1-Generator Generalized Quasi-Cyclic Codes over Z4

Cryptography and Communications manuscript No. (will be inserted by the editor)

1-Generator Generalized Quasi-Cyclic Codes over Z4 Tingting Wu · Jian Gao · Fang-Wei Fu

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Abstract In this short paper, we determine the minimal generating set of 1generator generalized quasi-cyclic codes over Z4 . We also determine their rank and introduce a lower bound for the minimum distance of free 1-generator generalized quasi-cyclic codes. Further, we construct some new Z4 -linear codes and we obtain some good binary nonlinear codes using the usual Gray map. Mathematics Subject Classification (2000) 94B05 · 94B15 Keywords 1-generator generalized quasi-cyclic codes · New Z4 -linear codes · Gray map · Good binary nonlinear codes

1 Introduction Codes over Z4 have been studied by many authors due to the fact that certain good binary nonlinear codes can be constructed from Z4 codes via the Gray map [10]. Quasi-cyclic(QC) codes are an important class of linear codes which include cyclic codes as their subclass. Recently, the structural properties of QC codes over Z4 with odd length components have been studied in [2], where The first two authors contributed equally to this work. Tingting Wu Chern Institute of Mathematics and LPMC, Nankai University Tianjin, 300071, P. R. China E-mail: [email protected] Jian Gao School of Science, Shandong University of Technology Zibo, 255091, P. R. China E-mail: [email protected]; [email protected] Fang-Wei Fu Chern Institute of Mathematics and LPMC, Nankai University Tianjin, 300071, P. R. China E-mail: [email protected]

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the authors discovered some new binary nonlinear codes from QC codes over Z4 by applying the usual Gray map. In [13], QC codes over Z4 with even length components are studied. In their work, besides the best known binary nonlinear codes the authors found a new binary nonlinear code with better parameters than the best known code in the literature. The algebraic structure of generalized quasi-cyclic(GQC) codes over finite fields was first introduced in [14]. Since then, further properties of GQC codes have been investigated by many authors [4–6,8]. As a special class of GQC codes, the double cyclic codes over Z4 have been studied in [7]. Some good binary nonlinear codes are obtained from this family of codes. In this short paper, we investigate the structure of 1-generator GQC codes of length (m1 , m2 , . . . , mℓ ) over Z4 , where mi is an odd positive integer for each i = 1, 2, . . . , ℓ. The rest of paper is organized as follows. In Section 2, we recall some basic definitions and facts of linear codes over Z4 briefly. In Section 3, we give the structural properties on 1-generator GQC codes over Z4 , including their minimal generating set and the minimum distance bound for this family of codes. In Section 4, we construct some new Z4 -linear codes, moreover, we obtain some binary nonlinear codes from 1-generator GQC codes over Z4 by applying the usual Gray map. 2 Preliminaries A linear code of length n over Z4 is a Z4 -submodule of Zn4 . Any Z4 -linear code C is permutation equivalent to a code with a generator matrix of the form ( ) Ik 1 A B G= 0 2Ik2 2D where A, D are k1 × k2 , k2 × (n − k1 − k2 ) F2 -matrices, respectively, and B is a k1 × (n − k1 − k2 ) Z4 -matrix. Then we say that C is of type 4k1 2k2 and the size of C is 4k1 2k2 . A Z4 -linear code is not necessarily a free module, that is a module with a Z4 -basis. A code is free if and only if k2 = 0. The Lee weights of 0, 1, 2, 3 ∈ Z4 are 0, 1, 2, 1, respectively. The Lee weight of a vector c ∈ Zn4 is the rational sum of the Lee weights of its components. The map that is used to obtain binary codes from Z4 -codes is the usual Gray map ϕ, which is from Zn4 to F2n 2 and maps the elements 0, 1, 2, 3 of Z4 to (0, 0), (0, 1), (1, 1), (1, 0) over F2 , respectively. The Gray image ϕ(C) of a Z4 code C of length n will be a binary code of length 2n. In general, the Gray image of a Z4 -code is not necessarily a binary linear code, since the Gray map is not F2 -linear. The Gray map ϕ is an isometry from (Zn4 , Lee distance) to (F2n 2 , Hamming distance). Cyclic codes are an important subclass of linear codes. Some results of cyclic codes over Z4 which will be used in the rest of paper are stated below. Definition 1 [15] Let C be a linear code of length n over Z4 , i.e. a Z4 submodule of Zn4 . If for any codeword (c0 , c1 , . . . , cn−1 ) ∈ C we have that (cn−1 , c0 , c1 , . . . , cn−2 ) ∈ C, then we say that C is a Z4 -cyclic code.

