Construction of quasi-cyclic self-dual codes
Abstract
There is a one-to-one correspondence between l-quasi-cyclic codes over a finite field Fq and linear codes over a ring R = Fq[Y ]/(Y m − 1). Using this correspondence, we prove that every l-quasi-cyclic self-dual code of length ml over a finite field Fq can be obtained by the building-up construction, provided that char (Fq) = 2 or q ≡ 1 (mod 4), m is a prime p, and q is a primitive element of Fp. We determine possible weight enumerators of a binary l-quasi-cyclic self-dual code of length pl (with p a prime) in terms of divisibility by p. We improve the result of [3] by constructing new binary cubic (i.e., l-quasi-cyclic codes of length 3l) optimal selfdual codes of lengths 30, 36, 42, 48 (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When m = 5, we obtain a new 8-quasi-cyclic self-dual [40, 20, 12] code over F3 and a new 6-quasi-cyclic self-dual [30, 15, 10] code over F4. When m = 7, we find a new 4-quasi-cyclic self-dual [28, 14, 9] code over F4 and a new 6-quasi-cyclic self-dual [42, 21, 12] code over F4. Introduction Self-dual codes have been one of the most interesting classes of linear codes over finite fields and in general over finite rings. They interact with other areas including lattices [12, 13], invariant theory [41], and designs [1]. On the other hand, quasi-cyclic codes have been one of the most practical classes of linear codes. Linear codes which are quasi-cyclic and self-dual simultaneously are an interesting class of codes, and this class of codes is our main topic. We refer to [30] for a basic discussion of codes. From the module theory over rings, quasi-cyclic codes can be considered as modules over the group algebra of the cyclic group. For a special ring R = Fq[Y ]/(Y m − 1), Ling and Solé [36, 37] consider linear codes over a ring R, where m is a positive integer coprime to q, and they use a correspondence φ between (self-dual) quasi-cyclic codes School of Liberal Arts, Korea University of Technology and Education, Cheonan 330-708, South Korea, Email: sunghyu@kut.ac.kr Department of Mathematics, University of Louisville, Louisville, KY 40292, USA, Email: jl.kim@louisville.edu Department of Mathematics, Ewha Womans University, Seoul 120-750, South Korea, Email: hsllee@ewha.ac.kr Department of Mathematics, Ewha Womans University, Seoul 120-750, South Korea, Email: yoonjinl@ewha.ac.kr The author is a corresponding author and supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2010-0015201). 1 over Fq and (self-dual, respectively) linear codes over R. We call quasi-cyclic codes over Fq cubic, quintic, or septic codes depending on m = 3, 5, or 7, respectively. Bonnecaze et. al. [3] studied binary cubic self-dual codes, and Bracco et. al. [7] considered binary quintic self-dual codes. In this paper, we focus on construction and classification of quasi-cyclic self-dual codes over a finite field Fq under the usual permutation or monomial equivalence. We note that the equivalence under the correspondence φ may not be preserved; two inequivalent linear codes over a ring R under a permutation equivalence may correspond to two equivalent quasi-cyclic codes over a finite field Fq under a permutation or monomial equivalence. Hence, we first construct all self-dual codes over the ring R using a building-up construction. Rather than considering the equivalence of these codes over R, we consider the equivalence of their corresponding quasi-cyclic self-dual codes over Fq to get a complete classification of quasi-cyclic self-dual codes over Fq. We prove that every l-quasi-cyclic self-dual code of lengthml over Fq can be obtained by the building-up construction, provided that char(Fq) = 2 or q ≡ 1 (mod 4), m is a prime p, and q is a primitive element of Fp. Our result shows that the building-up construction is a complete method for constructing all l-quasi-cyclic self-dual codes of length ml over Fq subject to certain conditions of m and q. We determine possible weight enumerators of a binary l-quasi-cyclic self-dual code of length pl with p a prime in terms of divisibility by p. By employing our building-up constructions, we classify binary cubic self-dual codes of lengths up to 24, and we construct binary cubic optimal self-dual codes of lengths 30, 36, 42, 48 (Type I), 54 and 66. We point out that the advantage of our construction is that we can classify all binary cubic self-dual codes in a more efficient way without searching for all binary self-dual codes. We summarize our result on the classification of binary cubic extremal self-dual codes in Table 1. We also give a complete classification of all binary quintic self-dual codes of even lengths 5l ≤ 30, and construct such optimal codes of lengths 40, 50, and 60. For various values of m and q, we obtain quintic self-dual codes of length 5l over F3 and F4 and septic self-dual codes of length 7l over F2,F4, and F5 which are optimal or have the best known parameters. In particular, we find a new quintic self-dual [40, 20, 12] code over F3 and a new quintic self-dual [30, 15, 10] code over F4. We also obtain a new septic self-dual [28, 14, 9] code over F4 and a new septic self-dual [42, 21, 12] code over F4. This paper is organized as follows. Section 1 contains some basic notations and definitions, and Section 2 presents the building-up construction method of quasi-cyclic self-dual codes over finite fields. In Section 3, we construct binary quasi-cyclic self-dual codes, and we find the cubic codes and quintic codes. In Section 4, we construct quasicyclic self-dual codes over various fields such as F2, F3,F4, and F5, and we obtain the cubic codes, the quintic codes and the septic codes. We use Magma [8] for computations. 