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Self-Dual Z4-Codes of Type IV Generated by Skew-Hadamard Matrices and Conference Matrices

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Self-Dual Z4-Codes of Type IV Generated by Skew-Hadamard Matrices and Conference Matrices AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 42 (2008), Pages 177–188 Self-dual Z4-codes of Type IV generated by skew-Hadamard matrices and conference matrices Mieko Yamada Division of Mathematical and Physical Sciences Graduate School of Natural Science and Technology Kanazawa University Japan [email protected] Abstract In this paper, we give families of self-dual Z4-codes of Type IV-I and Type IV-II generated by conference matrices and skew-Hadamard matri- ces. Furthermore, we give a family of self-dual Z4-codes of Type IV-I generated by bordered skew-Hadamard matrices. 1 Introduction In 1994, Hammons et al. showed that certain binary nonlinear codes are the binary image of linear codes over the Galois ring GR(4,m), an extension ring of Z4 = Z/4Z [7]. Active research on Z4-codes has been undertaken since their paper was published. The distinct rows of an Hadamard matrix are orthogonal. If we recognize the entries 1 and −1 of an Hadamard matrix H4n of order 4n as the elements of Z4,then the Z4-code generated by H4n is self-orthogonal. In 1999, Charnes proved that if H1 and H2 are H-equivalent, then the Z4-codes generated by H1 and H2 are equivalent [3]. Sol´e showed that if an Hadamard matrix H4n has order 4n and n is odd, then the Z4-code generated by H4n is self-dual and equivalent to Klemm’s code [3]. Charnes and Seberry considered the Z4-code generated by a weighing matrix W (n, 4). They proved that if n is even, then it is a tetrad code and if it has type 4(n−4)/224,thenit is a self-dual code [4]. Self-dual Z4-codes of lengths up to 20 are classified [2, 5, 6, 8, 9]. A Type II Z4-code is a self-dual code which has the property that all Euclidean weights are divisible by 8. A self-dual Z4-code which is not a Type II code is called a Type I Z4-code. A Type IV Z4-code is a self-dual code with all codewords of even Hamming weight. A type IV code which is also Type I or Type II, is called a TypeIV-I, or a Type IV-II code respectively. Two infinite families of Type IV codes are known, that is, Klemm’s codes and Cm,r codes [1, 5]. 178 MIEKO YAMADA In this paper, we give families of self-dual Z4-codes of Type IV-I and Type IV-II generated by conference matrices and skew-Hadamard matrices. Furthermore we give a family of self-dual Z4-codes of Type IV-I generated by bordered skew-Hadamard matrices. 2 Self-dual Z4-codes n An additive subgroup of Z4 is called a Z4-code of length n. We define an inner n n product on Z4 by a · b = i=1 ai · bi for vectors a =(a1,a2, ··· ,an)andb = ⊥ ⊥ n (b1,b2, ··· ,bn). The dual code C of a Z4-code C is defined as C = {x ∈ Z4 |x·y = 0 for all y ∈ C}.IfC ⊆ C⊥,acodeC is called self-orthogonal and if C = C⊥, C is called self-dual. Two codes are permutation-equivalent if one can be obtained from the other by permuting coordinates. Any Z4-code is permutation-equivalent to a code with generator matrix of the form Ik AB G = 1 O 2Ik2 2D where the entries of A and D are in Z2 = {0, 1} and the entries of B are in Z4.Then it contains 4k1 2k2 codewords. We say that the code C has type 4k1 2k2 .Itisknown k1 k2 ⊥ n−k1−k2 k2 that if Z4-code C has type 4 2 , then the dual code C has type 4 2 . Let ni(a) be the number of components of a vector a that are congruent to i (mod 4),i =0, 1, 2, 3. The Hamming weight wtH (a)ofa is defined by wtH (a)= n1(a)+n2(a)+n3(a) and the Euclidean weight wtE(a)ofa is defined by wtE(a)= n1(a)+4n2(a)+n3(a). The Hammimg distance dH(a, b) is defined by wtH (a − b) and the Euclidean distance dE(a, b) is defined by wtE(a − b). The minimum Hamming distance dH of a Z4-code C is min{dH(a, b)|a, b ∈ C, a = b} and the minimum Euclidean distance