..., ) i i i iN u u u u = , and ( , ,..., ) j j j jN v v v v = where , , , N is is jt jt u u v v F ∈ , 1 ≤ s, t ≤ N. Then for unidirectional error bimodel, ( ) ( ) ( ) ( ) − − ⎧ ∪ ∪ ≠ ∪⎪∪ = ⎨ ∪⎪⎩ where ( ) N1 1 1i1 is iss 1 k max 0,u v = = − ∑ , ( ) N2 1 1i1 is iss 1 k max 0,v u = = − ∑ ( ) N1 2 2i2 is iss 1 k max 0,u v = = − ∑ and ( ) N2 2 2i2 is iss 1 k max 0,v u = = − ∑ ; where k if k 0d k if k 0 ⎧ =⎪= ⎨ =⎪⎩ k if k 0d k if k 0 ⎧ =⎪= ⎨ =⎪⎩ ( ) N 1 1is i2s 1 N1 1 1i is i1s 1 1 1 1 1is is i1 i2 u if k 0m N u if k 0max u ,N u if k k 0 = = ⎧ =⎪⎪⎪= ⎨ − =⎪⎪ Σ − Σ = =⎪⎩ ∑ ∑ and ( )
N 2 2is i2s 1 N2 2 2i is i1s 1 2 2 2 2is is i1 i2 u if k 0m N u if k 0max u ,N u if k k 0 = = ⎧ =⎪⎪⎪= ⎨ − =⎪⎪ Σ − Σ = =⎪⎩ ∑ ∑ for the asymmetric error bimodel ( ) ( ) ( ) ( ) − − ⎧ ∪ ∪ ≠ ∪⎪∪ = ⎨ ∪⎪⎩ where , i i i i d k d k = = and Ni iss m u = = ∑ , Ni iss m u = = ∑ for asymmetric (1 → ∪ (1 →
0) error bimodel and d k , d k = = , N1 1i iss 1 m N u = = − ∑ and N2 2i iss 1 m N u = = − ∑ for the asymmetric (1 → ∪ ( →
1) error bimodel.
The minimum bidistance of a fuzzy RD bicode. D EFINITION
Let n n n n1 2u 1 u 2 f f { f u V } { f u V } ∪ = ∈ ∪ ∈ . Let n n n n1 2 :V V f f ψ ψ ψ = ∪ ∪ → ∪ defined by ( u ) ( u ) f f ψ ψ ∪ = ∪ . Clearly ψ is a bijection and ψ is a bijection. Let C [n , k , d ] ∪ C [n , k , d ] be an RD bicode which is a subbispace of n n V V ∪ of bidimension k ∪ k and minimum rank bidistance d ∪ d . Then n n1 1 2 2 ( C ) ( C ) f f ψ ψ ∪ ⊆ ∪ is a fuzzy RD bicode. If c = c ∪ c ∈ C ∪ C then c c f f ∪ is a fuzzy bicodeword of ( C ) ( C ) ψ ψ∪ . For any fuzzy RD bicode ( C ) ( C ) ψ ψ∪ , its minimum bidistance is defined as, min 1 1 min 2 2 d ( C ) d ( C ) ψ ψ∪ = ( ) ( ) { } min d f , f f , f C ; f f ψ∈ ≠ ∪ ( ) { } min d( f , f ) f , f C ; f f ψ∈ ≠ . where ( ) ( ) d f , f d f , f ∪ = ( ) ( ) ( ) ( ) f z f z f z f z ∈ ∈ − ∪ − ∑ ∑ is a bimetric in n n f f ∪ . D EFINITION
If u ∪ u ∈ n n V V ∪ represents a received biword and c ∪ c ∈ C ∪ C then ( ) ( ) f u f u ∪ gives the biprobability that (c ∪ c ) was transmitted. θ (u ) ∪ θ (u ) = ( ) ( ) { } | ; , a a b f a C f u f u b C ∈ ≥ ∈ ∪ ( ) ( ) { } | ; , a a b f a C f u f u b C ∈ ≥ ∈ . A bicode for which | θ (u ) ∪ θ (u )| = | θ (u )| ∪ | θ (u )| = 1 ∪ ∈ n V and u ∈ n V ; is said to be uniquely bidecodable. In such a case u ∪ u is bicoded as ( ( )) ( ( )) u u ψ θ ψ θ − − ∪ . The notions related to m–covering radius of RD–codes can be analogously transformed from the notion of RD–bicodes. P ROPOSITION If C C ∪ and C C ∪ are RD bicodes with C C ∪ ⊆ C C ∪ (i.e., C ⊆ C and C ⊆ C ) then ( ) ( ) m m t C t C ∪ ≥ ( ) ( ) m m t C t C ∪ [ ( ) m t C ≥ ( ) m t C and ( ) ( ) m m t C t C ≥ ]. Proof:
Let n1 S V ⊆ with |S | = m and n2 S V ⊆ with |S | = m cov(C ,S ) cov(C ,S ) ∪ = { } { } min cov(x ,S ) x C min cov(x ,S ) x C ∈ ∪ ∈ ≤ { } { } min cov(x ,S ) x C min cov(x ,S ) x C ∈ ∪ ∈ cov(C ,S ) cov(C ,S ) = ∪ . Thus t (C ) t (C ) t (C ) t (C ) ∪ ≤ ∪ . P ROPOSITION
For any RD bicode C ∪ C and a pair of positive integers (m , m ), ( ) ( ) m m t C t C ∪ ≤ ( ) ( ) m m t C t C + + ∪ . Proof: Let n1 S V ⊆ with |S | = m and n2 S V ⊆ with |S | = m , n n V V ∪ is the rank bispace where S ∪ S is a bisubset of n n V V ∪ . Now m 1 m 2 t (C ) t (C ) ∪ = max{cov(C , S ) | S ⊆ n V ; |S | = m } ∪ max{cov(C , S ) | S ⊆ n V ; |S | = m } ≤ max{cov(C , S ) | S ⊆ n V ; |S | = m + 1} ∪ max{cov(C , S ) | S ⊆ n V ; |S | = m + 1} = m 1 1 m 1 2 t (C ) t (C ) + + ∪ . P ROPOSITION
For any biset of positive integers {n , m , k , K } ∪ {n , m , k , K }; [ , ] [ , ] m m t n k t n k ∪ ≤ [ , ] [ , ] m m t n k t n k + + ∪ and ( , ) ( , ) m m t n K t n K ∪ ( , ) ( , ) m m t n K t n K + + ≤ ∪ . Proof:
Given C [n , k ] ∪ C [n , k ] RD bicode, with n1 C V ⊆ and n2 C V ⊆ . Now m 1 1 m 2 2 t [n , k ] t [n , k ] ∪ = nm 1 1 1 1 min{ t (C ) C V ;dim C k } ⊆ = ∪ nm 2 2 2 2 min{t (C ) C V ;dim C k } ⊆ = ≤ nm 1 1 1 1 1 min {t (C ) C V ;dim C k } + ⊆ = ∪ nm 1 2 2 2 2 min{t (C ) C V ;dim C k } + ⊆ = m 1 1 1 m 1 2 2 t [n , k ] t [n , k ] + + = ∪ . Similarly we have m 1 1 m 2 2 t (n , K ) t (n , K ) ∪ m 1 1 1 m 1 2 2 t (n , K ) t (n , K ) + + ≤ ∪ . That is m 1 1 m 2 2 t (n , K ) t (n , K ) ∪ = nm 1 1 1 1 min{t (C ) C V ; C K } ⊆ = ∪ nm 2 2 2 2 min{t (C ) C V ; C K } ⊆ = nm 1 1 1 1 1 min{t (C ) C V ; C K } + ⊆ = ∪ nm 1 2 2 2 2 min{t (C ) C V ; C K } + ⊆ = m 1 1 1 m 1 2 2 t (n , K ) t (n , K ) + + ≤ ∪ . P ROPOSITION
For any biset of positive integers {n , m , k , K } ∪ {n , m , k , K }; [ , ] [ , ] m m t n k t n k ∪ [ , 1] [ , 1] m m t n k t n k ≥ + ∪ + and ( , ) ( , ) m m t n K t n K ∪ ( , 1) ( , 1) m m t n K t n K ≥ + ∪ + . Proof: Given C = C ∪ C is a RD bicode hence a bisubspace of n n V V ∪ . Consider m 1 1 m 2 2 t [n , k 1] t [n , k 1] + ∪ + = nm 1 1 1 1 min{t (C ) C V ;dim C k 1} ⊆ = + ∪ nm 2 2 2 2 min{t (C ) C V ;dim C k 1} ⊆ = + ≤ nm 1 1 1 1 min{t (C ) C V ;dim C k } ⊆ = ∪ nm 2 2 2 2 min{t (C ) C V ;dim C k } ⊆ = . (since for each C ∪ C ⊆ C ∪ C ; m 12 m 22 m 1 m 2 t (C ) t (C ) t (C ) t (C )) ∪ ≤ ∪ m 1 1 m 2 2 t (n , k ) t (n , k ) = ∪ . Similarly, m 1 1 m 2 2 t [n , K 1] t [n , K 1] + ∪ + m 1 1 m 2 2 t (n , K ) t (n , K ) ≤ ∪ . Using these results and the fact
1m 1 1 k [n , t ] denotes the smallest dimension of a linear RD code of length n and m -covering radius t and
1m 1 1 k [n , t ] denotes the least cardinality of the RD codes of length n and m -covering radius t . The following results can be easily proved. Result 1:
For any biset of positive integers {n , m , t } ∪ {n , m , t } and m 1 1 m 2 2 m 1 1 1 m 1 2 2 k (n , t ) k (n , t ) k (n , t ) k (n , t ) + + ∪ ≤ ∪ and m 1 1 m 2 2 K (n , t ) K (n , t ) ∪ ≤ m 1 1 1 m 1 2 2 K (n , t ) K (n , t ) + + ∪ . Result 2:
For any biset of positive integers {n , m , t } ∪ {n , m , t } we have m 1 1 m 2 2 m 1 1 m 2 2 k [n , t ] k [n , t ] k [n , t 1] k [n , t 1] ∪ ≥ + ∪ + and m 1 1 m 2 2 m 1 1 m 2 2 K (n , t ) K (n , t ) K (n , t 1) K (n , t 1) ∪ ≥ + ∪ + . We say a bifunction f ∪ f is a non-decreasing function in some bivariable say (x ∪ x ) if both f and f happen to be a non-decreasing function in the same variable x and x respectively. With this understanding we can say the (m , m )- covering biradius of a fixed RD bicode C ∪ C , m 1 1 m 2 2 t [n , k ] t [n , k ] ∪ , m 1 1 m 2 2 t (n , K ) t (n , K ) ∪ , m 1 1 m 2 2 k [n , t ] k [n , t ] ∪ and m 1 1 m 2 2 K (n , t ) K (n , t ) ∪ , are non decreasing bifunctions of (m , m ). The relationship between the multicovering biradii of two RD bicodes and bicodes that are built using them are described. Let i i i1 2 C C C = ∪ , i = 1, 2 be a [n , k , d ] [n , k , d ] ∪ , [n , k , d ] [n , k ,d ] ∪ RD bicodes over N F with n , n , n , n , n n , n n N + + ≤ . P ROPOSITION
Let = ∪
C C C and = ∪
C C C be RD bicodes described above. = × = × ∪ ×
C C C C C C C { } { } ( | ) , ( | ) , = ∈ ∈ ∪ ∈ ∈ x y x C y C x y x C y C . Then × C C is a [ , , + ∪ + + ∪ + n n n n k k k k min{ d ,d } ∪ min{ d ,d }] rank distance bicode over N F and ( ) ( ) × ∪ × t C C t C C ≤ ( ) ( ) ( ) ( ) + ∪ + t C t C t C t C . Proof: Let n n1
S V + ⊆ and n n2 S V + ⊆ and S {s , , s } = … and S {s ,...,s } = with
1i 1i 1i s (x y ) = and
2i 2i 2i s (x y ) = n n1i 1i x V , y V ∈ ∈ , n2i x V ∈ and n2i y V ∈ . Let S {x x }, S {y y } = = … … , S {x x } and S {y y } = = … … . Now
1m 1 t (C ) being the m -covering radius of (C ) there exists c C ∈ such that
1m 11
S B (C ) ⊆ . This implies
11 1i 1 r (x c ) + ≤ m 1 t (C ) for all
11i 1 x S ∈ . The same argument is true for (C ) . Now consider (C ) , this code has m covering radius
1m 2 t (C ) such that there exists c C ∈ such that
1m 22
S B (C ) ⊆ . This implies
12 2i 2 m 2 r (x c ) t (C ) + ≤ for all
12i 2 x S ∈ . Now C (C | C ) (C | C ) C C = ∪ = ∪ . Here r (s C ) r ((x | y ) (C | C )) + = + r (x C | y C ) r (x C ) r (y C ) = + + ≤ + + + t (C ) t (C ) ≤ + . Similarly we have, r (s C ) r ((x | y ) (C | C )) + = + r (x C | y C ) r (x C ) r (y C ) = + + ≤ + + + t (C ) t (C ) ≤ + . Thus t (C) t (C C ) t (C C ) = × ∪ × ≤ t (C ) t (C ) + ∪ t (C ) t (C ) + . For any positive integer r, the r-fold repetition RD code C is the code C = {(c | c | … | c) | c ∈ C } where the code word c is concatenated r-times. This is a [rn , k , d ] rank distance code. Note that here n ≤ N is choosen so that rn ≤ N. We proceed on to define (r, r)-fold repetition of RD bicode. D EFINITION
For any (r, r) (r any positive integer), the (r, r) repetition of RD bicode C ∪ D is the bicode C = {(c | c | … | c) | c ∈ C } ∪ D = {(d | d | … | d) | d ∈ D } where the bicode word c ∪ d is concatenated r-times this is a [rn , k , d ] ∪ [rn , k , d ] rank distance bicode word with n i ≤ N and rn i ≤ N; i = 1,
2. Thus any bicode word in C ∪ D would be of the form (c | c | … | c) ∪ (d | d | … | d) where c ∈ C and d ∈ D . We can also define (r , r ) fold repetition bicode (r ≠ r ). D EFINITION
Let C ∪ D be a [n , k , d ] ∪ [n , k , d ] RD-code. Let C = {(c | c | … | c) | c ∈ C } be a r -fold repetition RD code C and D = {(d | d | … | d) | d ∈ D } be a r -fold repetition RD code C (r ≠ r ). Then C ∪ D is defined as the (r , r )-fold repetition bicode. We prove the following interesting result. P ROPOSITION
For an (r, r) fold repetition RD-bicode , ( ) ( ) ( ) ( ) ∪ ∪ = ∪ m m m 1 m 1
C D t C t D t C t D . Proof:
Let n1 1 m S {v v } V = ⊆ … be such that cov(C , S ) = m 1 t (C ) . Let n2 1 m S {u u } V = ⊆ … be such that cov(D , S ) = m 1 t (D ) . Let
1i i i i v (v | v | | v ) = … . Let S {v | v | | v } = … be a set of m -vectors of length rn each. A r-fold repetition of any RD code word retains the same rank weight. Hence
11 m 1 (C,S ) t (C ) = . Since
1m 1 t (C) cov(C,S ) ≥ , it follows that m m 1 t (C) t (C ) ≥ ------ I Conversely let S = {v … m v } be a set of m-vectors of length rn with v (v | v | ... | v ) = ; n1i v V ∈ . Then there exists c ∈ C such that
1R i m 1 d (c, v ) t (C ) ≤ for every i (1 ≤ i ≤ m ). This implies R i m 1 d ((c | c | ... | c), v ) t (C ) ≤ for every i (1 ≤ i ≤ m ). Thus m m 1 t (C) t (C ) ≤ ------ II From I and II, m m 1 t (C) t (C ) = . On similar lines we can prove, ( ) ( ) m m 1 t D t D = where cov(D ,S ) t (D ) = . Hence m m m 1 m 1 t (C) t (D) t (C ) t (D ) ∪ = ∪ . Multi-covering bibounds for RD-bicodes is discussed and a few interesting properties in this direction are given. The (m , m ) covering biradius m 1 m 2 t (C ) t (C ) ∪ of a RD-bicode C = C ∪ C is a non-decreasing bifunction of m ∪ m (proved earlier). Thus a lower bi-bound for m 1 m 2 t (C ) t (C ) ∪ implies a bibound for m 1 1 m 1 2 t (C ) t (C ) + + ∪ . First bibound exhibits m ∪ m ≥ ∪ , m )-covering biradii is quite different for ordinary covering radii. P ROPOSITION
If m ∪ m ≥ ∪
2 then the (m , m )-covering biradii of a RD bicode C = C ∪ C of bilength (n , n ) is atleast ⎡ ⎤ ⎡ ⎤∪⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ n n2 2 . Proof: Let C = C ∪ C be a RD bicode of bilength (n , n ) over GF(2 N ). Let m ∪ m ≥ ∪
2, let t , t be the 2-covering biradii of the RD code C = C ∪ C . Let x = x ∪ x n n V V ∈ ∪ . Choose n n1 2 y y y V V = ∪ ∈ ∪ such that all the (n , n ) coordinates of x y (x y ) (x y ) − = − ∪ − are linearly independent, that is R R 1 2 1 2 d (x, y) d (x x , y y ) = ∪ ∪
R 1 1 R 2 2 d (x , y ) d (x , y ) = ∪ (R R R and d d d ) = ∪ = ∪ = n ∪ n . Then for any c c c C C C = ∪ ∈ = ∪ , R R d (x,c) d (c, y) + R 1 1 R 1 1 R 2 2 R 2 2 d (x ,c ) d (c , y ) d (x ,c ) d (c , y ) = + ∪ +
R 1 1 R 2 2 d (x , y ) d (x , y ) ≥ ∪ = n ∪ n . This implies that one of R 1 1 R 2 2 d (x ,c ) d (x ,c ) ∪ and R 1 1 R 2 2 d (c , y ) d (c , y ) ∪ is at least n n2 2 ∪ (that is one of R 1 1 d (x , c ) and R 1 1 d (c , y ) is atleast n2 and one of R 2 2 d (x , c ) and R 2 2 d (c , y ) is at least n2 ) and hence n nt t t 2 2 ⎡ ⎤ ⎡ ⎤= ∪ ≥ ∪⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ . Since t is non decreasing bifunction of m ∪ m it follows that n nt (C) t (C ) t (C ) 2 2 ⎡ ⎤ ⎡ ⎤= ∪ ≥ ∪⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ for m ∪ m ≥ ∪
2. Bibounds of the multi-covering biradius of n n
V V ∪ can be used to obtain bibounds on the multi covering biradii of arbitary bicodes. Thus a relationship between (m , m )-covering biradii of an RD bicode and that of its ambient bispace n n V V ∪ is established T HEOREM
Let C = C ∪ C be any RD-code of bilength n ∪ n over ∪ N N
F F . Then for any pair of positive integers (m , m ); ( ) ( ) ( ) ( ) ( ) ( ) ∪ ≤ + ∪ + n n1 2 1 1 2 2m 1 m 2 1 1 m 1 2 m t C t C t C t V t C t V . Proof:
Let n n1 2
S S S V V = ∪ ⊆ ∪ (i.e., n1 S V ⊆ and n2 S V ⊆ ) with |S| = |S | ∪ |S | = m ∪ m . Then there exists u = u ∪ u ∈ n n V V ∪ such that n n1 11 1 2 2 m m cov(u,S) cov(u ,S ) cov(u ,S ) t (V ) t (V ) = ∪ ≤ ∪ . Also there is a c c c C C = ∪ ∈ ∪ such that )C(t)C(t)u,c(d)u,c(d)u,c(d ∪≤∪= . Now cov(c, S) = cov(c , S ) ∪ cov(c , S ) { } { } R 1 1 1 1 R 2 2 2 2 max d (c , y ) y S max d (c , y ) y S = ∈ ∪ ∈ { }
R 1 1 R 1 1 1 1 max d (c , u ) d (u , y ) y S ≤ + ∈ ∪ { }
R 2 2 R 2 2 2 2 max d (c , u ) d (u , y ) y S + ∈
R 1 1 1 1 R 2 2 2 2 d (c , u ) cov(u ,S ) d (c , u ) cov(u ,S ) = + ∪ + n n1 1 2 21 1 m 1 2 m t (C ) t (V ) t (C ) t (V ) ≤ + ∪ + . Thus for every n n1 2
S S S V V = ∪ ⊆ ∪ with |S| = m = |S | ∪ |S | = m ∪ m one can find a c c c C C = ∪ ∈ ∪ such that, cov(c, S) = cov(c , S ) ∪ cov(c , S ) n n1 1 2 21 1 m 1 2 m t (C ) t (V ) t (C ) t (V ) ≤ + ∪ + . Since cov(c, S) = cov(c , S ) ∪ cov(c , S ) { } { } min cov(a ,S ) a C min cov(a ,S ) a C = ∈ ∪ ∈ { } { } n n1 1 2 21 1 m 1 2 m t (C ) t (V ) t (C ) t (V ) ≤ + ∪ + ; for all n n1 2 S S S V V = ∪ ⊆ ∪ with |S| = |S | ∪ |S | = m ∪ m , it follows that { }
11 2 n1 2m 1 m 2 1 1 1 1 1 t (C ) t (C ) max cov(C ,S ) S V ; S m ∪ = ⊆ = ∪ { } n2 2 2 2 2 max cov(C ,S ) S V ; S m ⊆ = { } { } n n1 1 2 21 1 m 1 2 m t (C ) t (V ) t (C ) t (V ) ≤ + ∪ + . P ROPOSITION
For any pair of integers (n , n ); n ∪ n ≥ ∪ ( ) ( ) ∪ ≤ n n1 22 2 t V t V n – 1 ∪ n – 1; where = n n2 V F , = n n2 V F ; n ≤ N and n ≤ N. Proof:
Let )x,...,x,x(x = , n1n12111 V)y,...,y,y(y ∈= and )x,...,x,x(x = , n2n22212 V)y,...,y,y(y ∈= . Let n n1 2 u u u V V = ∪ ∈ ∪ where u (x u u u y ) − = … and u (x u u u y ) − = … . Thus u = u ∪ u bicovers x ∪ x and n n1 2 y y V V ∪ ∈ ∪ with in a biradius n – 1 ∪ n – 1 as R 1 1 1 d (u , x ) n 1 ≤ − and R 2 2 2 d (u , x ) n 1 ≤ − and R 1 1 1 d (u , y ) n 1 ≤ − and R 2 2 2 d (u , y ) n 1 ≤ − . Thus for any pair of bivectors x ∪ x and y ∪ y in n n V V ∪ there always exists a bivector namely u = u ∪ u which bicovers x ∪ x and y ∪ y within a biradius n – 1 ∪ n – 1. Hence n n1 22 2 1 2 t (V ) t (V ) (n 1 n 1) ∪ ≤ − ∪ − . Now we proceed on to describe the notion of generalized sphere bicovering bibounds for RD bicodes. A natural question is for a given t ∪ t , m ∪ m and n ∪ n what is the smallest RD bicode whose m ∪ m bicovering biradius is atmost t ∪ t . As it turns out even for m ∪ m ≥ ∪
2, it is necessary that t ∪ t be atleast n n2 2 ∪ . Infact the minimal t ∪ t for which such a bicode exists is the (m , m ) bicovering biradius of C ∪ C = n n2 2 F F ∪ . Various external values associated with this notion are n n1 2m m t (V ) t (V ) ∪ the smallest (m , m )-covering biradius among bilength n ∪ n RD bicodes t (n , K ) t (n , K ) ∪ , the smallest (m , m ) covering biradius among all (n , K ) (n , K ) ∪ RD bicodes.