1-Generator Generalized Quasi-Cyclic Codes over Z4

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It is a well-known fact that under the usual identification of vectors with polynomials, cyclic codes of length n are precisely ideals in the ring Rn = Z4 [x]/(xn − 1). Lemma 1 [15] Suppose C is a Z4 -cyclic code of odd length n, then Rn is a principal ideal ring and C = (g(x), 2a(x)) = (g(x) + 2a(x)) where g(x), a(x) ∈ Z4 [x] with a(x)|g(x) and g(x)|(xn − 1). Definition 2 [15] Two polynomials f1 (x), f2 (x) ∈ Z4 [x] are said to be relatively prime, denoted by gcd(f1 (x), f2 (x)) = 1, in Z4 [x] if there exist polynomials p1 (x), p2 (x) ∈ Z4 [x] such that p1 (x)f1 (x) + p2 (x)f2 (x) = 1. Note that if gcd(f1 (x), f2 (x)) = 1 in Z4 [x] then gcd(f 1 (x), f 2 (x)) = 1 in F2 [x], where for i = 1, 2 we have f i (x) = fi (x)(mod2). For more details of structural properties of Z4 cyclic codes, please see the references [10–12, 15]. 3 Algebraic structure on 1-generator GQC codes over Z4 Definition 3 Let m1 , m2 , . . . , mℓ be positive integers and Ri = Z4 [x]/(xmi − 1), i = 1, 2, . . . , ℓ. Any Z4 [x]-submodule of R = R1 × R2 × · · · × Rℓ is called a GQC code of length (m1 , m2 , . . . , mℓ ). Remark 1 In the rest of the paper, we assume that m1 , m2 , . . . , mℓ are odd positive integers. Note that if C is a GQC code of length (m1 , m2 , . . . , mℓ ) with m = m1 = m2 = · · · = mℓ , then C is a QC code with length mℓ. Further, if ℓ = 1, then C is a cyclic code of length m. And if ℓ = 2, then C is called a double cyclic code of length (m1 , m2 ). A GQC code C of length (m1 , m2 , . . . , mℓ ) is called 1-generator if C = RF (x) for some F (x) ∈ R. Lemma 2 Let C be a 1-generator GQC code of length (m1 , m2 , . . . , mℓ ), let F (x) = (F1 (x), F2 (x), . . . , Fℓ (x)) ∈ R be a generator of C, where Fi (x) ∈ Ri for i = 1, 2, . . . , ℓ. Then Fi ∈ Ci , where Ci is a cyclic code of length mi over Ri . Hence, Fi can be selected to be of the form Fi (x) = fi (x)(gi (x) + 2ai (x)) where fi (x), gi (x), ai (x) ∈ Z4 [x] with ai (x)|gi (x) and gi (x)|(xmi − 1). Proof For all 1 ≤ i ≤ ℓ, we define the following projection map Πi : R → Ri such that Πi (f1 (x), f2 (x), . . . , fℓ (x)) = fi (x). The set Πi (C) is a cyclic code over Ri . By Lemma 1, there exist gi (x), ai (x) ∈ Z4 [x] such that Πi (C) = (gi (x) + 2ai (x)) where ai (x)|gi (x) and gi (x)|(xmi − 1). Since Fi (x) ∈ Πi (C), there exists fi (x) ∈ Z4 [x] such that Fi (x) = fi (x)(gi (x) + 2ai (x)). ⊔ ⊓