dE of C is min{dE(a, b)|a, b ∈ C, a = b}. The highest minimum Hamming weights and the highest minimum Euclidean weights of Type IV self-dual codes of lengths up to 40, Type IV-I codes of length 56 and Type IV-II codes of lengths 48,56,64 were determined [2]. Klemm’s code Kn is given as Kn = Rn +2Pn =2Pn ∪ (e +2Pn) where Rn is a repetition code, Pn is its dual code, and e is the all-one vector. For 3r ≤ m − 1, Cm,r code is given as Cm,r = RM(r, m)+2RM(m − r − 1,m) where RM(r, m) is a Reed-Muller code. SELF-DUAL Z4-CODES 179 3 Self-dual Z4-codes of Type IV generated by conference matrices If a square matrix H of order n with entries ±1satisfiesHHt = nI, then it is called t an Hadamard matrix. An Hadamard matrix H = H0 + I such that H0 = −H0 is called a skew-Hadamard matrix. The distinct rows of an Hadamard matrix are orthogonal. If we recognize the entries 1 and −1 of an Hadamard matrix H4m of order 4m as the elements of Z4, then the Z4-code generated by rows of H4m is self-orthogonal. In this section, we give families of self-dual Z4-codes of Type IV-I and Type IV-II generated by conference matrices and skew-Hadamard matrices. Let q be an odd prime power and GF(q) be a finite field with q elements. Let χ be a quadratic character of GF(q). We let χ(0) = 0. A Paley matrix P =(pij )of order q is the matrix whose entry pij is defined by pij = χ(ai − aj) where ai and aj (0 ≤ i, j ≤ q −1) are elements of GF(q). Denote a conference matrix of order q +1byQ,thatis 0 e Q = χ(−1)et P where e is the all-one vector. The followings relations are well-known. QQt = qI, Qt = χ(−1)Q (1) where I is the unit matrix. In what follows, we recognize the entries 1, −1 and 0 of a matrix as the elements of Z4. We construct families of self-dual Z4-codes of Type IV-I and of Type IV-II. Theorem 1. Put N = Q +2I. Define the matrix GQ as follows: ⎛ ⎞ IN N I ⎝ ⎠ GQ = O 2I 2(J − I)2J OO 2I 2(J − I) where J is the all-one matrix and O is the zero matrix. Then CQ with generator matrix GQ is a self-dual Z4-codeofTypeIV. Proof. From the relations (1), we have NNt =(Q+2I)(Qt +2I)=QQt +2(Q+Qt)= qI,2NNt +2I = O,2N =2(J −I), 2N +2(J −I)=O and 2N +2n(J −I)+2J = O. 180 MIEKO YAMADA Hence we have, ⎛ ⎞ ⎛ ⎞ IO O IN N I ⎜ t ⎟ t ⎝ ⎠ ⎜N 2IO⎟ GQG = O 2I 2(J − I)2J Q ⎝N t 2(J − I)2I ⎠ OO 2I 2(J − I) I 2J 2(J − I) ⎛ ⎞ 2I +2NNt 2N +2N(J − I)+2J 2N +2(J − I) = ⎝2N t +2(J − I)N t +2JO O⎠ 2N t +2(J − I) OO = O. q+1 2(q+1) Since the number of codewords of CQ is 4 2 , the number of codewords of ⊥ 4(q+1)−(q+1)−2(q+1) 2(q+1) q+1 2(q+1) the dual code CQ is 4 2 =4 2 . Hence CQ is a self-dual Z4-code. ⎛ ⎞ G1 ⎝ ⎠ Next we shall prove CQ is a Type IV code. Put n = q +1. LetGQ = G2 G3 where G1 =(I,N,N,I), G2 =(O, 2I,⎛2(J −⎞I), 2J)and⎛ ⎞ ⎛ ⎞ u1 v1 w1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜u2⎟ ⎜v2⎟ ⎜w2⎟ G3 =(O, O, 2I,2(J − I)). Put G1 = ⎜ . ⎟, G2 = ⎜ . ⎟ and G3 = ⎜ . ⎟.Then ⎝ . ⎠ ⎝ . ⎠ ⎝ . ⎠ un vn wn the codeword c of CQ is written as n n n c = αkuk + βkvk + γkwk k=1 k=1 k=1 where αk ∈ Z4 and βk,γk ∈ Z2 (1 ≤ k ≤ n). Let s = |{αk : αk = ±1}| and t = |{αk : αk =2}|. Wemayassumethefirst s coefficients α1, ··· ,αs are odd, and the next t coefficients αs+1, ··· ,αs+t are all 2, and other coefficients are all zero by permuting rows and columns of G1.The matrices G2 and G3 do not change by further suitable permuting rows and columns of G2 and G3. Then the codeword is written as s t n n c = αkuk + αkuk+s + βkvk + γkwk k=1 k=1 k=1 k=1 ± ≤ ≤ ≤ ≤ where αk = 1, (1 k s)andαk =2(1 k t). Denote the (k,l) entry of the s t matrix N by N(k,l). Let the vector g1 = k=1 αkuk+ k=1 αkuk =(g1,g2, ··· ,g4n). Then we have • gi = αi, (1 ≤ i ≤ s), SELF-DUAL Z4-CODES 181 • gi+s = αi, (1 ≤ i ≤ t), • gi+s+t =0, (1 ≤ i ≤ n − (s + t)), s t • gi+n = gi+2n = k=1 αkN(k,i)+ k=1 αkN(s + k,i), (1 ≤ i ≤ n), • gi+3n = αi, (1 ≤ i ≤ s), • gi+3n+s = αi, (1 ≤ i ≤ t), • gi+3n+s+t =0, (1 ≤ i ≤ n − (s + t)).
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