K (n , t ) K (n , t ) ∪ is the smallest bicardinality of bilength n ∪ n RD bicode with m ∪ m covering biradius t ∪ t and so on. It is the latter quality that is studied in the book for deriving new lower bibounds. From the earlier results K (n , t ) K (n , t ) ∪ is undefined if n nt t 2 2 ∪ < ∪ . When this is the case, it is accepted to say K (n , t ) K (n , t ) ∪ = ∞ ∪ ∞ . There are other circumstances when K (n , t ) K (n , t ) ∪ is undefined. For instance ( ) ( ) Nn Nn1 2
K n , n 1 K n , n 1 − ∪ − = ∞∪∞ . Also K (n , t ) K (n , t ) ∪ = ∞∪∞ , ( )
11 1 m V n , t > and ( )
22 2 m V n , t > , since in this case no biball of biradius t ∪ t covers any biset of m ∪ m distinct bivectors. More generally one has the fundamental issue of whether K (n , t ) K (n , t ) ∪ is bifinite for a given n , m , t and n , m , t . This is the case if and only if t)V(t ≤ and t)V(t ≤ , since )V(t)V(t n2mn1m ∪ lower bibounds the (m , m ) covering biradii of all other bicodes of bidimension n ∪ n . When t ∪ t = n ∪ n every bicode word bicovers every bivector, so a bicode of size 1 ∪
1 will (m , m ) bicover n n V V ∪ for every m ∪ m . Thus K (n , n ) K (n , n ) ∪ = 1 ∪ ∪ m . If t ∪ t is n – 1 ∪ n – 1 what happens to K (n , t ) ∪ K (n , t ) ? When m = m = 1, K (n , n 1) K (n , n 1) 1 L (n ) 1 L (n ) − ∪ − ≤ + ∪ + . For 0 0 ∪ = (0 … 0) ∪ (0 … 0) will cover all bivectors of birank binorm less than or equal n – 1 ∪ n – 1 within biradius n – 1 ∪ n – 1. That is 0 0 ∪ = (0, 0, …, 0) ∪ (0, 0, …, 0) will bicover all binorm n – 1 ∪ n – 1 bivectors within the biradius n – 1 ∪ n – 1. Hence remaining bivectors are rank n ∪ n bivectors. Thus 0 0 ∪ = (0, 0, …, 0) ∪ (0, 0, …, 0) and these birank-(n ∪ n ) bivectors can bicover the ambient bispace within the biradius n – 1 ∪ n – 1. Therefore K (n , n 1) K (n , n 1) 1 L (n ) 1 L (n ) − ∪ − ≤ + ∪ + . P ROPOSITION
For any RD bicode of bilength n ∪ n over ∪ N N
F F ( , ) ( , ) ( ) ( ) − ∪ − ≤ + ∪ +
K n n 1 K n n 1 m L n 1 m L n 1 provided m ∪ m is such that ( ) ( ) + ∪ + ≤ + n n1 n 1 2 n 2 m L n 1 m L n 1 V V . Proof:
Consider a RD-bicode C = C ∪ C such that |C| = |C | ∪ |C | = m L (n ) 1 m L (n ) 1 + ∪ + . Each bivector in n V ∪ n V has n 1 n 2
L (n ) L (n ) ∪ rank complements, that is from each bivector n n1 2 v v V V ∪ ∈ ∪ ; there are n 1 L (n ) ∪ n 2 L (n ) bivectors at rank bidistance n ∪ n . This means for any set n n1 2 S S V V ∪ ⊆ ∪ of (m , m ) bivectors there always exists a c ∪ c ∈ C ∪ C which bicovers S ∪ S birank distance n – 1 ∪ n – 1. Thus, cov(c , S ) ∪ cov(c , S ) ≤ n – 1 ∪ n – 1 which implies cov(C , S ) ∪ cov(C , S ) ≤ n – 1 ∪ n – 1. Hence K (n , n 1) K (n , n 1) m L (n ) 1 m L (n ) 1. − ∪ − ≤ + ∪ +
By bounding the number of (m , m ) bisets that can be covered by a given bicode word, one obtains a straight forward generalization of the classical sphere bibound. T HEOREM (Generalized Sphere Bound for RD bicodes) For any (n , K ) ∪ (n , K ) RD bicode C = C ∪ C , ( , ( )) ( , ( )) ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞∪ ≥ ∪⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ n n1 21 2 N N1 m 1 2 m 21 21 2 1 2
V n t C V n t C 2 2K Km m m m . Hence for any n , t and m , n , t and m ( , ) ( , ) ( , ) ( , ) ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∪ ≥ ∪⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ n n1 21 2 N N1 21 2m 1 1 m 2 2 1 1 2 21 2 where ( , ) ( , ) ( ) ( ) = = ∪ = ∪ ∑ ∑ t t1 21 1 1 1 i 1 i 2i 0 i 0 V n t V n t L n L n , number of bivectors in a sphere of biradius t ∪ t and ( ) ( ) ∪ L n L n is the number of bivectors in ∪ n n V V whose rank binorm is i ∪ i . Proof: Each set of (m , m ) bivectors in n n V V ∪ must occur in a sphere of biradius m 1 m 2 t (C ) t (C ) ∪ around at least one code biword. Total number of such bisets is n n V V ∪ , choose m ∪ m , where n n1 2 1 2 N Nn n
V V 2 2 ∪ = ∪ . The number of bisets of (m , m ) bivectors in a neighborhood of biradius m 1 m 2 t (C ) t (C ) ∪ is V(n , t (C )) V(n , t (C )) ∪ . Choose m ∪ m . There are K code biwords. Hence n n1 21 2 N N1 m 1 2 m 21 21 2 1 2
V(n , t (C )) V(n , t (C )) 2 2K Km m m m ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞∪ ≥ ∪⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ . Thus for any n ∪ n , t ∪ t and m ∪ m , n n1 21 2 N N1 21 2m 1 1 m 2 2 1 1 2 21 2 ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∪ ≥ ∪⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ . C OROLLARY If ( , ) ( , ) ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞∪ > ∪⎢ ⎥ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ n n1 2 n n1 2 N N N N1 1 2 21 21 2
V n t V n t2 2 2 2m mm m , then ( , ) ( , ) ∪ = ∞ ∪ ∞
K n t K n t . Chapter Three R ANK D ISTANCE m-C
ODES
In this chapter we introduce the new notion of rank distance m-codes and describe some of their properties. D EFINITION
Let C = C [n , k ], C = C [n , k ], …, C m = C m [n m , k m ], be m distinct RD codes such that C i = C i [n i , k i ] ≠ C j = C j [n j , k j ] if i ≠ j and C i = C i [n i , k i ] ⊄ C j = C j [n j , k j ] or C j = C j [n j , k j ] ⊆ C i = C i [n i , k i ] for 1 ≤ i, j ≤ m if i ≠ j; be subspaces of the rank spaces , ,..., m nn n V V V over the field GF(2 N ) or N q F where n , n , …, n m ≤ N, i.e., each n i ≤ N for i = 1, 2, …, m. C = C [n , k ] ∪ C [n , k ] ∪ … ∪ C m [n m , k m ] is defined as the Rank Distance m-code (m ≥ ... ∪ ∪ ∪ m nn n V V V to be (n , n , …, n m ) dimensional vector m-space over the field N q F . So we can say C ∪ C ∪ … ∪ C m is a m-subspace of the vector m-space ... ∪ ∪ ∪ m nn n V V V . We represent any element of ∪ n n V V ∪ … ∪ m n V by x ∪ x ∪ … ∪ x n = ( , ,..., ) ( , ,..., ) ... ( , ,..., ) ∪ ∪ ∪ m m m mn n n x x x x x x x x x , where ∈ nj ii q x F ; 1 ≤ j ≤ m and 1 ≤ i j ≤ n j ; j = 1, 2, 3, …, m. Also N q F can be considered as a pseudo false m-space of dimension ( , , ) − …(cid:8)(cid:11)(cid:9)(cid:11)(cid:10) m times N N over F q . Thus elements x i, y i ∈ N q F has N - m-tuple representation as ∪ααα ),...,,( ...),...,,( ∪ααα ),...,,( mNjmi2mj1 m ααα∪ over F q with respect to some m-basis. Hence associated with each x ∪ x ∪ … ∪ x m ∈ n n n V V ... V ∪ ∪ ∪ (n i ≠ n j if i ≠ j, 1 ≤ i, j ≤ m) there is a m-matrix. m (x ) ... m (x ) ∪ ∪ = a a a aa a a aa a a a ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥∪⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ … …… …(cid:35) (cid:35) (cid:35) (cid:35)(cid:34) (cid:34) ∪ … ∪ mmm m m11 1nm m21 2nm mNm Nn a aa aa a ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ……(cid:35) (cid:35)(cid:34) where th th th1 2 m i i ... i ∪ ∪ ∪ m-column represents the th th th1 2 m i i ... i ∪ ∪ ∪ m–coordinate of x x ... x ∪ ∪ ∪ of x ∪ x ∪ … ∪ x m over F q . It is important and interesting to note in order to develop the new notion rank distance m-codes (m ≥
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,
Z Z ... Z ∪ ∪ ∪ is a false pseudo m-vector space over Z . Z Z Z Z Z Z ∪ ∪ ∪ ∪ ∪ is a false pseudo 6-vector space over Z . However we will in this book use only m-vector spaces over Z or N Z and denote it by GF(2) or GF(2 N ) or N F , unless otherwise specified. Now we see every x ∪ … ∪ x m in the m-vector space n n n V V ... V ∪ ∪ ∪ have an associated m-matrix m (x ) ∪ … ∪ m m (x m ). We proceed to define m- rank of the m- matrix over F q or GF(2). D EFINITION
The m-rank of an element x ∪ x ∪ … ∪ x m ∈ ... ∪ ∪ ∪ m nn n V V V is defined as the m-rank of the m-matrix m(x ) ∪ m(x ) ∪ … ∪ m(x m ) over GF(2) or F q [i.e., the m-rank of m(x ) ∪ m(x ) ∪ … ∪ m(x m ) is the rank of m(x ) ∪ rank of m(x ) ∪ … ∪ rank of m(x m )]. We shall denote the m-rank of x ∪ x ∪ … ∪ x m by r (x ) ∪ r (x ) ∪ … ∪ r m (x m ), we can in case of m-rank of a m-matrix prove the following; (i) For every x ∪ x ∪ … ∪ x m ∈ ... ∪ ∪ ∪ m nn n V V V (x i ∈ i n V , 1 ≤ i ≤ m) we have, r(x ∪ … ∪ x m ) = r (x ) ∪ … ∪ r m (x m ) ≥ ∪ ∪ … ∪ i (x i ) ≥ ∈ i ni x V ; i = 1, 2, 3, …, m). (ii) r(x ∪ x ∪ … ∪ x m ) = r (x ) ∪ r (x ) ∪ … ∪ r m (x m ) = ∪ ∪ … ∪ i = 0 for i = 1, 2, 3, …, m. [( ) ( ) ... ( )] + ∪ + ∪ ∪ + r x x x x x x ( ) ( ) ( ) ( ) ... ( ) ( ) ≤ + ∪ + ∪ ∪ + r x r x r x r x r x r x for every , ∈ i ni i1 2 x x V ; i = 1, 2, 3, …, m. That is we have, (iii) [( ) ( ) ... ( )] + ∪ + ∪ ∪ + r x x x x x x ( ) ( ) ... ( ) = + ∪ + ∪ ∪ + r x x r x x r x x ( ) ( ) ( ) ( ) ... ( ) ( ) ≤ + ∪ + ∪ ∪ + m mm m r x r x r x r x r x r x (as we have for every , ∈ i ni i x x V ; ( ) ( ) ( ) + ≤ + i i i ii i i r x x r x r x for i = 1, 2, 3, …, m). (iv) r (a x ) ∪ r (a x ) ∪ … ∪ r m (a m x m ) = |a |r (x ) ∪ |a |r (x ) ∪ … ∪ |a m |r m (x m ) for every a , a , …, a m ∈ F q or GF(2) and for every ∈ i ni x V ; i = 1, 2, 3, …, m. Thus the m-function x ∪ x ∪ … ∪ x m → r (x ) ∪ r (x ) ∪ … ∪ r m (x m ) defines a m-norm on ... ∪ ∪ ∪ m nn n V V V . D EFINITION
The m-metric induced by the m- rank m- norm is defined as the m- rank m-metric on ... ∪ ∪ ∪ m nn n V V V and is denoted by ∪ ∪ ∪ … m R R R d d d . If ... , ∪ ∪ ∪ m x x x ∪ y y ∪ … ∪ ∈ ∪ ∪ ∪ … m nn nm y V V V then the m-rank m-distance between ... ∪ ∪ ∪ x x x and ... ∪ ∪ ∪ y y y is ( , ) ( , ) ... ( , ) ∪ ∪ ∪ = d x y d x y d x y ( ) ( ) ... ( ) r x y r x y r x y − ∪ − ∪ ∪ − for every , ∈ i n1 1i i x y V , i = 1, 2, 3, …, m (Here ( , ) = i d x y ( ) − r x y for every , ∈ i n1 1i i x y V ; for i = 1, 2, 3, …, m). D EFINITION
A linear m-space ... ∪ ∪ ∪ m1 2 nn n
V V V over GF(2 N ), N > 1 of m-dimension n ∪ n ∪ … ∪ n m such that n i ≤ N for i = 1, 2, 3, …, m equipped with the m-rank m-metric is defined as the m-rank m-space. D EFINITION
A m-rank m-distance RD m-code of m-length n ∪ n ∪ … ∪ n m over GF(2 N ) is a m-subset of the m-rank m-space ∪ ∪ ∪ … m1 2 nn n V V V over GF(2 N ). D EFINITION
A linear [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] RD-m-code is a linear m-subspace of m-dimension k ∪ k ∪ … ∪ k m in the m-rank m-space ∪ ∪ ∪ … m1 2 nn n V V V . By C [n , k ] ∪ C [n , k ] ∪ … ∪ C m [n m , k m ] we denote a linear [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] RD m-code. We can equivalently define a RD m-code as follows: D EFINITION
Let , , , … m1 2 nn n V V V be rank spaces n i ≠ n j if i ≠ j over GF(2 N ), N >1. Suppose ⊆ i ni P V , i = 1, 2, 3, …, m be subset of the rank spaces over GF(2 N ). Then P ∪ P ∪ … ∪ P m ... ⊆ ∪ ∪ ∪ m1 2 nn n V V V is a rank distance m-code of m-length (n , n , …, n m ) over GF(2 N ). D EFINITION
A generator m-matrix of a linear [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] RD-m-code C ∪ C ∪ … ∪ C m is a k × n ∪ k × n ∪ … ∪ k m × n m , m-matrix over GF(2 N ) whose m-rows form a m-basis for C ∪ C ∪ … ∪ C m . A generator m-matrix G = G ∪ G ∪ … ∪ G m of a linear RD m-code C [n , k ] ∪ C [n , k ] ∪ … ∪ C m [n m , k m ] can be brought into the form G = G ∪ G ∪ … ∪ G m [ , ] × − = k k n k I A [ , ] × − ∪ k k n k I A ... [ , ] × − ∪ ∪ m m m m k k n k
I A where , ,..., m k k k I I I is the identity matrix and × − i i i k n k
A is some matrix over GF(2 N ); i = 1, 2, 3, …, m; this form of G = G ∪ G ∪ … ∪ G m is called the standard form. D EFINITION
If G = G ∪ G ∪ … ∪ G m is a generator m-matrix of C [n , k ] ∪ C [n , k ] ∪ … ∪ C m [n m , k m ] then a m-matrix H = H ∪ H ∪ … ∪ H m of order (n – k × n , n – k × n , …, n m – k m × n m ) over GF(2 N ) such that, GH T = ( G ∪ G ∪ … ∪ G m )(H ∪ H ∪ … ∪ H m ) T = ( G ∪ G ∪ … ∪ G m ) ( ... ) ∪ ∪ ∪ T T T1 2 m
H H H ... = ∪ ∪ ∪
T T T1 1 2 2 m m
G H G H G H = 0 ∪ ∪ … ∪
0 is called a parity check m-matrix of C [n , k ] ∪ C [n , k ] ∪ … ∪ C m [n m , k m ]. Suppose C = C ∪ C ∪ … ∪ C m is a linear [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] RD m-code with G = G ∪ G ∪ … ∪ G m and H = H ∪ H ∪ … ∪ H m as its generator and parity check m-matrix respectively, then C = C ∪ C ∪ … ∪ C m has two representations. (a) C = C ∪ C ∪ … ∪ C m is a row m-space of G = G ∪ G ∪ … ∪ G m (i.e., C i is the row space of G i for i = 1, 2, 3, …, m). (b) C = C ∪ C ∪ … ∪ C m is the solution m-space of H = H ∪ H ∪ … ∪ H m (i.e.; C i is the solution space of H i for i = 1, 2, 3, …, m). Now we proceed on to define the notion of minimum rank m-distance of the rank distance m-code C = C ∪ C ∪ … ∪ C m . D EFINITION
Let C = C ∪ C ∪ … ∪ C m be a rank distance m-code, the minimum rank m-distance d = d ∪ d ∪ … ∪ d m is defined by ,min ( , ) ⎧ ⎫∈⎪ ⎪= ⎨ ⎬≠⎪ ⎪⎩ ⎭ i i i ii R i i i i x y Cd d x y x y i = 1, 2, 3, …, m. That is d = d ∪ d ∪ … ∪ d m = ∈ ≠ ∪ min{ r ( x ) x C and x 0 } ∈ ≠ ∪ … min{ r ( x ) x C and x 0 } ∪ ∈ ≠ m m m m m min{ r ( x ) x C and x 0 } . If an RD m-code C = C ∪ C ∪ … ∪ C m has the minimum rank m-distance d = d ∪ d ∪ … ∪ d m then it can correct all m-errors e = e ∪ e ∪ … ∪ e m ∈ ∪ ∪ ∪ … m1 2N N N nn nq q q F F F with m-rank r(e) = ( r ∪ r ∪ … ∪ r m )( e ∪ e ∪ … ∪ e m ) = ∪ ∪ ∪ r ( e ) r ( e ) ... r ( e ) −− − ⎢ ⎥⎢ ⎥ ⎢ ⎥≤ ∪ ∪ ∪ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ n1 2 d 1d 1 d 1 ...2 2 2 . Let C = C ∪ C ∪ … ∪ C m denote an [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] RD m-code over N q F . A generator m-matrix G = G ∪ G ∪ … ∪ G m of C = C ∪ C ∪ … ∪ C m is a k × n ∪ k × n ∪ … ∪ k m × n m , m-matrix with entries from N q F whose rows form a m-basis for C = C ∪ C ∪ … ∪ C m . Then an (n – k ) × n ∪ (n – k ) × n ∪ … ∪ (n m – k m ) × n m m-matrix H = H ∪ H ∪ … ∪ H m with entries from N q F such that, GH T = ( G ∪ G ∪ … ∪ G m )(H ∪ H ∪ … ∪ H m ) T = ( G ∪ G ∪ … ∪ G m ) ∪ ∪ ∪ … T T T1 2 m ( H H H ) = ∪ ∪ ∪ … T T T1 1 2 2 m m
G H G H G H = 0 ∪ ∪ … ∪ ∪ C ∪ … ∪ C m . The result analogous to Singleton–Style bound in case of RD m-code is given in the following.