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In the following, we give the generating sets of 1-generator GQC codes over Z4 . Theorem 1 Let C be a 1-generator GQC code of length (m1 , m2 , . . . , mℓ ) over Z4 generated by G = (f1 g1 + 2q1 , f2 g2 + 2q2 , . . . , fℓ gℓ + 2qℓ ) where fi , gi , qi ∈ Z4 [x], gi |(xmi −1), i = 1, 2, . . . , ℓ. Assume that for all i ∈ {1, 2, . . . , ℓ}, fi gi +2qi mi is not a zero divisor of Ri . Let hi = gcd(fxi gi ,x−1 mi −1) , h = lcm(h1 , h2 , . . . , hℓ ) and x −1 deg(h) = r. Let vi = gcd(hq , v = lcm(v1 , v2 , . . . , vℓ ) and deg(v) = t. Let m i ,x i −1) ∪ B = (2hq1 , 2hq2 , . . . , 2hqℓ ). Then the minimal generating set of C is S1 S2 , where mi

S1 = {G, xG, . . . , xr−1 G}, S2 = {B, xB, . . . , xt−1 B}. Furthermore, C has 4r 2t codewords. Proof Let c(x) = f (x)G be a codeword in C, where f (x) ∈ Z4 [x]. Using the Euclidean division there are two unique polynomials Q1 (x), T1 (x) ∈ Z4 [x] such that f (x) = h(x)Q1 (x) + T1 (x), where T1 (x) = 0 or deg T1 (x) ≤ r − 1. Hence, c(x) = f (x)G = (h(x)Q1 (x) + T1 (x))(f1 g1 + 2q1 , f2 g2 + 2q2 , . . . , fℓ gℓ + 2qℓ ) = Q1 (x)B + T1 (x)G. Note that T1 (x)G ∈ Span(S1 ). Again using the Euclidean division there are two unique polynomials Q2 (x), T2 (x) ∈ Z4 [x] such that Q1 (x) = Q2 (x)v(x) + T2 (x), where T2 (x) = 0 or deg T1 (x) ≤ t − 1. Therefore, we have Q1 (x)B = (Q2 (x)v(x) + T2 (x))B = T2 (x)B. ∪ Obviously, T2 (x)B ∈ Span(S ∪ 2 ). It follows that c(x) ∈ Span(S1 ) Span(S2 ). Consequently, Span(S1 ) Span(S2 ) is a spanning set ∩ of C. In the following, we will prove that Span(S ) Span(S2 ) = {0}. Sup1 ∩ pose that e(x) ∈ Span(S1 ) Span(S2 ), where e(x) = (e1 (x), e2 (x), . . . , eℓ (x)), (1) ei (x) ∈ Ri . Since e(x) ∈ Span(S1 ), we have ei (x) = (fi gi + 2qi )Mi (x), where (1) Mi (x) = α0 + α1 x + · · · + αr−1 xr−1 . On the other hand, e(x) ∈ Span(S2 ) (2) (2) shows that ei (x) = 2hqi Mi , where Mi (x) = β0 +β1 x+· · ·+βt−1 xt−1 . Thus 2ei (x) = 0, which implies that αi = 0 or 2 for i = 0, 1, . . . , r − 1. Note that (1) (2) (1) (2) (fi gi + 2qi )Mi (x) = 2hqi Mi , which means (fi gi + 2qi )(Mi − hMi ) = 0. (1) (2) Since fi gi + 2qi is not a zero divisor of Ri , we must have Mi − hMi = 0, which results ∩ in αi = 0 and βj = 0 for i = 0, 1, . . . , r − 1, j = 0, 1, . . . , t − 1, i.e. Span(S1 ) Span(S2 ) = {0}. ⊔ ⊓ Corollary 1 For each i = 1, 2, . . . , ℓ, if (fi gi + 2qi )|(xmi − 1) over Z4 , then C is a free GQC code of rank r over Z4 . Furthermore, C has 4r codewords.


