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Type I Codes Over GF(4)

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Type I Codes Over GF(4) Type I Codes over GF(4) Hyun Kwang Kim¤ San 31, Hyoja Dong Department of Mathematics Pohang University of Science and Technology Pohang, 790-784, Korea e-mail: [email protected] Dae Kyu Kim School of Electronics & Information Engineering Chonbuk National University Chonju, Chonbuk 561-756, Korea e-mail: [email protected] Jon-Lark Kimy Department of Mathematics University of Louisville Louisville, KY 40292, USA e-mail: [email protected] Abstract It was shown by Gaborit el al. [10] that a Euclidean self-dual code over GF (4) with the property that there is a codeword whose Lee weight ´ 2 (mod 4) is of interest because of its connection to a binary singly-even self-dual code. Such a self-dual code over GF (4) is called Type I. The purpose of this paper is to classify all Type I codes of lengths up to 10 and extremal Type I codes of length 12, and to construct many new extremal Type I codes over GF (4) of ¤The present study was supported by Com2MaC-KOSEF, POSTECH BSRI research fund, and grant No. R01-2006-000-11176-0 from the Basic Research Program of the Korea Science & Engineering Foundation. ycorresponding author, supported in part by a Project Completion Grant from the University of Louisville. 1 lengths from 14 to 22 and 34. As a byproduct, we construct a new extremal singly-even self-dual binary [36; 18; 8] code, and a new ex- tremal singly-even self-dual binary [68; 34; 12] code with a previously unknown weight enumerator W2 for ¯ = 95 and γ = 1. Key Words. Binary self-dual code, Euclidean self-dual code over GF (4). 1 Introduction We briefly review basic de¯nitions. A linear [n; k] code C over GF (4) is a k-dimensional vector subspace of GF (4)n, where GF (4) is the Galois ¯eld with four elements 0; 1; !; and ! satisfying ! = !2 and ! = 1 + !. n The Hamming weight wtH (x) of x 2 GF (4) is the number of nonzero components of x. Let n0(x); n!(x); n!¹ (x), and n1(x) be the number of 0's, !'s, !'s, and 1's in a vector x 2 GF (4)n, respectively. The Lee weight n wtL(x) of x 2 GF (4) is de¯ned as 2n1(x) + n!(x) + n!¹ (x). Note that wtL(0) = 0; wtL(1) = 2; wtL(!) = 1, and wtL(!) = 1. Thus the Lee weight n wtL(x) of x 2 GF (4) is the rational sum of the Lee weights of all the coordinates of x. The minimum Lee weight dL (resp. minimum Hamming weight dH ) of C is the smallest Lee (resp. Hamming) weight among all non-zero codewords of C. Two codes C1 and C2 are (permutation) equivalent if there exists a co- ordinate permutation sending C1 onto C2 [2],[10]. The (permutation) au- tomorphism group PAut of C is the set of all coordinate permutations pre- serving C. The direct sum of two codes C1 and C2 is C1 © C2 = f(u; v)ju 2 n C1 and v 2 C2g. C denotes the direct sum of n copies of C. If D is equiv- alent to C1 © C2, it is called decomposable, otherwise indecomposable. The complete weight enumerator cweC (a; b; c; d) of C is X an0(c)bn! (c)cn!¹ (c)dn1(c): c2C The Lee weight enumerator of C is de¯ned as X wtL(c) 2 y = cweC (1; y; y; y ): c2C The Gray map Á from GF (4)n to GF (2)2n, ¯rst appeared in [17, pp. 508] and then in [10], is de¯ned as Á(!x + !y) = (x; y) for x; y 2 GF (2)n; where (x; y) is the binary vector of length 2n. 2 The Euclidean inner product is de¯ned as x ¢ y = x1y1 + ¢ ¢ ¢ + xnyn 2 n GF (4); for two vectors x = (x1; ¢ ¢ ¢ ; xn) and y = (y1; ¢ ¢ ¢ ; yn) in GF (4) . The dual code C? of C is de¯ned as C? = fx 2 GF (4)njx ¢ y = 0 for all y 2 Cg: If C = C?, then C is called a (Euclidean) self-dual code. A Euclidean self- dual code over GF (4) is called Type II if the Lee weight of every codeword is divisible by 4 and Type I if there is a codeword whose Lee weight ´ 2 (mod 4) [2],[10]. We remark that a Euclidean self-dual code over GF (4) can have a codeword of odd Hamming weight even though all codewords have even Lee weights. It was shown by Gaborit et al. [10] that if C is a Euclidean self-orthogonal code over GF (4), then Á(C) is a binary self-orthogonal code. So C is a Type I (resp. Type