Result: (Singleton–Style bound). The minimum rank m-distance d = d ∪ d ∪ … ∪ d m of any linear [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] RD m-code C = C ∪ C ∪ … ∪ C m ⊆ n n nq q q F F F ∪ ∪ ∪ … satisfies the following bounds d = d ∪ d ∪ … ∪ d m ≤ n – k + 1 ∪ n – k + 1 ∪ … ∪ n m – k m + 1. Based on this notion we now proceed on to define the new notion of Maximum Rank Distance m-codes. D EFINITION
An [n , k , d ] ∪ [n , k , d ] ∪ … ∪ [n m , k m , d m ] RD-m-code C = C ∪ C ∪ … ∪ C m is called a Maximum Rank Distance (MRD) m-code if the Singleton Style bound is reached that is d = d ∪ d ∪ … ∪ d m = n – k + 1 ∪ n – k + 1 ∪ … ∪ n m – k m + 1. Now we proceed on to briefly give the construction of MRD m-code. Let [s] = [s ] ∪ [s ] ∪ … ∪ [s m ] = s s s q q q ∪ ∪ ∪ … for any m integers s , s , …, s m . Let {g , , g } {h , , h } ∪ … … m ... {p , , p } ∪ ∪ … be any m-set of elements in N q F that are linearly independent over F q . A generator m-matrix G = G ∪ G ∪ … ∪ G m of an MRD m-code C = C ∪ C ∪ … ∪ C m is defined by G = G ∪ G ∪ … ∪ G m [ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ] g g g h h hg g g h h hg g g h h hg g g h h h − − − − − − ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= ∪⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ … …… …… …(cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35)… … [ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ] mmmm m mm p p pp p pp p pp p p − − − ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥∪ ∪ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ……… …(cid:35) (cid:35) (cid:35)… . It can be easily proved that the m-code C = C ∪ C ∪ … ∪ C m given by the generator m-matrix G = G ∪ G ∪ … ∪ G m has the rank m-distance d = d ∪ d ∪ … ∪ d m . Any m-matrix of the above form is called a Frobenius m-matrix with generating m-vector c c c c g g g ... g = ∪ ∪ ∪ (g , g , , g ) (h , h , , h ) ... (p , p , , p ) = ∪ ∪ ∪ … … … . Interested reader can prove the following theorem: T HEOREM
Let C[n, k] = C [n , k ] ∪ C [n , k ] ∪ … ∪ C m [n m , k m ] be the linear [n , k , d ] ∪ [n , k , d ] ∪ … ∪ [n m , k m , d m ] MRD m-code with d i = 2t i + 1 for i = 1, 2, 3, …, m. Then C[n, k] = C [n , k ] ∪ C [n , k ] ∪ … ∪ C m [n m , k m ] m-code corrects all m-errors of m-rank atmost t = t ∪ t ∪ … ∪ t m and detects all m-errors of m-rank greater than t = t ∪ t ∪ … ∪ t m . Consider the Galois field GF(2 N ) ; N > 1 . An element ... ( ) ( ) ... ( ) − = ∪ ∪ ∪ ∈ ∪ ∪ ∪ (cid:8)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10) N N N1 2 m m times
GF 2 GF 2 GF 2 α α α α can be denoted by m-N-tuple ( , ,..., ) ( , ,..., ) ... ( , ,..., ) − − − ∪ ∪ ∪ a a a a a a a a a as well as by the m-polynomial ... ... − −− − + + + ∪ + + + a a x a x a a x a x ... ... −− ∪ ∪ + + + m m m N 10 1 N 1 a a x a x over GF(2) We now proceed on to define the new notion of circulant m-transpose. D EFINITION
The circulant m-transpose ... = ∪ ∪ ∪
T T T T of a m-vector α = α ∪ α ∪ … ∪ α m = ( , ,..., ) ( , ,..., ) ( , ,..., ) − − − ∪ ∪ ∪ … a a a a a a a a a ∈ GF(2 N ) is defined as, = ∪ ∪ ∪ … m1 2 CC CC m1 2 TT TT 1 2 m α α α α = ( , ,..., ) ( , ,..., ) ( , ,..., ) ∪ ∪ ∪ … a a a a a a a a a If ... ( ) ( ) ( ) − = ∪ ∪ ∪ ∈ ∪ ∪ ∪ …(cid:8)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10) N N N1 2 m m times
GF 2 GF 2 GF 2 α α α α has the m-polynomial representation ... ... − −− − + + + ∪ + + + a a x a x a a x a x ... ... −− ∪ ∪ + + + m m m N 10 1 N 1 a a x a x in ( )[ ] ( )[ ] ( )[ ]...( ) ( ) ( ) ∪ ∪ ∪+ + + N N N
GF 2 x GF 2 x GF 2 xx 1 x 1 x 1 then by α = α ∪ α ∪ … ∪ α m , we denote the m-vector corresponding to the m-polynomial −− + + + ⋅ + ∪ [ a a x ... a x x ] mod( x 1 ) −− + + + ⋅ + ∪ ∪ … … [ a a x a x x ] mod( x 1 ) −− + + + ⋅ + … m m m N 1 i N0 1 N 1 [ a a x a x x ] mod( x 1 ) for i = 0, 1, 2, 3, …, N –1 ( α = α ∪ α ∪ … ∪ α m = α ο ). We now proceed onto define the m-word generated by α = α ∪ α ∪ … ∪ α m . D EFINITION
Let f = f ∪ f ∪ … ∪ f m : − ∪ ∪ ∪ ⎯⎯→ …(cid:8)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10) N N Nm times
GF( 2 ) GF( 2 ) GF( 2 ) − ∪ ∪ ∪ …(cid:8)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10) N N N N N Nm times [ GF( 2 )] [ GF( 2 )] [ GF( 2 )] be defined as f( α ) = f ( α ) ∪ f ( α ) ∪ … ∪ f m ( α m ) − − = ∪ ∪ … … … T T T T T T10 11 1N 1 20 21 2N 1 ( , , , ) ( , , , ) α α α α α α − ∪ … m m mC C Cm m m T T Tm0 m1 mN 1 ( , , , ) α α α . We call f( α ) = f ( α ) ∪ f ( α ) ∪ … ∪ f m ( α m ) as the m-code word generated by α = α ∪ α ∪ … ∪ α m . We analogous to the definition of Macwilliams and Solane define circulant m-matrix associated with a m-vector in GF(2 N ) ∪ GF(2 N ) ∪ … ∪ GF(2 N ). D EFINITION
A m-matrix of the form − −− − − − ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥∪⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ … …… …(cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35)… …
N NN N N N a a a a a aa a a a a aa a a a a a ... −− − ⎡ ⎤⎢ ⎥⎢ ⎥∪ ∪ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ……(cid:35) (cid:35) (cid:35)… m m mNm m mN Nm m m a a aa a aa a a is called the circulant m-matrix associated with the m-vector ( , ,..., ) ( , , , ) ( , , , ) − − − ∪ ∪ ∪ ∈ … … … m m mN N N a a a a a a a a a GF(2 N ) ∪ GF(2 N ) ∪ … ∪ GF(2 N ). Thus with each α = α ∪ α ∪ … ∪ α m ∈ GF(2 N ) ∪ GF(2 N ) ∪ … ∪ GF(2 N ) we can associate a circulant m-matrix whose i th m-columns represents T T T1i 2i mi ... α ∪ α ∪ ∪ α ; i = 0, 1, 2, …, N–1. f = f ∪ f ∪ … ∪ f m is nothing but a m-mapping of GF(2 N ) ∪ GF(2 N ) ∪ … ∪ GF(2 N ) onto the pseudo false m-algebra of all N × N circulant m-matrices over GF(2). Denote the m-space of f(GF(2 N )) = f (GF(2 N )) ∪ f (GF(2 N )) ∪ … ∪ f m (GF(2 N )) by N N Nm times
V V ... V − ∪ ∪ ∪ (cid:8)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:10) . We define the m-norm of a m-word N N N1 2 m m times v v v ... v V V ... V − = ∪ ∪ ∪ ∈ ∪ ∪ ∪ (cid:8)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:10) as follows : D EFINITION
The m-norm of a m-word v = v ∪ v ∪ … ∪ v m ∈ V N ∪ V N ∪ … ∪ V N is defined as the m-rank of v = v ∪ v ∪ … ∪ v m over GF(2) [By considering it as a circulant m-matrix over GF(2)]. We denote the m-norm of v = v ∪ v ∪ … ∪ v m by r(v) = r (v ) ∪ r (v ) ∪ … ∪ r m (v m ), we prove the following theorem: T HEOREM
Suppose ... ( ) ( ) ... ( ) − = ∪ ∪ ∪ ∈ ∪ ∪ ∪ (cid:8)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10) N N N1 2 m m times
GF 2 GF 2 GF 2 α α α α has the m-polynomial representation g(x) = g (x) ∪ g (x) ∪ … ∪ g m (x) over GF(2) such that gcd(g i (x), x N +1) has degree N – k i for i = 1, 2, 3, …, m; 1 ≤ k , k , …, k m ≤ N. Then the m-norm of the m-word generated by α = α ∪ α ∪ … ∪ α m is k ∪ k ∪ … ∪ k m . Proof:
We know the m-norm of the m-word generated by α = α ∪ α ∪ … ∪ α m is the m-rank of the circulant m-matrix T T T T T T10 11 1N 1 20 21 2N 1 ( , ,..., ) ( , ,..., ) − − α α α ∪ α α α ∪ ∪ … m m mC C Cm m m T T Tm0 m1 mN 1 ( , ,..., ) − α α α where T T TTi 1i 2i mi α = α ∪ α ∪ ∪ α … represents a m-polynomial i N i N1 2 [x g (x)]mod(x 1) [x g (x)]mod(x 1) + ∪ + i Nm ... [x g (x)]mod(x 1) ∪ ∪ + over GF(2). Suppose the m-GCD {(g (x), x N + 1) ∪ (g (x), x N + 1) ∪ … ∪ (g m (x), x N + 1)} has m-degree N – k ∪ N – k ∪ … ∪ N – k m . To prove that the m-word generated by α = α ∪ α ∪ … ∪ α m has m-rank k ∪ k ∪ … ∪ k m . It is enough to prove that the m-space generated by the N-polynomials {g (x) mod(x N + 1), x ⋅ g (x) mod(x N + 1), …, x N–1 ⋅ g (x) mod(x N + 1)} ∪ {g (x) mod(x N + 1), x ⋅ g (x) mod(x N + 1), …, x N–1 ⋅ g (x) mod(x N + 1)} ∪ … ∪ {g m (x) mod(x N + 1), x ⋅ g m (x) mod(x N + 1), …, x N–1 ⋅ g m (x) mod(x N + 1)} has m-dimension k ∪ k ∪ … ∪ k m . We will prove that the m-set of k ∪ k ∪ … ∪ k m , m-polynomials {g (x) mod x N + 1, x ⋅ g (x) mod x N + 1, …, x N–1 ⋅ g (x) mod x N + 1} ∪ {g (x) mod x N + 1, x ⋅ g (x) mod x N + 1, …, x N–1 ⋅ g (x) mod x N + 1} ∪ … ∪ {g m (x) mod x N + 1, x ⋅ g m (x) mod x N + 1, …, x N–1 ⋅ g m (x) mod x N + 1} forms a m-basis for the m-space. If possible let, k 11 1 10 1 1 1 k 1 1 a (g (x)) a x(g (x)) ... a (x g (x)) −− + + + ∪ k 12 2 20 2 1 2 k 1 2 a (g (x)) a x(g (x)) ... a (x g (x)) ... − − + + + ∪ ∪ mm k 1m m m0 m 1 m k 1 m a (g (x)) a x(g (x)) ... a (x g (x)) −− + + + = 0 ∪ ∪ … ∪ N + 1); where iji a ∈ GF(2), 1 ≤ j i ≤ k i – 1 and i = 1, 2, …, m. This implies (cid:11)(cid:11)(cid:11)(cid:11) (cid:10)(cid:11)(cid:11)(cid:11)(cid:11) (cid:9)(cid:8) timesm 1N1N1N x...xx − +++ ∪∪∪ m-divides k 11 1 10 1 k 1 1 (a a x ... a x )g (x) −− + + + ∪ k 12 2 20 1 k 1 2 (a a x ... a x )g (x) −− + + + ∪ ∪ … mm k 1m m m0 1 k 1 m (a a x ... a x )g (x) −− + + + . Now if g (x) ∪ g (x) ∪ … ∪ g m (x) = p (x)a (x) ∪ p (x)a (x) ∪ … ∪ p m (x)a m (x) where p i (x) is the gcd(g i (x), x N + 1); i = 1, 2, …, m, then (a i (x), x N + 1) = 1. Thus x N + 1 m-divides k 11 1 10 1 k 1 1 (a a x ... a x )g (x) ... −− + + + ∪ ∪ mm k 1m m m0 1 k 1 m (a a x ... a x )g (x) −− + + + implies the m-quotient N N1 m (x 1) (x 1)...p (x) p (x) + +∪ ∪ m-divides k 11 1 10 1 k 1 1 (a a x ... a x )a (x) −− + + + ∪ k 12 2 20 1 k 1 2 (a a x ... a x )a (x) ... −− + + + ∪ ∪ mm k 1m m 10 1 k 1 m (a a x ... a x )a (x) −− + + + . That is N N N1 2 m (x 1) (x 1) (x 1)...p (x) p (x) p (x) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ + +∪ ∪ ∪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ m-divides k 11 1 10 1 k 1 (a a x ... a x ) −− + + + ∪ k 12 2 20 1 k 1 (a a x ... a x ) ... −− + + + ∪ ∪ mm k 1m m 10 1 k 1 (a a x ... a x ) −− + + + which is a contradiction, as N N N1 2 m (x 1) (x 1) (x 1)...p (x) p (x) p (x) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ + +∪ ∪ ∪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ has m-degree (k , k , . . ., k m ) where as the m-polynomial k 11 1 10 1 k 1 (a a x ... a x ) −− + + + ∪ k 12 2 20 1 k 1 (a a x ... a x ) ... −− + + + ∪ ∪ mm k 1m m 10 1 k 1 (a a x ... a x ) −− + + + has m-degree atmost ((k – 1), (k – 1), …, (k m – 1)). Hence the m-polynomials {g (x) mod(x N + 1), x ⋅ g (x) mod(x N + 1), …, k 1 x − ⋅ g (x) mod(x N + 1)} ∪ {g (x) mod(x N + 1), x ⋅ g (x) mod(x N + 1), …, k 1 x − ⋅ g (x) mod(x N + 1)} ∪ … ∪ {g m (x) mod(x N + 1), x ⋅ g m (x) mod(x N + 1), …, m k 1 x − ⋅ g m (x) mod(x N + 1)} are m-linearly independent over GF(2). We will prove, {g (x) mod(x N + 1), x ⋅ g (x) mod(x N + 1), …, k 1 x − ⋅ g (x) mod(x N + 1)} ∪ {g (x) mod(x N + 1), x ⋅ g (x) mod(x N + 1), …, k 1 x − ⋅ g (x) mod(x N + 1)} ∪ … ∪ {g m (x) mod(x N + 1), x ⋅ g m (x) mod(x N + 1), …, m k 1 x − ⋅ g m (x) mod(x N + 1)} generate the m-space. For this it is enough to prove that x i g (x) ∪ x i g (x) ∪ … ∪ x i g m (x) is a linear combination of these m-polynomials for k j ≤ i ≤ N – 1; j = 1, 2, 3, …, m.