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Proof fi , gi , qi could be denoted by fi = ai + 2a′i , gi = bi + 2b′i , qi = ci + 2c′i , where ai , a′i , bi , b′i , ci , c′i ∈ F2 [x]. It follows that fi gi +2qi = ai bi +2(ai b′i +a′i bi + ci ), we let fi′ = ai , gi′ = bi , qi′ = ai b′i + a′i bi + ci . Therefore, we could always write fi gi + 2qi as fi′ gi′ + 2qi′ where fi′ , gi′ , qi′ ∈ F2 [x], without loss of generality, we may assume that fi , gi , qi ∈ F2 [x] for i = 1, 2, . . . , ℓ. If (fi gi + 2qi )|(xmi − 1) mi over Z4 , then fi gi |(xmi − 1) over F2 . Let hi = x fi g−1 . Then there exists a i mi x −1 polynomial wi ∈ F2 [x] such that fi gi +2qi = hi +2wi over Z4 . Hence, there exists a polynomial s ∈ F2 [x] such that h+2s = lcm{h1 +2w1 , h2 +2w2 , . . . , hℓ +2wℓ }, where h = lcm(h1 , h2 , . . . , hℓ ). Thus, (h + 2s)(fi gi + 2qi ) = 0. Therefore, 2hqi = −2sfi gi = −2s(fi gi + 2qi ), i.e. 2hqi is a multiple of fi gi + 2qi for all i = 1, 2, . . . , ℓ. So the set S2 ⊆ Span(S1 ). This means that the code C is free and its minimal generating set is S1 = {G, xG, . . . , xr−1 G}. Thus, the rank of C is r and it has 4r codewords. ⊔ ⊓ In the following, we give a lower bound on the minimum distance of free 1-generator GQC codes over Z4 . Theorem 2 Let C be a free 1-generator GQC code as in Corollary 1. Suppose mi −1 hi = fxi gi +2q , i = 1, 2, . . . , ℓ, and h = lcm{h1 , h2 , . . . , hℓ }. Then i ∑ (i) dmin (C) ≥ i∈K / di , where K ⊆ {1, 2, . . . , ℓ} is a set of maximum size for which lcm{hi , i ∈ K} ̸= h and di = dmin (Πi (C)); ∑ℓ (ii) if h1 = h2 = · · · = hℓ , then dmin (C) ≥ i=1 di . Proof Let c(x) ∈ C be a nonzero codeword. Then there exists a polynomial f (x) ∈ Z4 [x] such that c(x) = f (x)G. For each i = 1, 2, . . . , ℓ, since (fi gi + 2qi )|(xmi − 1), it follows that the i-th component is zero if and only if (xmi − 1)|f (x)(fi gi +2qi ), that is, if and only if hi |f (x). Therefore, c(x) = 0 if and only if h|f (x). So c(x) ̸= 0 if and only if h - f (x). This implies that c(x) ̸= 0 has the most number of zero blocks whenever h ̸= lcmi∈K hi , lcmi∈K hi |f (x), and ∑ K is a maximal subset of {1, 2, . . . , ℓ} having this property. Thus, dmin (C) ≥ i∈K / di , where di = dmin (Πi (C)). Clearly, K = ∅ if and only if h1 = h2 = · · · = hℓ ∑ℓ ⊔ ⊓ which deduces that dmin (C) ≥ i=1 di . Example 1 Let C be a 1-generator GQC code of length (3, 31) over Z4 generated by G = (f1 g1 + 2q1 , f2 g2 + 2q2 ) where f1 = 1, g1 = q1 = x2 + x + 1, f2 = 1, g2 = q2 = x25 + 3x24 + 2x23 + 2x22 + 3x21 + 2x20 + x19 + 3x18 + x16 + x15 + 3x14 + x13 + x11 + 3x9 + 2x7 + 2x6 + 3x5 + 2x3 + 3x2 + 3x + 3 such that (f1 g1 +2q1 )|(x3 −1), (f2 g2 +2q2 )|(x31 −1). It follows that Π1 (C) is a cyclic code generated by 3g1 with minimum Lee distance dmin (Π1 (C)) = 3 and Π2 (C) is a cyclic code generated by 3g2 with minimum Lee distance dmin (Π2 (C)) = 26. By Theorem 2, we get K = {1}, which implies that dmin (C) ≥ d(Π2 (C)) = 26. In fact, C is a (34, 46 , 28) free linear code over Z4 . This code is not only much better than the comparable best known quaternary linear code (34, 46 , 20), but also has a larger minimum Lee distance than the best known quaternary nonlinear code with parameters (34, 46 , 27) (see [1]). Its Gray image is a (68, 212 , 28) binary nonlinear code, which has smaller codewords than the best