II) code over GF (4) if and only if Á(C) is a singly- even (resp. doubly-even) binary self-dual code. As a binary self-dual code contains all one vector 1, any Euclidean self-dual code over GF (4) contains all one vector. There has been a classi¯cation of Type II codes of lengths 4; 8; and 12. It is known that there are only one Type II code of length 4 and exactly two Type II codes of length 8 [10], and that there are exactly seven Type II codes of length 12, one of which is extremal [2]. Several examples of extremal Type I codes are in [2],[10]. Our paper is the ¯rst attempt to classify Type I codes over GF (4). We classify all Type I (and Type II) codes of lengths up to 10 and extremal Type I (and Type II) codes of length 12, and construct many new extremal Type I codes over GF (4) of lengths from 14 to 22 and 34. We also give their corresponding binary singly-even self-dual codes whenever possible. As a byproduct, we construct a new extremal singly-even self-dual binary [36; 18; 8] code with a previously unknown group order and a new extremal singly-even self-dual binary [68; 34; 12] code with a previously unknown weight enumerator W2 for ¯ = 95 and γ = 1 [14]. We also prove that a Euclidean self-dual [12; 6] code over GF (4) with minimum Hamming weight 6 is unique; it is permutation equivalent to the extended quadratic residue [12; 6] code over GF (4). We summarize the currently known status of extremal or optimal (with respect to Lee weight) Euclidean self-dual codes over GF (4) of even lengths n (2 · n · 22) and n = 34 in Table 1. Here dL(I) and dL(II) denote the highest minimum Lee weight of Type I and Type II codes, respectively. The number of Type I codes and that of Type II codes are separated by ; and entries without reference are obtained from this paper. A period indicates that the list of codes is complete. The column with (dH ; no.) gives the number of Euclidean self-dual codes with highest minimum Hamming weight dH of lengths n · 12 and the last column with dH for our codes 3 gives the minimum Hamming weight of our Type I codes. The attainable Hamming weight of our Euclidean self-dual codes over GF (4) is better than the Pless-Pierce bound [20] for 8 · n · 20 and n = 32 , and slightly weaker than the Table 6 of [9] for n ¸ 14. 2 Preliminaries and Methods The following lemmas are straightforward by the de¯nition of the Gray map. Lemma 2.1 ([10]). The Gray map Á is a GF (2)-linear isometry from (GF (4)n, Lee distance) onto (GF (2)2n, Hamming distance) where the Lee distance of two codewords x and y is the Lee weight of x ¡ y. The Lee weight enumerator of a code C over GF (4) is the same as the Hamming weight enumerator of Á(C). Lemma 2.2 ([10]). If C1 and C2 are equivalent Euclidean self-dual codes over GF (4), then Á(C1) and Á(C2) are equivalent. The converse is not true. We now give an upper bound for the minimum Lee weights of self-dual codes over GF (4) by using Rains' bound [22] for binary self-dual codes. Lemma 2.3 ([10]). Let dL(I; n) and dL(II; n) be the highest minimum Lee weights of a Type I code and a Type II code, respectively, of length n. Then j n k d (I; n) · 4 + 4 (n ´ 0 (mod 2)) (1) L 12 j n k d (II; n) · 4 + 4 (n ´ 0 (mod 4)): (2) L 12 A Type I (resp. Type II) code of length n satisfying the above bound is called extremal. An optimal Type I code has the highest minimum Lee weight among all Type I codes of that length. We now give a building-up construction method of Euclidean self-dual codes over GF (4) from smaller length self-dual codes. Theorem 2.4 (Building-up). Let G0 = (LjR) = (lijri) be a generator matrix (may not be in standard form) of a Euclidean self-dual code C0 over GF (4) of length 2n, where li and ri are rows of n £ n matrices L and R respectively for 1 · i · n. Let x = (x1; ¢ ¢ ¢ ; xn; xn+1; ¢ ¢ ¢ ; x2n) be a vector 2n in GF (4) with x ¢ x = 1. Suppose that yi := (x1; ¢ ¢ ¢ ; xn; xn+1; ¢ ¢ ¢ ; x2n) ¢ (lijri) for 1 · i · n under the Euclidean inner product. Then the following matrix 2 3 1 0 x1 ¢ ¢ ¢ xn xn+1 ¢ ¢ ¢ x2n 6 7 6 y1 y1 7 G = 6 .
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