N N Nm times x 1 x 1 x 1 − + ∪ + ∪ ∪ + …(cid:8)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10) = p (x)b (x) ∪ p (x)b (x) ∪ … ∪ p m (x)b m (x) where b i (x) ii ki i i0 1 k b b x ... b x = + + + ; i = 1, 2, 3, …, m. (Note that i i i0 k b b 1 = = since b i (x) divides x N + 1, i = 1, 2, 3, …, m). Also we have g i (x) = p i (x)a i (x) for i = 1, 2, 3, …, m. Thus i iN i g (x)b (x)x 1 a (x) ⎛ ⎞+ = ⎜ ⎟⎝ ⎠ for i = 1, 2, 3, …, m. That is ii ki i ii 0 1 k Ni g (x)(b b x ... b x ) 0 mod(x 1)a (x) + + + = + (i = 1, 2, 3, …, m) that is i ii k 1i i i ki 0 1 k 1 Nii i g (x)(b b x ... b x ) g (x) x mod(x 1)a (x) a (x) −− + + + ⎡ ⎤⋅= +⎢ ⎥⎣ ⎦ true for i = 1, 2, 3, ..., m. Hence i ii k k 1i i i Ni 0 i 1 i k 1 i x g (x) (b g (x) b x g (x) ... b x g (x)) mod(x 1) −− = + + + + a linear combination of {g i (x) mod(x N + 1), [x ⋅ g i (x)] mod(x N + 1), …, i k 1 x − ⋅ g m (x) mod(x N + 1)} over GF(2). This is true for each i, for i = 1, 2, 3, …, m. Now it can be easily proved that x i g (x) ∪ x i g (x) ∪ … ∪ x i g m (x) is a m-linear combination of {g (x) mod(x N + 1), x ⋅ g (x) mod(x N + 1), …, k 1 x − ⋅ g (x) mod(x N + 1)} ∪ {g (x) mod(x N + 1), x ⋅ g (x) mod(x N + 1), …, k 1 x − ⋅ g (x) mod(x N + 1)} ∪ … ∪ {g m (x) mod(x N + 1), x ⋅ g m (x) mod(x N + 1), …, m k 1 x − ⋅ g m (x) mod(x N + 1)} for i > k j ; j = 1, 2, …, m. Hence the m-space generated by the m-polynomials {g (x) mod(x N + 1), x ⋅ g (x) mod(x N + 1), …, k 1 x − ⋅ g (x) mod(x N + 1)} ∪ {g (x) mod(x N + 1), x ⋅ g (x) mod(x N + 1), …, k 1 x − ⋅ g (x) mod(x N + 1)} ∪ … ∪ {g m (x) mod(x N + 1), x ⋅ g m (x) mod(x N + 1), …, m k 1 x − ⋅ g m (x) mod(x N + 1)} has m-dimension (k , k , …, k m ). That is the m-rank of a m-word generated by α = α ∪ α ∪ … ∪ α m is (k , k , …, k m ). C OROLLARY If α = α ∪ α ∪ … ∪ α m ∈ (2 ) ... (2 ) − ∪ ∪ (cid:8)(cid:11)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10) N Nm times
GF GF then the m-norm of the m-word generated by α = α ∪ α ∪ … ∪ α m is (N, N, …, N) and hence f( α ) = f ( α ) ∪ f ( α ) ∪ … ∪ f m ( α m ) is m-invertible. (We say f( α ) is m-invertible if each f i ( α )is invertible for i = 1, 2, 3, …, m). Proof: The corollary follows immediately from the theorem since gcd(g i (x), x N +1) = 1 has degree 0 for i = 1, 2, 3, …, m; hence the m-rank of f( α ) = f ( α ) ∪ f ( α ) ∪ … ∪ f m ( α m ) is (N, N, …, N). D EFINITION
The m-distance between two m-words u, v ∈ V N ∪ … ∪ V N is defined as, d(u, v) = d (u , v ) ∪ … ∪ d m (u m , v m ) = r (u +v ) ∪ … ∪ r m (u m + v m ) where u = u ∪ u ∪ … ∪ u m and v = v ∪ v ∪ … ∪ v m . D EFINITION
Let C = C ∪ C ∪ … ∪ C m be a circulant rank m-code of m-length N ∪ N ∪ … ∪ N m which is a m-subspace of ... ∪ ∪ ∪ m NN N
V V V equipped with the m-distance m-function d (u , v ) ∪ d (u , v ) ∪ … ∪ d m (u m , v m ) = r (u +v ) ∪ r (u +v ) ∪ … ∪ r m (u m + v m ) where , ,..., m NN N
V V V are rank spaces defined over GF(2 N ) with N i ≠ N j if i ≠ j. C = C ∪ C ∪ … ∪ C m is defined as the circulant m-code of m-length N ∪ N ∪ … ∪ N m defined as a m-subspace of ... ∪ ∪ ∪ m NN N
V V V equipped with the m-distance m-function. D EFINITION
A circulant m-rank m-code of m-length N ∪ N ∪ … ∪ N m is called m-cyclic if whenever ( ,..., ) ∪ N v v ( ,..., ) ... ( ,..., ) ∪ ∪ m m mN N v v v v is a m-code word then it implies ( , ,..., , ) ( , ,..., , ) ... ∪ ∪ ∪ N N v v v v v v v v ( , ,..., , ) m m m m mN v v v v is also a m-code word. Now we proceed on to define quasi MRD-m-codes. D EFINITION
Let C = C ∪ C ∪ … ∪ C m be a RD rank m-code where each C i ≠ C j if i ≠ j. If some of the C i ’s are MRD codes and others are RD codes then we call C = C ∪ C ∪ … ∪ C m to be a quasi MRD m-code. Note:
If r are MRD codes and r are RD codes r + r = m, (r ≥
1, r ≥ , r ) MRD m-code. We can say C , …, r C are C i [n i , k i ] RD-codes; i = 1, 2, 3, …, r and C j [n j , k j , d j ] are MRD codes for j = 1, 2, 3, …, r with r + r = m. Thus C C [n , k ] ... C [n , k ] = ∪ ∪
C [n , k ,d ] ... C [n , k , d ] ∪ ∪ ∪ is a quasi (r , r ) MRD m-code. Any m-code word in C would be of the form C = C ∪ C ∪ … ∪ C m . 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. D EFINITION
Let C , C , …, C m be m distinct codes some circulant rank codes and others linear RD-codes defined over GF(2 N ). C = C ∪ C ∪ … ∪ C m is defined as the quasi circulant m-code of type I. If in the quasi circulant m-code of type I some of the C i ’s are MRD codes i.e., C , C , …, C m is a collection of RD codes, MRD codes and circulant rank codes then we define C = C ∪ C ∪ … ∪ C m to be a quasi circulant m-code of type II. We can define also mixed quasi circulant rank m-codes. D EFINITION
Let C , C , …, C m be a collection of m-codes, all of them distinct C i ≠ C j if i ≠ j and C i ⊄ C j or C j ⊄ C i if i ≠ j. If this collection of codes C , C , …, C m 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 = C ∪ C ∪ … ∪ C m to be a mixed quasi circulant rank m-code. D EFINITION
If C = C ∪ C ∪ … ∪ C m be a collection of distinct circulant codes some C i ’s are circulant codes and some 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 m-codes will be appropriate. Now we proceed on to define the notion of almost maximum rank distance m-codes. D EFINITION
Let C [n , k ] ∪ C [n , k ] ∪ … ∪ C m [n m , k m ] be a collection of m distinct almost maximum rank distance codes with minimum distances greater than equal to n – k ∪ n – k ∪ … ∪ n m – k m defined over GF(2 N ). C is defined as the Almost Maximum Distance Rank-m-code or (AMRD-m-code) over GF(2 N ). An AMRD m-code is called a AMRD – tricode, if m = 3. An AMRD m-code whose minimum distance is greater than n – k ∪ n – k ∪ … ∪ n m – k m 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. T HEOREM
When (n – k ) ∪ (n – k ) ∪ … ∪ (n m – k m ) AMRD m-code C = C [n , k ] ∪ C [n , k ] ∪ … ∪ C m [n m , k m ] is such that each (n i – k i ) is odd for i = 1, 2, 3, …, m; then 1. The error correcting capability of the [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] AMRD m-code is equal to that of an [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] MRD m-code. 2. An [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] AMRD m-code is better than any [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] m-code in Hamming metric for error correction. Proof: (1) Suppose C = C ∪ C ∪ … ∪ C m is a [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] AMRD m-code such that (n – k ) ∪ (n – k ) ∪ … ∪ (n m – k m ) is an odd m-integer (i.e., n i – k i ≠ n j – k j if i ≠ j are odd integer 1 ≤ i, j ≤ m). The maximum number of m-errors corrected by C = C ∪ C ∪ … ∪ C m is given by (n k 1) (n k 1) (n k 1)...2 2 2 − − − − − −∪ ∪ ∪ . But (n k 1) (n k 1) (n k 1)...2 2 2 − − − − − −∪ ∪ ∪ is equal to the error correcting capability of an [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ]; MRD m- code (since (n – k ), (n – k ), … , (n m – k m ) are odd). That is, (n – k ) ∪ (n – k ) ∪ … ∪ (n m – k m ) is said to be m-odd if each n i – k i is odd for i = 1, 2, 3, …, m. Thus a [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] AMRD m-code is as good as an [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] MRD m-code. Proof: (2)
Suppose C = C ∪ C ∪ … ∪ C m is a [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] AMRD m-code such that (n – k ) ∪ (n – k ) ∪ … ∪ (n m – k m ) are odd; then each m-code word of C can correct r 1 r 2 r m r L (n ) L (n ) ... L (n ) L (n) ∪ ∪ ∪ = error m-vectors where r = r ∪ r ∪ … ∪ r m (n k 1) (n k 1) (n k 1)...2 2 2 − − − − − −= ∪ ∪ ∪ and r r 1 r 2 r m L (n) L (n ) L (n ) ... L (n ) = ∪ ∪ ∪ n 1 N N i 1i 1 n1 (2 1)...(2 2 )i −= ⎡ ⎤= + − − ∪⎢ ⎥⎣ ⎦ ∑ n 2 N N i 1i 1 n1 (2 1)...(2 2 )i −= ⎡ ⎤+ − − ∪⎢ ⎥⎣ ⎦ ∑ m n m N N i 1i 1 n... 1 (2 1)...(2 2 )i −= ⎡ ⎤∪ + − −⎢ ⎥⎣ ⎦ ∑ . Consider the same [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] m-code in Hamming metric. Let it be denoted by D = D ∪ D ∪ … ∪ D m then the minimum m-distance of D is atmost (n – k + 1) ∪ (n – k + 1) ∪ … ∪ (n m – k m + 1). The error correcting capability of D is n k 1 1 n k 1 1 n k 1 1...2 2 2 − + − − + − − + −⎢ ⎥ ⎢ ⎥ ⎢ ⎥∪ ∪ ∪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ = r ∪ r ∪ … ∪ r m (since (n – k ) ∪ (n – k ) ∪ … ∪ (n m – k m ) are odd). Hence the number of error m-vectors corrected by the m-code word is given by r r r1 2 mN i N i N ii 0 i 0 i 0 n n n(2 1) (2 1) ... (2 1)i i i = = = ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− ∪ − ∪ ∪ −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ∑ ∑ ∑ which is clearly less than r 1 r 2 r m L (n ) L (n ) ... L (n ) ∪ ∪ ∪ . Thus the number of error m-vectors that can be corrected by the [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] AMRD m-code is much greater than that of the same m- code considered in Hamming metric. For a given m-length n = n ∪ n ∪ … ∪ n m a single error correcting AMRD m-code is one having m-dimension (n – 3) ∪ (n – 3) ∪ … ∪ (n m – 3) and the minimum m-distance greater than or equal to 3 ∪ ∪ … ∪
3. We now proceed on to give a characterization of a single error correcting AMRD m-codes in terms of its parity check m-matrices. The characterization is based on the condition for the minimum distance proved by Gabidulin in [24, 27]. T HEOREM
Let H = H ∪ H ∪ … ∪ H m = ( ) ( ) ∪ ij ij α α ∪ … ∪ ( ) mij α be a (3 × n ) ∪ (3 × n ) ∪ … ∪ (3 × n m ) m-matrix of m-rank 3 over GF(2 N ); n ≤ N and n ≤ N which satisfies the following condition. For any two distinct, non empty m-subsets P , P , …, P m where ... = ∪ ∪ ∪ P P P P and ... = ∪ ∪ ∪
P P P P of {1, 2, 3, …, n } and {1, 2, 3, …, n } respectively; there exists = ∪ i i i , = ∪ i i i , …, = ∪ m mm 1 2 i i i ∈ {1, 2, 3} ∪ {1, 2, 3} ∪ … ∪ {1, 2, 3} such that ∈ ∈ ∈ ∈ ⎛ ⎞ ⎛ ⎞α ⋅ α ∪ α ⋅ α⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ∑ ∑ ∑ ∑ m m m m1 1 2 1m m m m1 1 1 2 m mi j i kj P k P ... ∈ ∈ ⎛ ⎞∪ ∪ α ⋅ α⎜ ⎟⎜ ⎟⎝ ⎠ ∑ ∑ ∈ ∈ ∈ ∈ ⎛ ⎞ ⎛ ⎞≠ α ⋅ α ∪ α ⋅ α⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ∑ ∑ ∑ ∑ m m m m2 1 1 1m m m m1 1 1 2 m mi j i kj P k P ... ∈ ∈ ⎛ ⎞∪ ∪ α ⋅ α⎜ ⎟⎜ ⎟⎝ ⎠ ∑ ∑ . Then H = H ∪ H ∪ … ∪ H m as a parity check m-matrix defines a (n , n – 3) ∪ (n , n – 3) ∪ … ∪ (n m , n m – 3) single m-error correcting AMRD m-code over GF(2). Proof:
Given H = H ∪ H ∪ … ∪ H m is a (3 × n ) ∪ (3 × n ) ∪ … ∪ (3 × n m ) m-matrix of m-rank 3 ∪ ∪ … ∪ N ), so that H = H ∪ H ∪ … ∪ H m as a parity check m-matrix defines a (n , n – 3) ∪ (n , n – 3) ∪ … ∪ (n m , n m – 3) RD m-code, where n T1 1 C {x V xH 0} = ∈ = , n T2 2 C {x V xH 0} = ∈ = , …, m n Tm m C {x V xH 0} = ∈ = . It remains to prove that the minimum m-distance of C = C ∪ C ∪ … ∪ C m is greater than or equal to 3 ∪ ∪ … ∪
3. We will prove that no non zero m-word of C = C ∪ C ∪ … ∪ C m has m-rank less than 3 ∪ ∪ … ∪
3. The proof is by the method of contradiction. Suppose there exists a non zero m-code word x = x ∪ x ∪ … ∪ x m such that r (x ) ≤
2, r (x ) ≤
2, …, r m (x m ) ≤
2, then x = x ∪ x ∪ … ∪ x m can be written as x = x ∪ x ∪ … ∪ x m = ( y ∪ y ∪ … ∪ y m )(M ∪ M ∪ … ∪ M m ) where y (y , y ) = , y (y , y ) = , …, m mm 1 2 y (y , y ) = ; i i N1 2 y , y GF(2 ) ∈ ; 1 ≤ i ≤ m and M = M ∪ M ∪ … ∪ M m )m(...)m()m( mij2ij1ij ∪∪∪= is a (2 × n ) ∪ (2 × n ) ∪ … ∪ (2 × n m ) m-matrix of m-rank 2 ∪ ∪ … ∪ T = T T T1 1 1 2 2 2 m m m y M H y M H ... y M H ∪ ∪ ∪ = 0 ∪ ∪ … ∪ T ) = T T T1 1 1 2 2 2 m m m y (M H ) y (M H ) ... y (M H ) ∪ ∪ ∪ = 0 ∪ ∪ … ∪
0. Since y = y ∪ y ∪ … ∪ y m is non zero y(MH T ) = (0 ∪ ∪ … ∪
0) implies
Ti i i y (M H ) 0 = for i = 1, 2, 3, …, m; that is the 2 ×
3 m-matrix
T T T1 1 2 2 m m
M H M H ... M H ∪ ∪ ∪ has m-rank less than 2 over GF(2 N ). Now let P P P ... P = ∪ ∪ ∪ {j such that m 1} {j such that m 1} ... = = ∪ = ∪ mm m mm 1 j {j such that m 1} ∪ = and P P P ... P = ∪ ∪ ∪ {j such that m 1} {j such that m 1} ... = = ∪ = ∪ mm m mm 2 j {j such that m 1} ∪ = . Since M = M ∪ M ∪ … ∪ M m )m(...)m()m( mij2ij1ij ∪∪∪= is a 2 × n ∪ × n ∪ … ∪ × n m m-matrix of m-rank 2 ∪ ∪ … ∪ and P are disjoint non empty m-subsets of {1, 2, …, n } ∪ {1, 2, …, n } ∪ … ∪ {1, 2, …, n m } respectively and T T T T1 1 2 2 m m
MH M H M H ... M H = ∪ ∪ ∪ ∈ ∈ ∈∈ ∈ ∈ ⎛ ⎞α α α⎜ ⎟⎜ ⎟= ∪⎜ ⎟⎜ ⎟α α α⎜ ⎟⎝ ⎠ ∑ ∑ ∑∑ ∑ ∑ ... ∈ ∈ ∈∈ ∈ ∈ ⎛ ⎞α α α⎜ ⎟⎜ ⎟ ∪ ∪⎜ ⎟⎜ ⎟α α α⎜ ⎟⎝ ⎠ ∑ ∑ ∑∑ ∑ ∑ m m m1j 2 j 3jj P j P j Pm m m1j 2 j 3jj P j P j P ∈ ∈ ∈∈ ∈ ∈ ⎛ ⎞α α α⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟α α α⎜ ⎟⎝ ⎠ ∑ ∑ ∑∑ ∑ ∑ . But the selection of H = H ∪ H ∪ … ∪ H m is such that there exists p p1 2 i ,i ∈ {1, 2, 3}; p = 1, 2, …, m such that i j i k i j i kk P k Pj P j P ... ∈ ∈∈ ∈ α ⋅ α ∪ α ⋅ α ∪ ∪ ∑ ∑ ∑ ∑ m 1 m1 1 2 m2 m 22 1 i j i kk Pj P ∈∈ α ⋅ α ∑ ∑ i j i k i j i kk P k Pj P j P ... ∈ ∈∈ ∈ ≠ α ⋅ α ∪ α ⋅ α ∪ ∪ ∑ ∑ ∑ ∑ m 1 m2 1 1 m1 m 21 1 i j i kk Pj P ∈∈ α ⋅ α ∑ ∑ . Hence in MH T there exists a 2 ×
2 m-submatrices whose determinant is non zero; i.e.,
T T T T1 1 1 2 2 2 m m m r(MH ) r (M H ) r (M H ) ... r (M H ) = ∪ ∪ ∪ over GF(2 N ). But this is a contradiction to fact that, T T T T1 1 2 2 m m rank(MH ) rank(M H ) rank(M H ) ... rank(M H ) = ∪ ∪ ∪ < 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 n ∪ n ∪ … ∪ n m . D EFINITION
Let C ∪ C ∪ … ∪ C n be a RD-m-code where C i is a constant rank code of length n i , i = 1, 2, …, m (Each C i is a subset of the rank space i n V ; i = 1, 2, …, m) then C = C ∪ C ∪ … ∪ C m is a constant m-rank code of m-length n ∪ n ∪ … ∪ n m ; that is every m-code word has same m-rank. D EFINITION
A(n , r , d ) ∪ A(n , r , d ) ∪ … ∪ A(n m , r m , d m ) is defined as the maximum number of m-vectors in ... ∪ ∪ ∪ m nn n V V V of constant m-rank, r ∪ r ∪ … ∪ r m and m-distance between any two m-vectors is at least d ∪ d ∪ … ∪ d m [By (n , r , d ) ∪ (n , r , d ) ∪ … ∪ (n m , r m , d m ) m-set we mean a m-subset of m-vectors of ... ∪ ∪ ∪ m nn n V V V having constant m-rank r ∪ r ∪ … ∪ r m and m-distance between any two m-vectors is atleast d ∪ d ∪ … ∪ d m ]. We analyze the m-function A(n , r , d ) ∪ A(n , r , d ) ∪ … ∪ A(n m , r m , d m ) by the following theorem: T HEOREM A(n , r , 1) ∪ A(n , r , 1) ∪ … ∪ A(n m , r m , 1) = ( ) ( ) ... ( ) ∪ ∪ ∪ m r r r m L n L n L n , the number of m-vectors of m-rank r ∪ r ∪ … ∪ r m in ... ∪ ∪ ∪ m nn n V V V . 2.