1 Preliminaries We briefly introduce some basic notions about quasi-cyclic self-dual codes. For more detailed description, we refer to [36, 37]. 2 Table 1: Binary extremal cubic self-dual codes of lengths up to 66 length n highest min. wt No. of extremal Ref. cubic self-dual codes 6 2 1 Sec. 3 12 4 1 Sec. 3 18 4 1 Sec. 3 24 8 1 Sec. 3 30 6 8 Sec. 3, [3], [40] 36 8 13 Sec. 3, [3], [15], [27] 42 8 1569 Sec. 3, [3], [5], [6] 48 10 ≥ 4 Sec. 3, [3] 54 10 ≥ 7 Sec. 3, [3] 60 12 ≥ 3 [3] 66 12 ≥ 7 Sec. 3, [3] Let R be a commutative ring with identity. A linear code C of length n over R is defined to be an R-submodule of R; in particular, if R is a finite field Fq of order q, then C is a vector subspace of Fnq over Fq. The dual of C is denoted by C , C is selforthogonal if C ⊆ C, and self-dual if C = C. We denote the standard shift operator on R by T . A linear code C is said to be quasi-cyclic of index l or l-quasi-cyclic if it is invariant under T . A 1-quasi-cyclic code means a cyclic code. Throughout this paper, we assume that the index l divides the code length n. Let m be a positive integer coprime to the characteristic of Fq, Fq[Y ] be a polynomial ring, and R := R(Fq, m) = Fq[Y ]/(Y m − 1). Then it is shown [36] that there is a oneto-one correspondence between l-quasi-cyclic codes over Fq of length lm and linear codes over R of length l, and the correspondence is given by the map φ defined as follows. Let C be a quasi-cyclic code over Fq of length lm and index l with a codeword c denoted by c = (c00, c01, . . . , c0,l−1, c10, . . . , c1,l−1, . . . , cm−1,0, . . . , cm−1,l−1). Let φ be a map φ : Fq lm → R defined by φ(c) = (c0(Y ), c1(Y ), . . . , cl−1(Y )) ∈ R , where cj(Y ) = ∑m−1 i=0 cijY i ∈ R, for j = 0, . . . , l − 1. We denote by φ(C) the image of C under φ. A conjugationmap − onR is defined as the map that sends Y to Y −1 = Y m−1 and acts as the identity map on Fq, and it is extended Fq-linearly. On R , we define the Hermitian inner product by 〈x,y〉 = ∑l−1 j=0 xjyj for x = (x0, . . . , xl−1) and y = (y0, . . . , yl−1). It is proved [36] that for a,b ∈ F q , T (a) · b = 0 for all 0 ≤ k ≤ m − 1 if and only if 〈φ(a), φ(b)〉 = 0, where · denotes the standard Euclidean inner product. From this fact, it follows that φ(C) = φ(C), where φ(C) is the dual of φ(C) with respect to the Hermitian inner product, and C is the dual of C with respect to the Euclidean inner product. In particular, a quasi-cyclic code C over Fq is self-dual with respect to 3 the Euclidean inner product if and only if φ(C) is self-dual over R with respect to the Hermitian inner product [36]. Two linear codes C1 and C2 over R are equivalent if there is a permutation of coordinates of C1 sending C1 to C2. Similarly, two linear codes over Fq are equivalent if there is a monomial mapping sending one to another. Note that the equivalence of two linear codes C1 and C2 over R implies a permutation equivalence of quasi-cyclic linear codes φ(C1) and φ (C2) over Fq, but not conversely in general. 2 Construction of quasi-cyclic self-dual codes Throughout this paper, let R = Fq[Y ]/(Y −1), and self-dual (or self-orthogonal) codes over R means self-dual (or self-orthogonal) codes with respect to the Hermitian inner product. We begin with the following lemma regarding the length of self-dual codes. Lemma 2.1. Let R = Fq[Y ]/(Y m − 1). (i) If char(Fq) = 2 or q ≡ 1 (mod 4), then there exists a self-dual code over R of length l if and only if 2 | l. (ii) If q ≡ 3 (mod 4), then there exists a self-dual code over R of length l if and only if 4 | l. Proof. To prove (i) and (ii), we observe the following. Suppose C is a self-dual code of length l over R. We may assume that C1 in the decomposition of C in [36, Theorem 4.2] is a Euclidean self-dual code over Fq of length l. For (i), suppose that char(Fq) = 2 or q ≡ 1 (mod 4). By the above observation, 2 | l. Conversely, let l = 2k. We take a Euclidean self-dual code over Fq of length 2 using the following generator matrix: [1 c], where c = −1. We can see that this matrix generates a self-dual code C over R of length 2. Then the direct sum of the k copies of C is a self-dual code over R of length l = 2k. For (ii), let q ≡ 3 (mod 4). It is well known [46, p. 193] that if q ≡ 3 (mod 4) then a self-dual code of length n exists if and only if n is a multiple of 4. Hence by the above observation, 4 | l. Conversely, let l = 4k for some positive integer k. It is known [31, p. 281] that if q is a power of an odd prime with q ≡ 3 (mod 4), then there exist nonzero α and β in Fq such that α 2 + β + 1 = 0 in Fq. We take a Euclidean self-dual code over Fq of length 4 with the following generator matrix: G = [ 1 0 α β 0 1 −β α ] , where α + β + 1 = 0 in Fq. We can see that this matrix generates a self-dual code C over R of length 4. Then the direct sum of the k copies of C is a self-dual code over R of length l = 4k. The following theorem is the building-up constructions for self-dual codes over R, equivalently, l-quasi-cyclic self-dual codes over Fq for any odd prime power q. The proof is similar to that of [32], so the proof is omitted. 4 Theorem 2.2. Let C0 be a self-dual code over R of length 2l and G0 = (ri) be a k× 2l generator matrix for C0, where ri is the i-th row of G0, 1 ≤ i ≤ k. (i) Assume that char (Fq) = 2 or q ≡ 1 (mod 4). Let c be in R such that cc = −1, x be a vector in R with 〈x,x〉 = −1, and yi = −〈ri,x〉 for 1 ≤ i ≤ k. Then the following matrix G =