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known binary nonlinear code (68, 213 , 28). But it has the same parameters as the optimal binary linear code [68, 12, 28] (see [9]). By Corollary 1 and Theorem 2, we have the following results directly. Corollary 2 Let C be a 1-generator GQC code generated by G = ((f1 g1 + 2q1 )k1 , (f2 g2 +2q2 )k2 , . . . , (fℓ gℓ +2qℓ )kℓ ), where for each i = 1, 2, . . . , ℓ we have (fi gi + 2qi )|(xmi − 1). Suppose that for each i = 1, 2, . . . , ℓ, deg(fi gi + 2qi ) = mi −1 deg(fi gi ), hi = fxi gi +2q , gcd(hi , ki ) = 1 and h = lcm{h1 , h2 , . . . , hℓ }. Then i (i) C is a free code of rank deg(h) and |C| = 4deg(h) ; ∑ (ii) dmin (C) ≥ i∈K / di , where K ⊆ {1, 2, . . . , ℓ} is a set of maximum size for which lcm{hi , i ∈ K} ̸= h and di = dmin (Πi (C)); ∑ℓ (iii) if h1 = h2 = · · · = hℓ , then dmin (C) ≥ i=1 di . Corollary 3 Let C1 be a free 1-generator GQC code as in Theorem 2 (ii) and n −1 C2 = (f g + 2q) be a free cyclic code of length n over Z4 and h = fxg+2q . Let C be a code obtained by concatenating of C1 and C2 . Then ∑ℓ (i) if gcd(h, hi ) = 1, then C is a free 1-generator GQC code of length i=1 mi + n with rank deg(hi h) and dmin (C) ≥ min{dmin (C1 ), dmin (C2 )}; ∑ℓ (ii) if h|hi , then C is a free 1-generator GQC code of length i=1 mi + n with rank deg(hi ) and dmin (C) ≥ dmin (C1 ) Example 2 Let C1 be a free 1-generator GQC code of length (21, 63) over Z4 generated by G = (f1 g1 + 2q1 , f2 g2 + 2q2 ) where f1 = 1, g1 = q1 = x11 + 2x9 + 3x8 +x7 +2x5 +2x4 +x2 +2x+3, f2 = 1, g2 = q2 = x53 +2x51 +3x50 +x49 +2x47 + 2x46 + 3x44 + 2x43 + 3x42 + x32 + 2x30 + 3x29 + x28 + 2x26 + 2x25 + 3x23 + 2x22 + 3x21 +x11 +2x9 +3x8 +x7 +2x5 +2x4 +3x2 +2x+3 such that (f1 g1 +2q1 )|(x21 −1), (f2 g2 + 2q2 )|(x63 − 1). And h1 = h2 = 3x10 + 2x8 + 3x7 + x6 + x4 + x2 + 2x + 3. Then C1 is a free (84, 410 , 32) GQC code. Moreover, C1 is a linear code and it has better parameters than the best known quaternary linear code (84, 410 , 29) (see [1]). (i) If C2 is a free cyclic code of length 17 over Z4 and its parity-check polynomial is h(x) = 3x8 +2x6 +x5 +3x4 +x3 +2x2 +3 satisfying gcd(h, hi ) = 1, then C2 is a free cyclic code generated by x9 +2x7 +x6 +3x5 +x4 +3x3 +2x2 +3 with minimum Lee distance 8. Concatenation of C1 and C2 forms a GQC code C as defined in Corollary 3. By Corollary 3(i), we get dmin (C) ≥ 8. In fact, C is a (101, 418 , 8) free 1-generator GQC code over Z4 . (ii) If C2 is a free cyclic code of length 7 over Z4 and its parity-check polynomial is h(x) = 3x3 + x2 + 2x + 1 satisfying h|hi , then C2 is a free cyclic code generated by x4 + x3 + 3x2 + 2x + 1 with minimum Lee distance 6. Concatenation of C1 and C2 forms a 1-generator GQC code C as defined in Corollary 3. By Corollary 3(ii), we get dmin (C) ≥ 32. In fact, C is a (91, 410 , 38) free 1-generator GQC code over Z4 and it has better parameters than the best known quaternary linear code (91, 410 , 32) (see [1]).