A(n , r , d ) ∪ A(n , r , d ) ∪ … ∪ A(n m , r m , d m ) = 0 ∪ ∪ … ∪ i > 0 or d i > n i and d i > 2r i (i = 1, 2, 3, … , m), Proof: (1) Follows from the fact that r 1 r 2 r m L (n ) L (n ) ... L (n ) ∪ ∪ ∪ is the number of m-vectors of m-length n ∪ n ∪ … ∪ n m , constant m-rank r ∪ r ∪ … ∪ r m and m-distance between any two distinct m-vectors in a m-rank space n n n V V ... V ∪ ∪ ∪ is always greater than or equal to 1 ∪ ∪ … ∪
1. (2) Follows immediately from the definition of A(n , r , d ) ∪ A(n , r , d ) ∪ … ∪ A(n m , r m , d m ). T HEOREM
A(n , 1, 2) ∪ A(n , 1, 2) ∪ … ∪ A(n m , 1, 2) = − ∪ n − n .... ∪ ∪ − m n N ). Proof: Denote by V ∪ V ∪ … ∪ V m the set of m-vectors of m-rank 1 ∪ ∪ … ∪ n n n V V ... V ∪ ∪ ∪ ; we know each non zero element α ∪ α ∪ … ∪ α m ∈ GF(2 N ), there exists n n n (2 1) (2 1) ... (2 1) − ∪ − ∪ ∪ − m-vectors of m-rank one having α ∪ α ∪ … ∪ α m as a coordinate. Thus the m-cardinality of V ∪ V ∪ … ∪ V m is nN N (2 1)(2 1) (2 1) − − ∪ − n nN (2 1) ... (2 1)(2 1) − ∪ ∪ − − . Now m-divide V ∪ V ∪ … ∪ V m into n n n (2 1) (2 1) ... (2 1) − ∪ − ∪ ∪ − blocks of (2 N –1) ∪ (2 N –1) ∪ … ∪ (2 N –1) m-vectors such that each block consists of the same pattern of all nonzero m-elements of GF(2 N ) ∪ GF(2 N ) ∪ … ∪ GF(2 N ). 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 (n , 1, 2) ∪ (n , 1, 2) ∪ … ∪ (n m , 1, 2) m-set. Also it is always possible to construct such a m-set. Thus A(n , 1, 2) ∪ A(n , 1, 2) ∪ … ∪ A(n m , 1, 2) = n n (2 1) (2 1) − ∪ − m n ... (2 1) ∪ ∪ − . T HEOREM
A(n , n , n ) ∪ A(n , n , n ) ∪ … ∪ A(n m , n m , n m ) = (2 N –1) ∪ (2 N –1) ∪ … ∪ (2 N –1) (i.e., A(n i , n i , n i ) = 2 N – 1; i = 1, 2, 3, …, m) over GF(2 N ). Proof:
Denote by n n n
V V ... V ∪ ∪ ∪ the m-set of all m-vectors of m-rank n ∪ n ∪ … ∪ n m in the m-space n n V V ∪ ∪ … m n V ∪ . We know the m-cardinality of n n n V V ... V ∪ ∪ ∪ is (2 N – 1)(2 N – 2) … n 1N (2 2 ) − − ∪ (2 N – 1) (2 N – 2) … n 1N (2 2 ) − − ∪ … ∪ (2 N – 1) (2 N – 2) … m n 1N (2 2 ) − − and by the definition in a (n , n , n ) ∪ (n , n , n ) ∪ … ∪ (n m , n m , n m ) m-set, the m-distance between any two m-vector should be n ∪ n ∪ … ∪ n m . Thus no two m-vectors can have a common symbol at a co-ordinate place i ∪ i ∪ … ∪ i m ; (1 ≤ i ≤ n , 1 ≤ i ≤ n , …, 1 ≤ i m ≤ n m ). This implies that A(n , n , n ) ∪ A(n , n , n ) ∪ … ∪ A(n m , n m , n m ) ≤ (2 N –1) ∪ (2 N –1) ∪ … ∪ (2 N –1). Now we construct a (n , n , n ) ∪ (n , n , n ) ∪ … ∪ (n m , n m , n m ) m-set as follows: Select N m-vectors from n n n V V ... V ∪ ∪ ∪ such that i.
Each m-basis m-elements of n n
GF(2 ) GF(2 ) ∪ ∪ … ∪ m n GF(2 ) should occur (can be as a m-combination) atleast once in each m-vector. ii.
If the th th th1 2 m (i ,i ,...,i ) m-vector is choosen th1 ((i 1) , + th th2 m (i 1) ,..., (i 1) ) + + m-vector should be selected such that its m-rank m-distance from any m-linear combination of the previous (i , i , … , i m ) m-vectors is n ∪ n ∪ … ∪ n m . Now the set of all m-linear combinations of these N ∪ N ∪ … ∪ N, m-vectors over GF(2) ∪ GF(2) ∪ … ∪ GF(2) will be such that the m-distance between any two m-vectors is n ∪ n ∪ … ∪ n m . Hence it is (n , n , n ) ∪ (n , n , n ) ∪ … ∪ (n m , n m , n m ) m-set. Also the m-cardinally of this (n , n , n ) ∪ (n , n , n ) ∪ … ∪ (n m , n m , n m ) m-sets is (2 N –1) ∪ (2 N –1) ∪ … ∪ (2 N –1) (we do not count all zero m-linear combinations). Thus A(n , n , n ) ∪ A(n , n , n ) ∪ … ∪ A(n m , n m , n m ) = (2 N –1) ∪ (2 N –1) ∪ … ∪ (2 N –1). Recall a [n, 1] repetition RD code is a code generated by the matrix G = (1, 1, …, 1) over N F . Any non zero code word has rank 1. D EFINITION
A [n , 1] ∪ [n , 1] ∪ … ∪ [n m , 1] repetition RD m-code is a m-code generated by the m-matrix G = G ∪ G ∪ … ∪ G m = (1 1 … 1) ∪ (1 1 … 1) ∪ … ∪ (1 1 … 1) (G i ≠ G j if i ≠ j; 1 ≤ i, j ≤ m) over N F . Any non zero m-code word has m-rank 1 ∪ ∪ … ∪ D EFINITION
Let C = C ∪ C ∪ … ∪ C m be a linear [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] RD m-code defined over N F . The covering m-radius of C = C ∪ C ∪ … ∪ C m is defined as the smallest m-tuple of integers (r , r , …, r m ) such that all m-vectors in the rank m-space ... ∪ ∪ ∪ mN N N nn n F F F are with in the rank m-distance r ∪ r ∪ … ∪ r m of some m-code word. The covering m-radius of C = C ∪ C ∪ … ∪ C m is denoted by t(C ) ∪ t(C ) ∪ … ∪ t(C m ) = t(C) = ∈ ∈ + +⎧ ⎫ ⎧ ⎫∪⎨ ⎬ ⎨ ⎬∈ ∈⎩ ⎭ ⎩ ⎭ n n1 21 2N N2 2 min( r ( x C )) min( r ( x C ))max maxc C c C ∈ +⎧ ⎫∪ ∪ ⎨ ⎬∈⎩ ⎭ nmm N2 m m mx F m m min( r ( x C ))... max c C . T HEOREM
The linear [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k m ] RD-m-code C = C ∪ C ∪ … ∪ C m satisfies t(C) = t(C ) ∪ t(C ) ∪ … ∪ t(C m ) ≤ (n – k ) ∪ (n – k ) ∪ … ∪ (n m – k m ). Proof: Let C = C ∪ C ∪ … ∪ C n be a (n , k ) ∪ … ∪ (n m , k m ) RD-m-code. Consider the m-generator m-matrix G = G ∪ G ∪ … ∪ G m = k k ,n k k k ,n k k k ,n k (I , A ) (I , A ) (I , A ) − − − ∪ ∪ ∪ … . Suppose x = x ∪ x ∪ … ∪ x m = (x , x , , x , x , , x ) + … … ∪ … ∪ m m m m m m m m1 2 k k 1 n (x , x , , x , x , , x ) + … … be any m vector in n n V V ∪ ∪ … . Let C = (C ∪ C ∪ … ∪ C n )(G ∪ G ∪ … ∪ G m ) = C G ∪ C G ∪ … ∪ C n G m = (x x ) … G ∪ … ∪ m m m1 k (x x ) … G m . Then C is a m-code word of C and r(x +c) = r (x + c ) ∪ … ∪ r m (x m + c m ) ≤ n – k ∪ … ∪ n m – k m . Hence the proof. For any [n , k ] ∪ [n , k ] ∪ … ∪ [n m , k ] repetition RD m-code generated by the m-matrix G = G ∪ G ∪ … ∪ G m = (1 1 … 1) ∪ (1 1 … 1) ∪ … ∪ (1 1 … 1) (G i ≠ G j if i ≠ j; 1 ≤ i, j ≤ m) over N F . A non zero m-code word of it has m-rank 1 ∪ ∪ … ∪
1. We proceed on to define the notion of covering m-radius. T HEOREM
The covering m-radius of a [n , 1] ∪ [n , 1] ∪ … ∪ [n m , 1] repetition RD m-code over N F is (n – 1) ∪ (n – 1) ∪ … ∪ (n m – 1). Proof: The Cartesian m-product of 2 linear RD m-codes
C C [n , k ] C [n , k ] ... C [n , k ] = ∪ ∪ ∪ and
D D [n , k ] D [n , k ] ... D [n , k ] = ∪ ∪ ∪ over N F is given by C × D = C × D ∪ C × D ∪ … ∪ C m × D m {(a , b ) a C and b D } = ∈ ∈ ∪ {(a , b ) a C and b D } ∈ ∈ ∪ ∪ … m m m m1 1 1 m 1 m {(a , b ) a C and b D } ∈ ∈ . C × D is a {(n n ) (n n ) ... (n n ), + ∪ + ∪ ∪ + (k k ) + ∪ (k k ) ... (k k )} + ∪ ∪ + linear RD m-code. (We assume n n N + ≤ for i = 1, 2, … , m).
Now the reader is expected to prove the following theorem: T HEOREM
If C = C ∪ C ∪ … ∪ C m and D = D ∪ D ∪ … ∪ D m be two linear RD m-codes then t(C × D) ≤ (t(C ) + t(D )) ∪ (t(C ) + t(D )) ∪ … ∪ (t(C m ) + t(D m )). Hint:
If C = C ∪ C ∪ … ∪ C m and D = D ∪ D ∪ … ∪ D m then C × D = {C × D } ∪ {C × D } ∪ … ∪ {C m × D m } and t(C × D) = t(C × D ) ∪ t(C × D ) ∪ … ∪ t(C m × D m ) ≤ {t(C ) + t(D )} ∪ {t(C ) + t(D )} ∪ … ∪ {t(C m ) + t(D m )}. 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 D EFINITION
C = C [n , k , d ] ∪ C [n , k , d ] ∪ … ∪ C m [n m , k m , d m ] be a linear RD m-code over N q F , n i ≤ N, 1 ≤ i ≤ m and N > 1. If there exists (m , m , …, m m ) (m i > 1; i = 1, 2, …, m) such that ii i mr ( c ; q ) ; 1 ≤ i ≤ n i ; i = 1, 2, …, m for all c i ∈ C i ; then we say the m-code C is m-divisible. T HEOREM
Let C = C [n , 1, n ] ∪ C [n , 1, n ] ∪ … ∪ C m [n m , 1, n m ] (n i ≠ n j , i ≠ j; 1 ≤ i, j ≤ m) be a MRD m-code for all n i ≤ N, 1 ≤ i ≤ m. Then C is a m-divisible m-code. Proof: Since there cannot exists m-code words of m-rank greater than (n , n , …, n m ) in an [n , 1, n ] ∪ [n , 1, n ] ∪ … ∪ [n m , 1, n m ] MRD m-code. C is a m-divisible m-code. D EFINITION
Let C i = [n i , k i ] be a linear RD-code, i = 1, 2, 3, …, m and C j = [n j , k j , d j ] linear divisible RD codes, j = 1, 2, 3, …, m defined over GF(2 N ). Let m = m + m , then the RD linear m-code C = C ∪ C ∪ … ∪ C m is defined as quasi divisible RD m-code, n i ≤ N, 1 ≤ i ≤ m. D EFINITION
Let C i = C i [n i , k i , d i ] be a MRD code which is not divisible and C j = C j [n j , k j , d j ] be a divisible MRD code defined over GF (2 N ); i = 1, 2, 3, …, m and j = 1, 2, 3, …, m such that m = m + m . C = C ∪ C ∪ … ∪ C m is defined to be a quasi divisible MRD m-code. D EFINITION
Let C , C , …, m C be circulant rank codes and C j [n j , k j , d j ] a divisible RD-code defined over GF(2 N ); j = 1, 2, …, m such that m + m = m. Then C = C ∪ C ∪ … ∪ C m is defined to be a quasi divisible circulant rank m-code. D EFINITION
Let C , C , …, m C be AMRD codes and C j [n j , k j , d j ] be a divisible RD code defined over GF(2 N ), j = 1, 2, …, m such that m + m = m. Then C = C ∪ C ∪ … ∪ C m 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. D EFINITION
Let C i = C i [n i , k i , d i ], i = 1, 2, …, m be MRD codes defined over N q F , n i ≤ N; i = 1, 2, …, m with n i ≠ n j if i ≠ j, 1 ≤ i, j ≤ m. ∪ ∪ ∪ s 1 1 s 2 2 s m m A [ n ,d ] A [ n ,d ] ... A [ n ,d ] be the number of m-code words with rank m-norms s i in the linear [n i , k i , d i ] MRD-code 1 ≤ i ≤ m. Then m-spectrum of the MRD m-code C = C ∪ C ∪ … ∪ C m is described by the formulae A (n , d ) ∪ A (n , d ) ∪ … ∪ A (n m , d m ) = 1 ∪ ∪ … ∪ + + + ∪ ∪ ∪ d m 1 1 d m 2 2 d m m m A ( n ,d ) A ( n ,d ) ... A ( n ,d ) + − − − −+= +⎡ ⎤ ⎡ ⎤= − ∪⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦ ∑ j 111 1 1 11 1 11 ( m j )( m j 1 )( Q 1 )m1 1 1j m 2j 01 1 1 1 n d m( 1 ) qd m d j + − − − −+= +⎡ ⎤ ⎡ ⎤− ∪ ∪⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦ ∑ j 122 2 2 22 2 22 ( m j )( m j 1 )( Q 1 )m2 2 2j m 2j 02 2 2 2 n d m( 1 ) q ...d m d j + − − − −+= +⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦ ∑ j 1mm m m mm m mm ( m j )( m j 1 )( Q 1 )mm m mj m 2j 0m m m m n d m( 1 ) qd m d j where Q = q N ; −− ⎡ ⎤ − − −=⎢ ⎥ − − −⎣ ⎦ i i i ii i i i n n n m 1i m m m m 1i n ( q 1 )( q 2 )...( q q )m ( q 1 )( q 2 )...( q q ) ; i = 1, 2, …, m. Using the m-spectrum of a MRD m-code we prove the following theorem: T HEOREM
All MRD m-codes C [n , k , d ] ∪ C [n , k , d ] ∪ … ∪ C m [n m , k m , d m ] with d i < n i (i.e., with k i ≥
2) 1 ≤ i ≤ m are non m-divisible. Proof: This is proved by making use of the m-spectrum of the MRD m-code. Clearly d 1 1 d 2 2 d m m
A (n , d ) A (n ,d ) ... A (n , d ) 0 0 ... 0. ∪ ∪ ∪ ≠ ∪ ∪ ∪
If the existence of a m-code word with m-rank (d +1) ∪ (d +1) ∪ … ∪ (d m +1) is established then the proof is complete as the m-gcd {(d , d +1) ∪ (d , d +1) ∪ … ∪ (d m , d m +1)} = 1 ∪ ∪ … ∪
1. So the proof is to show that, d 1 1 1 d 1 2 2 d 1 m m
A (n , d ) A (n ,d ) ... A (n ,d ) + + + ∪ ∪ ∪ is non zero (i.e., i d 1 i i A (n ,d ) 0 + ≠ ; i = 1, 2, …, m). Now d 1 1 1 d 1 2 2 d 1 m m A (n , d ) A (n ,d ) ... A (n ,d ) + + + ∪ ∪ ∪ n d 1 (Q 1) (Q 1)d 1 d ⎛ ⎞+⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − − + − ∪⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎜ ⎟+⎣ ⎦ ⎣ ⎦⎝ ⎠ n d 1 (Q 1) (Q 1) ...d 1 d ⎛ ⎞+⎡ ⎤ ⎡ ⎤ ⎡ ⎤− − + − ∪ ∪⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎜ ⎟+⎣ ⎦ ⎣ ⎦⎝ ⎠ m m 2m m n d 1 (Q 1) (Q 1)d 1 d ⎛ ⎞+⎡ ⎤ ⎡ ⎤ ⎡ ⎤− − + −⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎜ ⎟+⎣ ⎦ ⎣ ⎦⎝ ⎠ n d 1(Q 1) Q 1d 1 d ⎛ ⎞+⎡ ⎤ ⎡ ⎤= − + − ∪⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟+⎣ ⎦ ⎣ ⎦⎝ ⎠ n d 1(Q 1) Q 1 ...d 1 d ⎛ ⎞+⎡ ⎤ ⎡ ⎤− + − ∪ ∪⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟+⎣ ⎦ ⎣ ⎦⎝ ⎠ m m m m n d 1(Q 1) Q 1d 1 d ⎛ ⎞+⎡ ⎤ ⎡ ⎤− + −⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟+⎣ ⎦ ⎣ ⎦⎝ ⎠ . Suppose d 1Q 1 d +⎡ ⎤+ − ∪⎢ ⎥⎣ ⎦ d 1Q 1 ...d +⎡ ⎤+ − ∪ ∪⎢ ⎥⎣ ⎦ m m d 1Q 1 d +⎡ ⎤+ − ⎢ ⎥⎣ ⎦ = 0 ∪ ∪ … ∪
0; i.e., d 1N q 1q 1 q 1 + −+ = − i.e., dN 1 q 1q 1 q − −− = . Clearly, dN 1 q 1 1q − − < For, d N 1 dN 1 q 1if 1 then q q 1q −− − ≥ < − which is impossible as d < n ≤ N. Thus q – 1 < 1 which implies q < 2 a contradiction. Hence, d 1 1 1 d 1 2 2 d 1 m m
A (n , d ) A (n , d ) ... A (n , d ) + + + ∪ ∪ ∪ is non zero. Thus expect C (n , 1, n ) ∪ C (n , 1, n ) ∪ … ∪ C m (n m , 1, n m ) MRD m-codes all C [n , k , d ] ∪ C [n , k , d ] ∪ … ∪ C m [n m , k m , d m ] MRD m-codes with d i ≤ n i ; i = 1, 2, …, m are non divisible. Now finally we define the fuzzy rank distance m-codes (m ≥ ≥
3) when (m = ≥ D EFINITION
Let ∪ ∪ ∪ m1 2 nn n
V V ... V denote the (n , n , …, n m ) dimensional vector m-space of (n , n , …, n m )-tuples over N F ; n i ≤ N and N >1; 1 ≤ i ≤ m. Let u i , v i ∈ i n V ; i = 1, 2, …, m, where, = i i i ii 1 2 n u ( u ,u ,...,u ) and = i i i ii 1 2 n v ( v ,v ,...,v ) with ∈ N i ij j 2 u , v F ; 1 ≤ j ≤ n i ; 1 ≤ i ≤ m. A fuzzy RD m-code word ∪ ∪ ∪ = ∪ ∪ ∪ f f f ... f is a fuzzy m-subset of ∪ ∪ ∪ m1 2 nn n V V ... V defined by, ∪ ∪ ∪ = ∪ ∪ ∪ f f f ... f = ∈ ∪ ∈ ∪ ∪ n n1 21 u 1 1 2 u 2 2 { ( v , f ( v )) v V } { ( v , f ( v )) v V } ... ∈ mm nmm u m m { ( v , f ( v )) v V } where ∪ ∪ ∪ f ( v ) f ( v ) ... f ( v ) is the membership m-function. D EFINITION
For the symmetric error m-model assume p ∪ p ∪ … ∪ p m to represent the m-probability that no transition (i.e., error) is made and q ∪ q ∪ … ∪ q m to represent the m-probability that a m-rank error occurs so that, p + q ∪ p + q ∪ … ∪ p m + q m = 1 ∪ ∪ … ∪
1, then ∪ ∪ ∪ … f ( v ) f ( v ) f ( v ) −− − = ∪ ∪ ∪ … m m m1 1 1 2 2 2 n r rn r r n r r1 1 2 2 m m p q p q p q where = − = − i i i i i i r r ( u v ,2 ) u v , i = 1, 2, …, m. D EFINITION
For unidirectional and asymmetric error m-models assume q ∪ q ∪ … ∪ q m to represent the probability that (1 → ∪ (1 → ∪ … ∪ (1 →
0) m-transition or (0 → ∪ (0 → ∪ … ∪ (0 →
1) m-transition occurs. Then ∪ ∪ ∪ f ( v ) f ( v ) ... f ( v ) = = = = ∪ ∪ ∪ ∏ ∏ ∏ m1 21 2 mi i i nn n1 1 2 2 m mi i iu u ui 1 i 1 i 1 f ( v ) f ( v ) ... f ( v ) where ∪ ∪ ∪ … f ( v ) f ( v ) f ( v ) inherits its definition from the unidirectional and asymmetric m-models respectively, since each ∪ ∪ ∪ … u u u or ∪ ∪ ∪ … v v v itself is an N-m-tuple over F . That is since ∈ N j ji i 2 u , v F ; j = 1, 2, …, m; each ∪ ∪ ∪ … u u u or ∪ ∪ ∪ … v v v itself is an N, m-tuple from F . = ∪ ∪ ∪ … j j j ji i1 i2 iN u ( u u u ) , = ∪ ∪ ∪ … j j j jk k1 k 2 kN v ( v v v ) where, ∈ j jip kl 2 u v F , 1 ≤ p, l ≤ N and 1 ≤ i ≤ m. Then for unidirectional error m-model ∪ ∪ ∪ = … f ( v ) f ( v ) f ( v ) − − − ⎧ ∪ ∪ ∪⎪ ∪ ∪ ≠ ∪ ∪⎨⎪ ∪ ∪ ∪⎩ … = = − ∑ Nj j jit is iss 1 k max( 0,u v ) where j = 1, 2, …, m and i = 1, 2, …, m. ⎧ =⎪= ⎨ =⎪⎩ j ji1 i2ji j ji2 i1 k if k 0d k if k 0 j = 1, 2, …, m. ( ) = = ⎧ =⎪⎪⎪= − =⎨⎪⎪ − = =⎪⎩ ∑ ∑∑ ∑
N j jis i2s 1 Nj j ji is i1s 1 j j j dis is i1 i2 u if k 0m N u if k 0max u ,N u if k k 0 j = 1, 2, …, m. For the asymmetric error m-model ∪ ∪ ∪ = f ( v ) f ( v ) ... f ( v ) − − − ⎧ ∪ ∪⎪ ∪ ∪ ≠ ∪ ∪⎨⎪ ∪ ∪ ∪⎩ … … …… = j ji i1 d k and j = 1, 2, …, m. = = ∑ Nj ji iss 1 m u , j = 1, 2, …, m for asymmetric (1 → ∪ (1 →
0) error m-model and = j ji i2 d k , j = 1, 2, …, m and = = − ∑ Nj ji iss 1 m N u , j = 1, 2, …, m for the asymmetric 0 → ∪ →
1 error m-model.
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 ≥ P ROPOSITION If ∪ ∪ ∪ … C C C and ∪ ∪ ∪ … C C C are RD m-codes with ∪ ∪ ∪ … C C C ⊆ ∪ ∪ ∪ … C C C (i.e., ⊆ C C ; j = 1, 2, …, m) then ∪ ∪ ∪ ≥ t ( C ) t ( C ) ... t ( C ) ∪ ∪ ∪ t ( C ) t ( C ) ... t ( C ) (i.e., i i t (C ) t (C ) ≥ ; i = 1, 2, …, m). Proof: Let i ni S V ⊆ with 1 ≤ j ≤ m; |S i | = m i ; i = 1, 2, …, m cov(C ,S ) cov(C ,S ) ... cov(C ,S ) ∪ ∪ ∪ =
21 1 1 1 min{cov(x ,S ); x C } ∈ ∪
22 2 2 2 min{cov(x ,S ); x C } ... ∈ ∪ ∪
2m m m m min{cov(x ,S ); x C } ∈
11 1 1 1 min{cov(x ,S ) x C } ≤ ∈ ∪
12 2 2 2 min{cov(x ,S ) x C } ... ∈ ∪ ∪
1m m m m min{cov(x ,S ) x C } ∈ cov(C ,S ) cov(C ,S ) ... cov(C ,S ) = ∪ ∪ ∪ . Thus t (C ) t (C ) ... t (C ) ∪ ∪ ∪ t (C ) t (C ) ... t (C ) ≤ ∪ ∪ ∪ . P ROPOSITION
For any RD m-code C = C ∪ C ∪ … ∪ C m and a m-tuple of positive integers (m , m , …, m m ) ∪ ∪ ∪ ≤ … m 1 m 2 m m t ( C ) t ( C ) t ( C ) + + + ∪ ∪ ∪ … m 1 1 m 1 2 m 1 m t ( C ) t ( C ) t ( C ) . Proof: Since i ni S V ⊆ ; i = 1, 2, …, m; S ∪ S ∪ … ∪ S m is a m-subset of n n n V V V ∪ ∪ ∪ … . Now m 1 m 2 m m t (C ) t (C ) t (C ) ∪ ∪ ∪ … n1 1 1 1 1 max{cov(C ,S ) S V , S m } = ⊆ = ∪ n2 2 2 2 2 max{cov(C ,S ) S V , S m } ... ⊆ = ∪ ∪ m nm m m m m max{cov(C ,S ) S V , S m } ⊆ = n1 1 1 1 1 max{cov(C ,S ) S V , S m 1} ≤ ⊆ = + ∪ n2 2 2 2 2 max{cov(C ,S ) S V , S m 1} ... ⊆ = + ∪ ∪ m nm m m m m max{cov(C ,S ) S V , S m 1} ⊆ = + m 1 1 m 1 2 m 1 m t (C ) t (C ) ... t (C ) + + + = ∪ ∪ ∪ . P ROPOSITION
For any m-set of positive integers {n , m , k , K } ∪ {n , m , k , K } ∪ … ∪ {n m , m m , k m , K m }; ∪ ∪ ∪ … m 1 1 m 2 2 m m m t [ n ,k ] t [ n ,k ] t [ n ,k ] + + + ≤ ∪ ∪ ∪ … m 1 1 1 m 1 2 2 m 1 m m t [ n ,k ] t [ n ,k ] t [ n ,k ] . Proof: Given C [n , k ] ∪ C [n , k ] ∪ … ∪ C m [n m , k m ] to be a RD m-code with i ni C V ⊆ ; i = 1, 2, …, m. Now m 1 1 m 2 2 m m m t [n , k ] t [n , k ] ... t [n , k ] ∪ ∪ ∪ nm 1 1 1 1 min{t (C ) C V , dim C k } = ⊆ = ∪ nm 2 2 2 2 min{t (C ) C V , dim C k } ... ⊆ = ∪ ∪ mm nm m m m m min{t (C ) C V ,dim C k } ⊆ = nm 1 1 1 1 1 min{t (C ) C V , dim C k } + ≤ ⊆ = ∪ nm 1 2 2 2 2 min{t (C ) C V , dim C k } ... + ⊆ = ∪ ∪ mm nm 1 m m m m min{t (C ) C V ,dim C k } + ⊆ = m 1 1 1 m 1 2 2 m 1 m m t [n , k ] t [n , k ] ... t [n , k ] + + + = ∪ ∪ ∪ . Similarly we have m 1 1 m 2 2 m m m t [n , K ] t [n , K ] ... t [n , K ] ∪ ∪ ∪ m 1 1 1 m 1 2 2 m 1 m m t [n , K ] t [n , K ] ... t [n , K ] + + + ≤ ∪ ∪ ∪ . That is m 1 1 m 2 2 m m m t [n , K ] t [n , K ] ... t [n , K ] ∪ ∪ ∪ nm 1 1 1 1 min{t (C ) C V , C K } = ⊆ = ∪ nm 2 2 2 2 min{t (C ) C V , C K } ... ⊆ = ∪ ∪ mm nm m m m m min{t (C ) C V , C K } ⊆ = nm 1 1 1 1 1 min{t (C ) C V , C K } + ≤ ⊆ = ∪ nm 1 2 2 2 2 min{t (C ) C V , C K } ... + ⊆ = ∪ ∪ mm nm 1 m m m m min{t (C ) C V , C K } + ⊆ = m 1 1 1 m 1 2 2 m 1 m m t [n , K ] t [n , K ] ... t [n , K ] + + + ≤ ∪ ∪ ∪ . P ROPOSITION
For any m-set of positive integers {n , m , k , K } ∪ {n , m , k , K } ∪ … ∪ {n m , m m , k m , K m }; ∪ ∪ ∪ m 1 1 m 2 2 m m m t [ n ,k ] t [ n ,k ] ... t [ n ,k ] ≥ + ∪ + ∪ ∪ + m 1 1 m 2 2 m m m t [ n ,k 1] t [ n ,k 1] ... t [ n ,k 1] . Proof:
Given C = C ∪ C ∪ … ∪ C m is a RD m-code, hence a m-subspace of n n n V V V ∪ ∪ ∪ … . Consider m 1 1 m 2 2 m m m t [n , k 1] t [n , k 1] ... t [n , k 1] + ∪ + ∪ ∪ + nm 1 1 1 1 min{t (C ) C V , dim C k 1} = ⊆ = + ∪ nm 2 2 2 2 min{t (C ) C V , dim C k 1} ... ⊆ = + ∪ ∪ mm nm m m m m min{t (C ) C V ,dim C k 1} ⊆ = + nm 1 1 1 1 min{t (C ) C V ,dim C k } ≤ ⊆ = ∪ nm 2 2 2 2 min{t (C ) C V , dim C k } ... ⊆ = ∪ ∪ mm nm m m m m min{t (C ) C V ,dim C k } ⊆ = since for each C ∪ C ∪ … ∪ C m ⊆ C ∪ C ∪ … ∪ C m2 m 12 m 22 m m2 t (C ) t (C ) ... t (C ) ∪ ∪ ∪ m 1 m 2 m m t (C ) t (C ) ... t (C ) ≤ ∪ ∪ ∪ m 1 1 m 2 2 t [n , k ] t [n , k ] = ∪ ∪ … ∪ m m m m t [n , k ]. . Similarly m 1 1 m 2 2 m m m t (n , K 1) t (n , K 1) ... t (n , K 1) + ∪ + ∪ ∪ + m 1 1 m 2 2 m m m t (n , K ) t (n , K ) ... t (n , K ) ≤ ∪ ∪ ∪ . Using these results and the fact i im i i k [n , t ] denotes the smallest dimension of a linear RD code of length n i and m i covering radius t i and i im i i K [n , t ] denotes the least cardinality of the RD codes of length n i and m i -covering radius t i the following results can be easily proved. Result 1:
For any m-set of positive integers {n , m , t } ∪ {n , m , t } ∪ … ∪ {n m , m m , t m }; and m 1 1 m 2 2 m m m k [n , t ] k [n , t ] ... k [n , t ] ∪ ∪ ∪ m 1 1 1 m 1 2 2 m 1 m m k [n , t ] k [n , t ] ... k [n , t ] + + + ≤ ∪ ∪ ∪ and m 1 1 m 2 2 m m m K (n , t ) K (n , t ) ... K (n , t ) ∪ ∪ ∪ m 1 1 1 m 1 2 2 m 1 m m
K (n , t ) K (n , t ) ... K (n , t ) + + + ≤ ∪ ∪ ∪ . Result 2:
For any m-set of positive integers {n , m , t } ∪ {n , m , t } ∪ … ∪ {n m , m m , t m }. we have, m 1 1 m 2 2 m m m k [n , t ] k [n , t ] ... k [n , t ] ∪ ∪ ∪ m 1 1 m 2 2 m m m k [n , t 1] k [n , t 1] ... k [n , t 1] ≥ + ∪ + ∪ ∪ + and m 1 1 m 2 2 m m m K (n , t ) K (n , t ) ... K (n , t ) ∪ ∪ ∪ m 1 1 m 2 2 m m m
K (n , t 1) K (n , t 1) ... K (n , t 1) ≥ + ∪ + ∪ ∪ + . We say a m-function f ∪ f ∪ … ∪ f m is a non decreasing m-function in some m-variable say x ∪ x ∪ … ∪ x m if each f i happen to be a non-decreasing function in the variable x i ; i = 1, 2, …, m. With this understanding we have for (m , m , … , m m )-covering m-radius of a fixed RD m-code C ∪ C ∪ … ∪ C m , m 1 1 m 2 2 m m m t [n , k ] t [n , k ] ... t [n , k ] ∪ ∪ ∪ , m 1 1 m 2 2 m m m k [n , t ] k [n , t ] ... k [n , t ] ∪ ∪ ∪ , m 1 1 m 2 2 m m m t (n , K ) t (n , K ) ... t (n , K ) ∪ ∪ ∪ and m 1 1 m 2 2 m m m K (n , t ) K (n , t ) ... K (n , t ) ∪ ∪ ∪ are non decreasing m-functions of (m , m , … , m m ). The relationships between the multi covering m-radii of two RD m-codes that are built using them are described. Let i i i i1 2 m C C C .... C = ∪ ∪ ∪ for i = 1, 2 be a [ n , k ,d ] [n , k , d ] ... [n , k ,d ] ∪ ∪ ∪ and [ n , k , d ] [n , k , d ] ... [n , k ,d ] ∪ ∪ ∪
RD m-codes over N F with i i i i1 2 1 2 n , n , n n N + ≤ for i = 1, 2, …, m. P ROPOSITION
Let = ∪ ∪ ∪
C C C ... C and = ∪ ∪ ∪
C C C ... C be RD m-codes described above = × = × ∪ × ∪ ∪ ×
C C C ( C C ) ( C C ) ... ( C C ) = ∈ ∈ ∪ { ( x y ) x C , y C } ∈ ∈ ∪ ∪ { ( x y ) x C , y C } ... ∈ ∈ { ( x y ) x C , y C } .
Then C × C is a + ∪ + ∪ ∪ + [ n n n n ... n n , + ∪ + ∪ ∪ + k k k k ... k k , ∪ ∪ ∪ min{ d ,d } min{ d ,d } ... min{ d ,d }] rank distance m-code over N F and × ∪ × ∪ ∪ × t ( C C ) t ( C C ) ... t ( C C ) ≤ + ∪ t ( C ) t ( C ) t ( C ) + ∪ ∪ + … t ( C ) t ( C ) t ( C ) . Proof:
Let i i1 2 n ni
S V + ⊆ ; for i = 1, 2, …, m. and i i ii 1 m S {s ,...,s } = for i = 1, 2, …, m with ji ji ji s (x y ) for i = 1, 2, …, m; j1 nji x V ∈ and j2 nji y V ∈ , 1 ≤ i ≤ m i ; 1 ≤ i ≤ m. Let S {x ,..., x }, S {y ,..., y } = = , S {x ,..., x }, S {y ,..., y } = = , …, m m
S {x ,..., x } and S {y ,..., y } = = . Now i im 1 t (C ) being the m i covering radius of i1 C ; i = 1, 2, …, m, there exists i i1 1 c C ∈ such that mi i i i1 t 1 S B (C ) ⊆ ; i = 1, 2, …, m. This implies as in case of RD bicodes j j j j jj ji j ji ji 1 2 j ji 1 ji 2 r (s C ) r ((x y ) (C C )) r (x C y C ) + = + = + + j j j j j jj ji 1 j ji 2 m 1 m 2 r (x C ) r (y C ) t (C ) t (C ) ≤ + + + ≤ + ; j = 1, 2, …, m. Thus t (C) t (C C ) t (C C ) ... t (C C ) = × ∪ × ∪ ∪ × ≤ t (C ) t (C ) t (C ) t (C ) ... t (C ) t (C ). + ∪ + ∪ ∪ + . For any m times (r, r,..., r) − (cid:8)(cid:11)(cid:9)(cid:11)(cid:10) (r a positive integer (m ≥
3) the (r, r, …, r) fold repetition RD m-code C ∪ C ∪ … ∪ C m is the m-code C {(c | c | ... | c ) c C } = ∈ ∪ {(c | c | ... | c ) c C } ... ∈ ∪ ∪ m m m m m {(c | c | ... | c ) c C } ∈ where the m-code word C ∪ C ∪ … ∪ C m is a concatenation of (r, r, …, r) times, this is a [rn , k , d ] ∪ [rn , k , d ] ∪ … ∪ [rn m , k m , d m ] rank distance m-code with n i ≤ N and rn i ≤ N; i = 1, 2, …, m. Thus any m-code word in C ∪ C ∪ … ∪ C m would be of the form {(c | c | ... | c )} {(c | c | ... | c )} ... {(c | c | ... | c )} ∪ ∪ ∪ such that x i ∈ C i for i = 1, 2, …, m. We can also define (r , r , …, r m ) fold repetition m-code (r i ≠ r j if i ≠ j; 1 ≤ i, j ≤ m). D EFINITION
Let C ∪ C ∪ … ∪ C m be a [n , k , d ] ∪ [n , k , d ] ∪ … ∪ [n m , k m , d m ] RD m-code. Let c i = { } − ∈ (cid:8)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:10) i i i i i ir times ( c | c | ...| c ) c C be a r i -fold repetition RD-code C i , i = 1, 2, …, m. Then C ∪ C ∪ … ∪ C m is defined as the (r , r , …, r m )-fold repetition m-code each r i n i < N for i = 1, 2, …, m.