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4 New Z4 -linear codes and good binary nonlinear codes Similar to the case of QC codes over Z4 , we are able to obtain many good binary nonlinear codes by taking advantage of the structure of GQC codes over Z4 . In the following, we construct some GQC codes over Z4 , which are new codes and have better parameters than the best known quaternary linear or nonlinear codes. Furthermore, most of these codes could yield good binary nonlinear codes via the Gray map, these good binary nonlinear codes are also shown in this section. The generator matrix of the 1-generator GQC code is determined by the first row alone, since every component of the generator matrix is a circulant matrix. Hence, we will only give the first row of the generator matrix, where the components will be separated by a comma. Example 3 Let C be a 1-generator GQC code of length (3, 73) over Z4 generated by G = (f1 g1 + 2q1 , f2 g2 + 2q2 ), where f1 = 1, g1 = q1 = x2 + x + 1, f2 = 1, 73 9 5 9 8 4 g2 = q2 = sx1 s2−1 s3 ( s1 = x+3, s2 = x +2x +x+3 and s3 = x +3x +2x +3) mi satisfying (fi gi + 2qi )|(x − 1), i = 1, 2. By Theorem 1, we get deg(h) = 19. Therefore, C is spanned by S1 = {G, xG, . . . , x18 G}. The first row of the generator matrix of C is given by 333, 3210101020330233112001223303033221002113320330201010123000000000 000000000. C is a (76, 419 , 40) free linear code over Z4 . It is a new linear code and it has the same parameters with the best known quaternary nonlinear code (76, 419 , 40) (see [1]). The Gray image of C is a (152, 238 , 40) binary nonlinear code. It has the same parameters with the best known binary linear code [152, 38, 40] (see [9]). Its Hamming weight enumerator is x152 + 9636x112 y 40 + 17082x110 y 42 + 88987x108 y 44 + 462382x106 y 46 + 1923039x104 y 48 + 7533162x102 y 50 + 26840056x100 y 52 + 87015708x98 y 54 + 250905380x96 y 56 + 645303794x94 y 58 + 1504871786x92 y 60 + 3144612386x90 y 62 + 10185859360x86 y 66 + 15733788112x84 y 68 + 5950320885x88 y 64 + 22113866202x82 y 70 + 28163643747x80 y 72 + 32534544516x78 y 74 + 34174694502x76 y 76 + 32534544516x74 y 78 + 28163643747x72 y 80 + 22113866202x70 y 82 + 15733788112x68 y 84 + 10185859360x66 y 86 + 5950320885x64 y 88 + 3144612386x62 y 90 + 1504871786x60 y 92 + 645303794x58 y 94 + 250905380x56 y 96 + 87015708x54 y 98 + 26840056x52 y 100 + 7533162x50 y 102 + 1923039x48 y 104 + 462382x46 y 106 + 88987x44 y 108 + 17082x42 y 110 + 9636x40 y 112 + y 152 . Furthermore, if we add an all-one vector to the first column of the generator matrix of C, then we can extend the code C to a code with parameters