We prove the following interesting theorem. T HEOREM
For an (r, r, …, r) fold repetition m-code C ∪ C ∪ … ∪ C m m 1 m 2 m m t ( C ) t ( C ) t ( C ) ∪ ∪ ∪ … t ( C ) t ( C ) t ( C ) = ∪ ∪ ∪ … . Proof:
Let ii ni i1 i2 im S {v , v ,..., v } V = ⊆ for i = 1, 2, …, m; such that i i i m i cov(C ,S ) t (C ) = ; i = 1, 2, …, m. Let i1 i1 i1 i1 v (v | v | ... | v ) ′ = i2 i2 i2 i2 v (v | v | ... | v ) ′ = and so on im im im im v (v | v | ... | v ) ′ = ; 1 ≤ i ≤ m. Let i i i1 i2 im S {v , v ,..., v } ′ ′ ′= be the set of m i -vectors of length m i for, i = 1, 2, …, m. An r fold repetition of any RD code word retains the same rank weight. Hence i i i m i (C ,S ) t (C ) ′ = true for i = 1, 2, …, m. Since i i im i t (C ) cov(C ,S ) ′= it follows that i i im m i t (C ) t (C ) ≥ for i = 1, 2, …, m; i.e., t (C ) t (C ) ... t (C ) ∪ ∪ ∪ ≥ m 1 m 2 m m t (C ) t (C ) ... t (C ) ∪ ∪ ∪ ----- I Conversely let i i i1 i2 im S {v , v ,..., v } = be a set of m i vectors i = 1, 2, …, m of length rn i with ij ij ij v (v | ... | v ) ′ ′= ; j = 1, 2, …, m i , i = 1, 2, …, m. i ni v V ′ ∈ , 1 ≤ i ≤ m i . Then there exists c i ∈ C i such that i i R i i m i d (c , v ) t (C ) ′ ≤ ; i = 1, 2, …, m i ; 1 ≤ i ≤ m. This implies i i R i i i ij m i d ((c | c | ... | c ), v ) t (C ) ≤ for every i(1 ≤ i ≤ m). Thus i i im m i t (C ) t (C ) ≤ , i = 1, 2, …, m; i.e., t (C ) t (C ) ... t (C ) ∪ ∪ ∪ ≤ m 1 m 2 m m t (C ) t (C ) ... t (C ) ∪ ∪ ∪ ------- II From I and II t (C ) t (C ) ... t (C ) ∪ ∪ ∪ = m 1 m 2 m m t (C ) t (C ) ... t (C ) ∪ ∪ ∪ . Now we proceed on to analyse the notion of multi covering m-bounds for RD m-codes. The (m , m , … , m m ) covering m-radius m 1 m 2 m m t (C ) t (C ) ... t (C ) ∪ ∪ ∪ of a RD m-code C = C ∪ C ∪ … ∪ C m is a non-decreasing m-function of m ∪ m ∪ … ∪ m m . Thus a lower m-bound for m 1 m 2 m m t (C ) t (C ) ... t (C ) ∪ ∪ ∪ implies a m-bound for m 1 1 m 1 2 m 1 m t (C ) t (C ) ... t (C ) + + + ∪ ∪ ∪ . First m-bound exhibits that for m ∪ m ∪ … ∪ m m ≥ ∪ ∪ … ∪
2 the situation of (m , m , …, m m ) covering m-radii is quite different for ordinary covering radii. P REPOSITION
If m ∪ m ∪ … ∪ m m > ∪ ∪ … ∪ , m , …, m m ) covering m-radii of a RD m-code C = C ∪ C ∪ … ∪ C m of m-length (n , n , …, n m ) is at least ⎡ ⎤⎡ ⎤ ⎡ ⎤∪ ∪ ∪ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ m1 2 nn n ...2 2 2 . Proof:
Let C = C ∪ C ∪ … ∪ C m be a RD m-code of m-length (n , n , … , n m ) over GF(2 N ). Let m ∪ m ∪ … ∪ m m = ∪ ∪ … ∪
2. Let t , t , …, t m be the 2-covering m-radii of the RD m-code C = C ∪ C ∪ … ∪ C m . Let x = x ∪ x ∪ … ∪ x m ∈ n n n V V ... V ∪ ∪ ∪ . Choose y = y ∪ y ∪ … ∪ y m ∈ n n n V V ... V ∪ ∪ ∪ such that all the (n , n , …, n m ) coordinate of x y (x y ) (x y ) ... (x y ) − = − ∪ − ∪ ∪ − are linearly independent that is R R 1 2 m 1 2 m d (x, y) d (x x ... x , y y ... y ) = ∪ ∪ ∪ ∪ ∪ ∪
R 1 1 R 2 2 R m m d (x , y ) d (x , y ) ... d (x , y ) = ∪ ∪ ∪ (R = R ∪ R ∪ … ∪ R m and R R R R d d d ... d = ∪ ∪ ∪ ) = n ∪ n ∪ … ∪ n m. Then for any c = c ∪ c ∪ … ∪ c m ∈ C ∪ C ∪ … ∪ C m . R R d (x c) d (c y) + + +
R 1 1 R 1 1 R 2 2 R 2 2 d (x , c ) d (c , y ) d (x , c ) d (c , y ) ... = + ∪ + ∪ ∪ m m
R m m R m m d (x , c ) d (c , y ) + R 1 1 R 2 2 R m m d (x , y ) d (x , y ) ... d (x , y ) ≥ ∪ ∪ ∪ = n ∪ n ∪ … ∪ n m ; this implies that one of R 1 1 R 2 2 R m m d (x ,c ) d (x , c ) ... d (x , c ) ∪ ∪ ∪ and
R 1 1 R 2 2 R m m d (c , y ) d (c , y ) ... d (c , y ) ∪ ∪ ∪ is at least n n n...2 2 2 ∪ ∪ ∪ . (That is one of i R i i d (x , c ) and i R i i d (c , y ) is atleast i n2 ; i = 1, 2, …, m) and hence t = t ∪ t ∪ … ∪ t m ≥ n n n...2 2 2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤∪ ∪ ∪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . Since t is a non-decreasing m-function of m ∪ m ∪ … ∪ m m it follows that m m 1 m 2 m m t (C) t (C ) t (C ) ... t (C ) = ∪ ∪ ∪ ≥ n n n...2 2 2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤∪ ∪ ∪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ for m ∪ m ∪ … ∪ m m ≥ ∪ ∪ … ∪
2. m-bounds of the multi covering m-radius of n n n
V V ... V ∪ ∪ ∪ can be used to obtain m-bounds on the multi covering m-radii of arbitrary m-codes. Thus a relationship between (m , m , …, m m ) covering m-radii of an RD m-code and that of its ambient m-space n n n V V ... V ∪ ∪ ∪ is established. T HEOREM
Let C = C ∪ C ∪ … ∪ C m be RD m-code of m-length n ∪ n ∪ … ∪ n m over ∪ ∪ ∪ … N N N
F F F . Then for any positive m-integer tuple (m , m , …, m m ) ∪ ∪ ∪ t ( C ) t ( C ) ... t ( C ) ≤ + ∪ + ∪ ∪ n n1 1 2 21 1 m 1 2 m t ( C ) t (V ) t ( C ) t (V ) ... + mm nm m1 m m t ( C ) t (V ) . Proof: Let S = S ∪ S ∪ … ∪ S m n n n V V V ⊆ ∪ ∪ ∪ … (i.e., i ni S V ⊆ ; i = 1, 2, …, m) with |S| = |S | ∪ |S | ∪ … ∪ |S m | = m ∪ m ∪ … ∪ m m . Then there exists u = u ∪ u ∪ … ∪ u m n n n V V ... V ∈ ∪ ∪ ∪ such that cov(u,S) cov(u ,S ) cov(u ,S ) ... cov(u ,S ) = ∪ ∪ ∪ n n n1 2 mm m m t (V ) t (V ) ... t (V ) ≤ ∪ ∪ ∪ . Also there is a c = c ∪ c ∪ … ∪ c m ∈ C ∪ C ∪ … ∪ C m such that R R 1 1 R 2 2 R m m d (c, u) d (c , u ) d (c , u ) ... d (c , u ) = ∪ ∪ ∪ t (C ) t (C ) ... t (C ) ≤ ∪ ∪ ∪ . Now, cov(c,S) cov(c ,S ) cov(c ,S ) ... cov(c ,S ) = ∪ ∪ ∪ R 1 1 1 1 max{d (c , y ) y S } = ∈ ∪ R 2 2 2 2 max{d (c , y ) y S } ... ∈ ∪ ∪ m R m m m m max{d (c , y ) y S } ∈ R 1 1 R 1 1 1 1 max{d (c , u ) d (u , y ) y S } ≤ + ∈ ∪
R 2 2 R 2 2 2 2 max{d (c , u ) d (u , y ) y S } ... + ∈ ∪ ∪ m m
R m m R m m m m max{d (c , u ) d (u , y ) y S } + ∈ R 1 1 1 1 d (c , u ) cov(u ,S ) = + ∪ R 2 2 2 2 d (c , u ) cov(u ,S ) ... + ∪ ∪ m R m m m m d (c , u ) cov(u ,S ) + n n1 1 2 21 1 m 1 2 m t (C ) t (V ) t (C ) t (V ) ... ≤ + ∪ + ∪ ∪ mm nm m1 m m t (C ) t (V ) + . Thus for every S = S ∪ S ∪ … ∪ S m n n n V V ... V ⊆ ∪ ∪ ∪ with |S| = |S | ∪ |S | ∪ … ∪ |S m | = m = m ∪ m ∪ … ∪ m m one can find c = c ∪ c ∪ … ∪ c m ∈ C ∪ C ∪ … ∪ C m such that cov(c,S) cov(c ,S ) cov(c ,S ) ... cov(c ,S ) = ∪ ∪ ∪ n n1 1 2 21 1 m 1 2 m t (C ) t (V ) t (C ) t (V ) ... ≤ + ∪ + ∪ ∪ mm nm m1 m m t (C ) t (V ) + . Since cov(c,S) cov(c ,S ) cov(c ,S ) ... cov(c ,S ) = ∪ ∪ ∪ min{cov(a ,S ) a C } = ∈ ∪ min{cov(a ,S ) a C } ... ∈ ∪ ∪ m m m m min{cov(a ,S ) a C } ∈ n n1 1 2 21 1 m 1 2 m {t (C ) t (V )} {t (C ) t (V )} ... ≤ + ∪ + ∪ ∪ mm nm m1 m m {t (C ) t (V )} + for all S = S ∪ S ∪ … ∪ S m n n n V V ... V ⊆ ∪ ∪ ∪ with |S| = |S | ∪ |S | ∪ … ∪ |S m | = m ∪ m ∪ … ∪ m m , it follows that t (C ) t (C ) ... t (C ) ∪ ∪ ∪ n1 1 1 1 1 max{cov(C ,S ) S V , S m } = ⊆ = ∪ n2 2 2 2 2 max{cov(C ,S ) S V , S m } ... ⊆ = ∪ ∪ m nm m m m m max{cov(C ,S ) S V , S m } ⊆ = n n1 1 2 21 1 m 1 2 m t (C ) t (V ) t (C ) t (V ) ... ≤ + ∪ + ∪ ∪ mm nm m1 m m t (C ) t (V ) + . P ROPOSITION
For any m-tuple of integer (n , n , …, n m ); n ∪ n ∪ … ∪ n m ≥ ∪ ∪ … ∪ ∪ ∪ ∪ … m1 2 nn n1 2 m2 2 2 t (V ) t (V ) t (V ) ≤ − ∪ − ∪ ∪ − … n 1 n 1 n 1 where = i iN n n2 V F ; i = 1, 2, …, n i ; n i ≤ N, i = 1, 2, …, m. Proof:
Let i i i ii 1 2 n x (x , x ,..., x ) = and ii ni i ii 1 2 n y (y , y ,..., y ) V = ∈ ; i = 1, 2, …, m. Let u = u ∪ u ∪ … ∪ u m n n n V V ... V ∈ ∪ ∪ ∪ where i i i i i i ii 1 2 3 n 1 n u (x , u , u ,..., u , y ) − = ; i = 1, 2, …, m. Thus u = u ∪ u ∪ … ∪ u m m-covers x ∪ x ∪ … ∪ x m and y ∪ y ∪ … ∪ y m n n n V V ... V ∈ ∪ ∪ ∪ within a m-radius n –1 ∪ n –1 ∪ … ∪ n m –1 as i R i i i d (u , x ) n 1 ≤ − ; i = 1, 2, …, m. Thus for any pair of m-vectors x ∪ x ∪ … ∪ x m , y ∪ y ∪ … ∪ y m in n n n V V ... V ∪ ∪ ∪ there always exists a m-vector namely u = u ∪ u ∪ … ∪ u m which m-covers x ∪ x ∪ … ∪ x m and y ∪ y ∪ … ∪ y m within a m-radius n –1 ∪ n –1 ∪ … ∪ n m –1. Hence n n n1 2 m2 2 2 t (V ) t (V ) ... t (V ) ∪ ∪ ∪ n 1 n 1 ... n 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 t ∪ t ∪ … ∪ t m , m ∪ m ∪ … ∪ m m and n ∪ n ∪ … ∪ n m what is the smallest RD m-code whose m ∪ m ∪ … ∪ m m , m-covering m-radius is atmost t ∪ t ∪ … ∪ t m . As it turns out even for m ∪ m ∪ … ∪ m m ≥ ∪ ∪ … ∪
2, it is necessary that t ∪ t ∪ … ∪ t m be atleast n n n...2 2 2 ∪ ∪ ∪ . Infact the minimal t ∪ t ∪ … ∪ t m for which such a m-code exists is the (m , m , …, m m ), m-covering m-radius of C ∪ C ∪ … ∪ C m = n n n2 2 2 F F ... F ∪ ∪ ∪ . Various external values associated with this notion are n n n1 2 mm m m t (V ) t (V ) ... t (V ) ∪ ∪ ∪ , the smallest (m , m , …, m m ) covering m-radius among m-length n ∪ n ∪ … ∪ n m RD-m-codes t (n , K ) t (n , K ) ... t (n , K ) ∪ ∪ ∪ ; the smallest (m , m , …, m m ) covering m-radius among all (n , K ) ∪ (n , K ) ∪ … ∪ (n m , K m ) RD-m-codes. K (n , t ) K (n , t ) ... ∪ ∪ ∪ m m mm m K (n , t ) is the smallest m-cardinality of a m-length n ∪ n ∪ … ∪ n m . RD-m-code with m ∪ m ∪ … ∪ m m covering m-radius t ∪ t ∪ … ∪ t m and so on. It is the latter quantity that is studied in the book for deriving new lower m-bounds. From the earlier results K (n , t ) K (n , t ) ... ∪ ∪ ∪ m m mm m K (n , t ) is undefined if t ∪ t ∪ … ∪ t m n n n...2 2 2 < ∪ ∪ ∪ . When this is the case it is accepted to say K (n , t ) K (n , t ) ... ∪ ∪ ∪ m m mm m K (n , t ) = ∞ ∪ ∞ ∪ … ∪ ∞ . There are other circumstances when K (n , t ) K (n , t ) ... ∪ ∪ ∪ m m mm m K (n , t ) is undefined. For instance
N Nn n1 2
K (n , n 1) K (n , n 1) ... − ∪ − ∪ ∪
Nnm m m m2
K (n , n 1) − = ∞ ∪ ∞ ∪ … ∪ ∞. m V(n , t ), m V(n , t ),..., m V(n , t ) > > > ; since in this case no m-ball of m-radius t ∪ t ∪ … ∪ t m m-covers any m-set of m ∪ m ∪ … ∪ m m distinct m-vectors. More generally one has the fundamental issue of whether K (n , t ) K (n , t ) ... ∪ ∪ ∪ m m mm m K (n , t ) is m-finite for a given
11 1 n , m , t ,
22 2 n , m , t ,..., mm m n , m , t . This is the case if and only if n n n1 1 2 2 m mm m m t (V ) t , t (V ) t ,..., t (V ) t ≤ ≤ ≤ since n n n1 2 mm m m t (V ) t (V ) ... t (V ) ∪ ∪ ∪ lower m-bounds the (m , m , …, m m ) covering m-radius of all other m-codes of m-dimension n ∪ n ∪ … ∪ n m when t ∪ t ∪ … ∪ t m = n ∪ n ∪ … ∪ n m every m-code word m-covers every m-vector, so a m-code of size 1 ∪ ∪ … ∪ , m , …, m m ) m-cover n n n V V ... V ∪ ∪ ∪ for every m ∪ m ∪ … ∪ m m . Thus K (n , n ) K (n , n ) ... ∪ ∪ ∪ m mm m m K (n , n ) = 1 ∪ ∪ … ∪ for every m ∪ m ∪ … ∪ m m . If t ∪ t ∪ … ∪ t m = n – 1 ∪ n – 1 ∪ … ∪ n m – 1 what happens to K (n , t ) K (n , t ) ... ∪ ∪ ∪ m m mm m K (n , t ) ? When m = m = … = m m , K (n , n 1) K (n , n 1) ... − ∪ − ∪ ∪ m2 m m
K (n , n 1) − n 1 n 2 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 n – 1 ∪ n – 1 ∪ … ∪ n m – 1 within m-radius n – 1 ∪ n – 1 ∪ … ∪ n m – 1. That is 0 0 ... 0 ∪ ∪ ∪ = (0, 0, …, 0) ∪ (0, 0, …, 0) ∪ (0, 0, …, 0) will m-cover all m-norm n – 1 ∪ n – 1 ∪ … ∪ n m – 1 m-vectors within the m-radius n – 1 ∪ n – 1 ∪ … ∪ n m – 1. Hence remaining m-vectors are m-rank n ∪ n ∪ … ∪ n m m-vectors. Thus 0 0 ... 0 ∪ ∪ ∪ = (0, 0, …, 0) ∪ (0, 0, …, 0) ∪ (0, 0, …, 0) and these m-rank (n ∪ n ∪ … ∪ n m ) m-vector can m-cover the ambient m-space within the m-radius n – 1 ∪ n – 1 ∪ … ∪ n m – 1. Therefore, K (n , n 1) K (n , n 1) ... − ∪ − ∪ ∪ m2 m m
K (n , n 1) − n 1 n 2 n m ≤ + ∪ + ∪ ∪ + . P ROPOSITION
For any RD m-code of m-length n ∪ n ∪ … ∪ n m over ∪ ∪ ∪ … N N N
F F F , − ∪ − ∪ ∪ … K ( n ,n 1 ) K ( n ,n 1 ) − m2 m m K ( n ,n 1 ) ≤ + ∪ + ∪ ∪ + … m L ( n ) 1 m L ( n ) 1 m L ( n ) 1 provided m ∪ m ∪ … ∪ m m is such that + ∪ + ∪ ∪ + m L ( n ) 1 m L ( n ) 1 ... m L ( n ) 1 ≤ ∪ ∪ ∪ m1 2 nn n V V ... V . Proof:
Consider a RD m-code C = C ∪ C ∪ … ∪ C m such that C C C ... C = ∪ ∪ ∪ = m L (n ) 1 m L (n ) 1 m L (n ) 1 + ∪ + ∪ ∪ + … . Each m-vector in n n n V V ... V ∪ ∪ ∪ has n 1 n 2 n m
L (n ) L (n ) ... L (n ) ∪ ∪ ∪ rank complements, that is from each m-vector n n n1 2 m v v ... v V V ... V ∪ ∪ ∪ ∈ ∪ ∪ ∪ there are n 1 n 2 n m
L (n ) L (n ) ... L (n ) ∪ ∪ ∪ m-vectors at rank m-distance n ∪ n ∪ … ∪ n m . This means for any set n n n1 2 m S S ... S V V ... V ∪ ∪ ∪ ⊆ ∪ ∪ ∪ of (m , m , …, m m ) m-vectors there always exists a c ∪ c ∪ … ∪ c m ∈ C ∪ C ∪ … ∪ C m which m-covers S ∪ S ∪ … ∪ S m ; m-rank distance n – 1 ∪ n – 1 ∪ … ∪ n m – 1. Thus cov(u ,S ) cov(u ,S ) ... cov(u ,S ) ∪ ∪ ∪ ≤ n – 1 ∪ n – 1 ∪ … ∪ n m – 1 which implies cov(u ,S ) cov(u ,S ) ∪ n 1 n 1 ≤ − ∪ − . Hence K (n , n 1) K (n , n 1) ... − ∪ − ∪ ∪ m mm m m K (n , n 1) − m L (n ) 1 m L (n ) 1 ... m L (n ) 1 ≤ + ∪ + ∪ ∪ + . By bounding the number of (m , m , …, m m ) m-sets that can be covered by a given m-code word, one obtains a straight forward generalization of the classical sphere m-bound. T HEOREM (Generalized sphere bound for RD-m-codes): For any ∪ ∪ ∪ … ( n ,K ) ( n ,K ) ( n ,K ) RD m-code C = C ∪ C ∪ … ∪ C m , V ( n ,t ( C )) V ( n ,t ( C ))K Km m ⎛ ⎞ ⎛ ⎞∪⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ m m m mm m V ( n ,t ( C ))K m ⎛ ⎞∪ ∪ ⎜ ⎟⎜ ⎟⎝ ⎠ … ⎛ ⎞⎛ ⎞ ⎛ ⎞≥ ∪ ∪ ∪ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ … nn n m1 2 NN N m1 2
22 2 mm m . Hence for any i i i n ,t ,m triple i = 1, 2, …, m ∪ ∪ ∪
K ( n ,t ) K ( n ,t ) ... K ( n ,t ) ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠≥ ∪ ∪ ∪⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ nn n m1 2
NN N m1 21 1 2 2 m m1 2 m
22 2 mm m ...V ( n ,t ) V ( n ,t ) V ( n ,t )m m m where ∪ ∪ ∪
V ( n ,t ) V ( n ,t ) ... V ( n ,t ) = = = = ∪ ∪ ∪ ∑ ∑ ∑ m1 21 2 m1 2 m tt t1 2 mi 1 i 2 i mi 0 i 0 i 0
L ( n ) L ( n ) ... L ( n ) number of m-vectors in a sphere m-radius t ∪ t ∪ … ∪ t m and ∪ ∪ ∪ … L ( n ) L ( n ) L ( n ) is the number of m-vectors in ∪ ∪ ∪ … m1 2 nn n V V V whose rank m-norm is i ∪ i ∪ … ∪ i m . Proof: Each set of (m , m , …, m m ) m-vectors in n n n V V ... V ∪ ∪ ∪ must occur in a sphere of m-radius m 1 m 2 m m t (C ) t (C ) ... t (C ) ∪ ∪ ∪ around at least one code m-word. Total number of such m-sets is n n n
V V ... V ∪ ∪ ∪ choose m ∪ m ∪ … ∪ m m where n n n Nn Nn Nn V V ... V 2 2 ... 2 ∪ ∪ ∪ = ∪ ∪ ∪ . The number of m-sets of (m , m , …, m m ) m-vectors in a neighborhood of m-radius m 1 m 2 m m t (C ) t (C ) ... t (C ) ∪ ∪ ∪ is V(n , t (C )) V(n , t (C )) ... ∪ ∪ ∪ m m m m V(n , t (C )) choose m ∪ m ∪ … ∪ m m . There are K-code m-words. Hence V(n , t (C )) V(n , t (C ))K Km m ⎛ ⎞ ⎛ ⎞∪⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ m m m mm m V(n , t (C ))K m ⎛ ⎞∪ ∪ ⎜ ⎟⎝ ⎠ … n n n1 2 m N N N1 2 m ⎛ ⎞ ⎛ ⎞ ⎛ ⎞≥ ∪ ∪ ∪⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ . Thus for any n ∪ n ∪ … ∪ n m , t ∪ t ∪ … ∪ t m and m ∪ m ∪ … ∪ m m K (n , t ) K (n , t ) ... K (n , t ) ∪ ∪ ∪ ( ) ( ) ( ) n n n1 2 m
N N N1 2 m1 1 2 2 m m1 2 m ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠≥ ∪ ∪ ∪⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ . C OROLLARY If ⎛ ⎞⎛ ⎞ ⎛ ⎞∪ ∪ ∪ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ nn n m1 2 NN N m1 2
22 2 ... mm m ⎛ ⎞⎛ ⎞ ⎛ ⎞> ∪ ∪ ∪ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ nn n m1 2
NN N m m1 1 2 21 2 m
V ( n ,t )V ( n ,t ) V ( n ,t )2 2 ... 2m m m then ∪ ∪ ∪ = ∞ ∪ ∞ ∪ ∪ ∞
K ( n ,t ) K ( n ,t ) ... K ( n ,t ) ... .
Chapter Four A PPLICATIONS OF R ANK D ISTANCE m-C
ODES
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 m-codes can be used simultaneously in criss cross error corrections 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 m-codes 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. F URTHER R EADING Adams M.J.,
Subcodes and covering radius , IEEE Trans. Inform. Theory, Vol.IT-32, pp.700-701, 1986. 2.
Andrews G.E.,
The Theory of Partitions , Vol. 2 of Encyclopaedia of Mathematics and Its Applications, Addison – Wesley, Reading, Massachusetts, 1976. 3.
Assmuss Jr. E.F. and Pless V.,
On the Covering radius of extremal self dual codes , IEEE Trans. Inform. Theory, Vol.IT-29, pp.359-363, 1983. 4.
Bannai, E., and Ito, T.,
Algebraic Combinatorics I. Association Schemes , The Benjamin/Cummings Publishing Company, 1983. 5.
Berger, T. P.,
Isometries for rank distance and permutation group of Gabidulin codes , IEEE Transactions on Information Theory, Vol. 49, no. 11, pp. 3016–3019, 2003. 6.
Berlekamp E.R.,
Algebraic Coding Theory , McGraw Hill, 1968. Best M.R.,
Perfect codes hardly exist , IEEE Trans. Inform. Theory, Vol.IT-29, pp.349-351, 1983. 8.
Blahut, R.,
Theory and Practice of Error Control Codes . Addison- Wesley, 1983. 9.
Blaum M. and McElice R.J.,
Coding protection for magnetic tapes; a generalization of the Patel-Hong code , IEEE Trans. Inform. Theory, Vol.IT-31, pp.690-693, 1985. 10.
Blaum M.,
A family of efficient burst-correcting array codes , IEEE Trans. Inform. Theory, Vol.IT-36, pp.671-674, 1990. 11.
Blaum M., Farrel F.G. and Van Tilborg H.C.A.,
A class of error correcting array codes , IEEE Trans. Inform. Theory, Vol.IT-32, pp.836-839, 1989. 12.
Brouwer A.E.,
A few new constant weight codes , IEEE Trans. Inform. Theory, Vol.IT-26, p.366, 1980. 13.
Carlitz, L., q-Bernoulli numbers and polynomials , Duke Mathematical Journal, Vol. 15, no. 4, pp. 987–1000, 1948. 14.
Chabaud, F., and Stern, J.,
The cryptographic security of the syndrome decoding problem for rank distance codes , Advances in Cryptology: ASIACRYPT ’96, Volume 1163 of LNCS, pp. 368–381, Springer-Verlag, 1996. 15.
Chen, K.,
On the non-existence of perfect codes with rank distance , Mathematische Nachrichten, Vol. 182, no. 1, pp. 89–98, 1996. 16.
Clark, W.E., and Dunning, L.A.,
Tight upper bounds for the domination numbers of graphs with given order and minimum degree , The Electronic Journal of Combinatorics, Vol. 4, 1997. 17.
Cohen G.D.,
A non constructive upper bound on covering radius , IEEE Trans. Inform. Theory, Vol.IT-29, pp.352-353, 1983.
Cohen G.D., Karpovski M.R., Mattson Jr. H.F. and Schatz J.R.,
Covering Radius – Survey and Recent Results , IEEE Trans. Inform., Theory, Vol.IT-31, pp.328-343, 1985. 19.
Cohen G.D., Lobstein A.C. and Sloane N.J.A.,
Further results on the covering radius of codes , IEEE Trans. Inform. Theory, Vol.IT-32, pp.680-694, 1986. 20.
Delsarte P.,
Four fundamental parameters of a code and their combinatorial significance , Inform. and Control, Vol.23, pp.407-438, 1973. 21.
Delsarte, P.,
Bilinear forms over a finite field, with applications to coding theory , Journal of Combinatorial Theory A, Vol. 25, no. 3, pp. 226–241, 1978. 22.
Delsarte, P.,
Properties and applications of the recurrence F(i + 1, k + 1, n + 1) = qk+1F(i, k + 1, n) − qkF(i, k, n), SIAM Journal of Applied Mathematics, Vol. 31, no. 2, pp. 262–270, September 1976. 23.
Erickson T.,
Bounds on the size of a code , Lecture notes in Control and Inform. Sciences, Vol.128, pp.45-72, Springer – Verlag, 1989. 24.
Gabidulin E.M.,
Theory of codes with Maximum Rank Distance , Problemy Peredachi Informatsii, Vol.21, pp.17-27, 1985. 25.
Gabidulin, E. M. and Loidreau, P.,
On subcodes of codes in the rank metric , Proc. IEEE Int. Symp. on Information Theory, pp. 121–123, Sept. 2005. 26.
Gabidulin, E. M. and Obernikhin, V.A.,
Codes in the Vandermonde F -metric and their application , Problems of Information Transmission, Vol. 39, no. 2, pp. 159–169, 2003. 27. Gabidulin, E. M.,
Optimal codes correcting lattice-pattern errors , Problems of Information Transmission, Vol. 21, no. 2, pp. 3–11, 1985.
Gabidulin, E. M., Paramonov, A.V., and Tretjakov, O.V., Ideals over a non-commutative ring and their application in cryptology,” in
Proceedings of the Workshop on the Theory and Application of Cryptographic Techniques (EUROCRYPT ’91) , Vol. 547 of
Lecture Notes in Computer Science , pp. 482–489, Brighton, UK, April 1991. 29.
Gabidulin, E. M.,
Public-key cryptosystems based on linear codes over large alphabets: efficiency and weakness , Codes and Cyphers, pp. 17– 31, 1995. 30.
Gadouleau, M., and Yan, Z. , Packing and covering properties of rank metric codes,
IEEE Trans. Info. Theory, Vol. 54, no. 9,pp. 3873–3883, September 2008. 31.
Gadouleau, M., and Yan, Z., “Properties of codes with the rank metric,” in
Proceedings of IEEE Global Telecommunications Conference (GLOBECOM ’06) , pp. 1–5, San Francisco, Calif, USA, November 2006. 32.
Gadouleau, M., and Yan, Z.,
Construction and covering properties of constant-dimension codes , submitted to IEEE Trans. Info. Theory, 2009. 33.
Gadouleau, M., and Yan, Z., Decoder error probability of MRD codes, in
Proceedings of IEEE Information Theory Workshop (ITW ’06) , pp. 264–268, Chengdu, China, October 2006. 34.
Gadouleau, M., and Yan, Z.,
On the decoder error probability of bounded rank-distance decoders for maximum rank distance codes , to appear in IEEE Transactions on Information Theory. 35.
Gasper, G., and Rahman, M.,
Basic Hypergeometric Series , Vol. 96 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, New York, NY, USA, 2nd edition, 2004. 36.
Gibson, J.K.,
The security of the Gabidulin public-key cryptosystem , in EUROCRYPT’96, pp. 212-223. 1996 37.
Graham R.L. and Sloane N.J.A.,
Lower Bounds for constant weight codes , IEEE Trans. Inform. Theory, Vol.IT-26, pp.37-43, 1980. 38.
Graham R.L. and Sloane N.J.A.,
On the covering radius of codes , IEEE Trans. Inform. Theory, Vol.IT-31, pp.385-401, 1985. 39.
Grant, D. and Varanasi, M.,
Duality theory for space-time codes over finite fields , to appear in Advance in Mathematics of Communications. 40.
Grant, D. and Varanasi, M., Weight enumerators and a MacWilliams-type identity for space-time rank codes over finite fields, in
Proceedings of the 43rd Allerton Conference on Communication, Control, and Computing , pp. 2137–2146, Monticello, Ill, USA, October 2005. 41.
Hall, L.O., and Gur Dial,
On fuzzy codes for Asymmetric and Unidirectional error,
Fuzzy sets and Systems, 36, pp. 365-373, 1990. 42.
Harold N.Ward,
Divisible codes , Archiv der Mathematic, 36, pp. 485-494, 1981. 43.
Helgert H.J. and Stinaff R.D.,
Minimum distance bounds for binary linear codes , IEEE Trans. Inform. Theory, Vol.IT-19, pp.344-356, 1973. 44.
Helleseth T., Klove T. and Mykkeltveit J.,
On the covering radius of binary codes , IEEE Trans. Inform. Theory, Vol.IT-24, pp.627-628, 1978. 45.
Helleseth T.,
The weight distribution of the coset leaders for some classes of codes with related parity check matrices , Discrete Math., Vol.28, pp.161-171, 1979. 46.
Herstein I.N.,
Topics in Algebra , 2 nd , Wiley, New York, 1975. Hill R.,
A first course in coding theory , Clarendon, Oxford, 1986. 48.
Hoffman K. and Kunze R.,
Linear Algebra , 2 nd , Prentice Hall of India, New Delhi, 1990. 49. Hua, L.,
A theorem on matrices over a field and its applications , Chinese Mathematical Society, Vol. 1, no. 2, pp. 109–163, 1951. 50.
Janwa H.L.,
Some new upper bounds on the covering radius of binary linear codes , IEEE Trans. Inform. Theory, Vol.IT-35, pp.110-122, 1989. 51.
Johnson S.M.,
Upper bounds for constant weight error correcting codes , Discrete Math., Vol.3, pp.109-124, 1972. 52.
Koetter, R., and Kschischang, F.R.,
Coding for errors and erasures in random network coding , IEEE Trans. Info. Theory, Vol. 54, no. 8, pp. 3579–3591, August 2008. 53.
Kshevetskiy, A. and Gabidulin, E. M.,
The new construction of rank codes , Proc. IEEE Int. Symp. on Information Theory, pp. 2105–2108, Sept. 2005. 54.
Lidl R. and Niederreiter H.,
Finite fields , Addison-Wesley, Reading, Massachusetts, 1983. 55.
Lidl R. and Pilz G.,
Applied Abstract Algebra , Springer-Verlag, New York, 1984. 56.
Loidreau, P.,
A Welch-Berlekamp like algorithm for decoding Gabidulin codes , in Proceedings of the 4th International Workshop on Coding and Cryptography (WCC ’05), Vol. 3969, pp. 36–45, Bergen, Norway, March 2005. 57.
Loidreau, P.,
Etude et optimisation de cryptosyst`emes `a cl´e publique fond´es sur la th´eorie des codes correcteurs , Ph.D. Dissertation, Ecole Polytechnique, Paris, France, May 2001.
Loidreau, P.,
Properties of codes in rank metric , http://arxiv.org/pdf/cs.DM/0610057/. 59.
Lusina, P., Gabidulin, E., and Bossert, M.,
Maximum Rank Distance codes as space-time codes , IEEE Trans. Info. Theory, Vol. 49, pp. 2757– 2760, Oct. 2003. 60.
MacLane S. and Birkhoff G.,
Algebra , Macmillan, New York, 1967. 61.
MacWilliams, F., and Sloane, N.,
The Theory of Error-Correcting Codes , North-Holland, Amsterdam, The Netherlands, 1977. 62.
Manoj, K.N., and Sundar Rajan, B.,
Full Rank Distance codes , Technical Report, IISc Bangalore, Oct. 2002. 63.
Mattson Jr. H.F.,
An improved upper bound on the covering radius , Lecture notes in Computer Science, 228, Springer – Verlag, New York, pp.90-106, 1986. 64.
Mattson Jr. H.F.,
An upper bound on covering radius , Ann. Discrete Math., Vol.17, pp.453-458, 1983. 65.
Mattson Jr. H.F.,
Another upper bound on covering radius , IEEE Trans. Inform. Theory, Vol.IT-29, pp.356-359, 1983. 66.
McEliece R.J., and Swanson, L.,
On the decoder error probability for Reed-Solomon codes , IEEE Trans. Info. Theory, Vol. 32, no. 5, pp. 701–703, Sept. 1986. 67.
McEliece, R.J.,
A public-key cryptosystem based on algebraic coding theory , Jet Propulsion Lab. DSN Progress Report, pp. 114–116, 1978. 68.
Ourivski, A. V. and Gabidulin, E. M.,
Column scrambler for the GPT cryptosystem , Discrete Applied Mathematics, Vol. 128, pp. 207–221, 2003.
Peterson W.W. and Weldon Jr. E.J.,
Error correcting codes , M.I.T. Press, Cambridge, Massachusetts, 1972. 70.
Pless, V.,
Power moment identities on weight distributions in error correcting codes , Information and Control, Vol. 6, no. 2, pp. 147–152, 1963. 71.
Prunsinkiewicz P. and Budkowski S.,
A double track error correction code for magnetic tape , IEEE Trans. Comput., Vol.C-25, pp.642-645, 1976. 72.
Richter, G., and Plass, S., Fast decoding of rank-codes with rank errors and column erasures,” in
Proceedings of IEEE International Symposium on Information Theory (ISIT ’04) , p.398, Chicago, Ill, USA, June-July 2004. 73.
Roth R.M.,
Maximum Rank Array codes and their application to crisscross error correction , IEEE Trans. Inform. Theory, Vol.IT-37, pp.328-336, 1991. 74.
Roth, R. M.,
Probabilistic criss cross error correction , IEEE Transactions on Information Theory, Vol. 43, no. 5, pp. 1425–1438, 1997. 75.
Schonheim J.,
On linear and non-linear single error correcting q-nary perfect codes , Inform. and Control, Vol.12, pp.23-26, 1968. 76.
Schwartz, M., and Etzion, T.,
Two-dimensional cluster correcting codes , IEEE Transactions on Information Theory, Vol. 51, no. 6, pp. 2121–2132, 2005. 77.
Selvaraj, R.S.,
Metric properties of rank distance codes , Ph.D. Thesis, Thesis Guide; Dr. W. B. Vasantha Kandasamy, Department of Mathematics, Indian Institute of Technology, Madras, .2006. 78.
Shannon C.E.,
A mathematical theory of communication , Bell. Syst. Tech., Jl.27, pp.379-423 and pp.623-656, 1948.
Silva, D., Kschischang, F. R. and Koetter, R.,
A rank-metric approach to error control in random network coding , IEEE Trans. Info. Theory, Vol. 54, no. 9, pp. 3951–3967, September 2008. 80.
Sripati, U., and Sundar Rajan, B.,
On the rank distance of cyclic codes , Proc. IEEE Int. Symp. on Information Theory, p. 72, July 2003. 81.
Strang G.,
Linear Algebra and its applications , 2 nd , Academic Press, New York, 1980. 82. Suresh Babu, N., S tudies on Rank Distance Codes , Ph.D. Thesis. Thesis Guide; Dr. W.B.Vasantha Kandasamy, Department of Mathematics, Indian Institute of Technology, Madras, 1995. 83.
Tarokh, V., Seshadri, N., and Calderbank, A.R.,
Space-time codes for high data rate wireless communication: performance criterion and code construction , IEEE Transactions on Information Theory, Vol. 44, no. 2, pp. 744–765, 1998. 84.
Tietavanien A.,
On the non-existence of perfect codes over finite fields , Siam. Jl. Appl. Math., Vol.24, 1973. 85.
Van Lint J.H. and Wilson R.M,
A course in combinatories , Cambridge University Press, 1992. 86.
Van Lint J.H.,
A Survey of perfect codes , Rocky Moundain Jl. of Math., Vol.5, pp.199-224, 1975. 87.
Van Lint J.H.,
Coding Theory , Springer-Verlag, New York, 1971. 88.
Vasantha Kandasamy, W.B., and Raja Durai, R. S.,
Maximum rank distance codes with complementary duals , Mathematics and Information Theory, Recent topics and Applications, pp. 86-90, 2002.
Vasantha Kandasamy, W.B., and Selvaraj, R. S.,
Multi-covering radii of codes with rank metric , Proc. Information Theory Workshop, p. 215, Oct. 2002. 90.
Vasantha Kandasamy, W.B., and Suresh Babu, N.,
On the covering radius of rank-distance codes,
Ganita Sandesh, Vol. 13, pp. 43–48, 1999. 91.
Vasantha Kandasamy, W.B.,
Bialgebraic structures and Smarandache bialgebraic structures , American Research Press, Rehoboth, 2003. 92.
Vasantha Kandasamy, W.B., Smarandache, Florentin and K. Ilanthenral,
Introduction to bimatrices , Hexis, Phoenix, 2005. 93.
Vasantha Kandasamy, W.B., Smarandache, Florentin and K. Ilanthenral,
Introduction to Linear Bialgebra , Hexis, Phoenix, 2005. 94.
Vasantha, W. B. and R.S.Selvaraj,
Multi-covering Radii of Codes with Rank Metric , Proceedings of 2002 IEEE Information Theory Workshop, Indian Institute of Science, Bangalore, October 20-25, p. 215, 2002. 95.
Vasantha, W.B. and Selvaraj R.S.,
Divisible MRD Codes,
Proceedings of National Conference on Challenges of the 21 st century in Mathematics and its Allied Topics, University of Mysore, Karnataka, India, pp., 242-246, February 3-4, 2001. 96. Vasantha, W.B. and Selvaraj, R.S.
Fuzzy RD Codes with Rank Metric and Distance Properties , Proceedings of the International Conference on Recent Advances in Mathematical Sciences, Indian Institute of Technology Kharagpur, India, pp., 341-348, Narosa Publishing House, New Delhi, December 20-22, 2000 97.
Vasantha, W.B. and Selvaraj, R.S.
Multicovering radius of Codes for Rank Metric , Proceedings of 4 th Asia Europe Workshop on Information Theory Concepts, Viareggio, Italy, October 6-8, pp., 96-98, 2004.
Vasantha Kandasamy, W.B., and Smarandache, Florentin,
Rank distance m-codes and their properties (To appear) , Maths Tiger. 99.
Von Kaenel, P.A,
Fuzzy codes and distance properties , Fuzzy sets and systems, 8, pp., 199-204, 1982. 100.
Zhen Zhang,
Limiting efficiencies of Burst correcting Array codes , IEEE Trans. Inform. Theory, Vol.IT-37, pp.976-982, 1991. 101.
Zimmermann K.H.
The weight distribution of indecomposable cyclic codes over 2-groups , Jour. of Comb. Theory, Series A, Vol.60, 1992. 102.
Zinover V.A. and Leontev V.K.,
The non-existence of perfect codes over Galois fields,
Problems of Control and Info. Theory, Vol.2, pp.123-132, 1973. I NDEX (m, n) dimensional vector bispace, 28 (m , m ) bicovering biradius, 69 (m , m ) covering biradii, 69 (m , m , m , …, m m ) covering m-radii, 120-2 (r, r) repetition RD bicode, 67 (r , r ) 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
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 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 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 Q Quasi (r , r ) 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 r i - 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
Solution bispace, 29 Solution m- space, 80-2 Symmetric error bimodel, 59-60 Symmetric error m-model, 110-1 A BOUT THE A UTHORS
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 46 th 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 prize-money 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.