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(77, 419 , 40). The extended code of C is a new linear code and much better than the best known quaternary linear code (77, 419 , 14), but it has a bit smaller minimum Lee distance than the best known quaternary nonlinear code (77, 419 , 41) (see [1]). Moreover, its Gray image is a (154, 238 , 40) binary nonlinear code. It has the same parameters with the best known binary linear code [154, 38, 40] (see [9]). Example 4 Let C be a 1-generator GQC code of length (3, 23) over Z4 generated by G = (f1 g1 + 2q1 , f2 g2 + 2q2 ), where f1 = 1, g1 = q1 = x2 + x + 1, f2 = 1, g2 = q2 = x11 + 3x10 + 2x7 + x6 + x5 + x4 + x2 + 2x + 3. The first row of generator matrix is given by 333, 12303332001300000000000. C is a (26, 412 , 12) free linear code over Z4 . It is a new linear code and it has the same parameters with the best known quaternary nonlinear code (26, 412 , 12) (see [1]). The Gray image of C is a (52, 224 , 12) binary nonlinear code. It gives the parameters of the best known linear code with parameters [52, 24, 12]. The Hamming weight enumerator is x52 + 7084x40 y 12 + 32384x38 y 14 + 125534x36 y 16 + 500480x34 y 18 + 1084105x32 y 20 + 2040192x30 y 22 + 2977580x28 y 24 + 3242496x26 y 26 + 2977580x24 y 28 + 2040192x22 y 30 + 1084105x20 y 32 + 500480x18 y 34 + 125534x16 y 36 + 32384x14 y 38 + 7084x12 y 40 + y 52 . Furthermore, if we add an all-one vector to the first column of the generator matrix of C, then we can extend the code C to a code with parameters (27, 412 , 12). The extended code of C is a new linear code and gives the same parameters with the best known quaternary nonlinear code (27, 412 , 12) (see [1]). Moreover, its Gray image is a (54, 224 , 12) binary nonlinear code. It has the same parameters with the best known binary linear code [54, 24, 12] (see [9]). Example 5 Let C be a 1-generator GQC code of length (3, 63) over Z4 generated by G = (f1 g1 + 2q1 , f2 g2 + 2q2 ), where f1 = 1, g1 = q1 = x2 + x + 1, 63 f2 = 1, g2 = q2 = s1xs2 s−1 , where s1 = x + 3, s2 = x6 + 3x5 + 2x4 + x2 + x + 1, 3 s4 s3 = x6 + 3x5 + x3 + x2 + 2x + 1, s4 = x6 + x5 + 3x4 + 3x2 + 2x + 1. The first row of the generator matrix of C is given by 333, 300022221130303031313002212103120230331330123000000000000000000. C is a (66, 419 , 32) free linear code over Z4 . It is a new linear code and much better than the best known quaternary linear code (66, 419 , 11), but it has a bit smaller minimum Lee distance than the best known quaternary nonlinear code (66, 419 , 34) (see [1]). The Gray image of C is a (132, 238 , 32) binary nonlinear


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code. It has a smaller minimum Hamming distance than the best known binary linear code [132, 38, 34] (see [9]). Its Hamming weight enumerator is x132 + 3465x100 y 32 + 6552x98 y 34 + 104139x96 y 36 + 384048x94 y 38 + 2612736x92 y 40 + 8872416x90 y 42 + 42806736x88 y 44 + 119039760x86 y 46 + 422195424x84 y 48 + 965472480x82 y 50 + 2520139104x80 y 52 + 4795171248x78 y 54 + 9388716624x76 y 56 + 14949331488x74 y 58 + 22246970592x72 y 60 + 34095927915x68 y 64 + 36902650640x66 y 66 + 34095927915x64 y 68 + 29429873424x62 y 70 + 22246970592x60 y 72 + 14949331488x58 y 74 + 9388716624x56 y 76 + 4795171248x54 y 78 + 2520139104x52 y 80 + 965472480x50 y 82 + 422195424x48 y 84 + 119039760x46 y 86 + 42806736x44 y 88 + 8872416x42 y 90 + 2612736x40 y 92 + 384048x38 y 94 + 104139x36 y 96 + 6552x34 y 98 + 3465x32 y 100 + y 132 . Remark 2 All the examples are computed by the computer algebra system MAGMA [3]. Acknowledgements This research is supported by the National Key Basic Research Program of China (973 Program Grant No. 2013CB834204), the National Natural Science Foundation of China (Nos. 61171082 and 61301137)

References 1. Aydin, N., Asamov, T.: The Z4 Database [Online]. Available: http://z4codes.info/. accessed on 28.08.2015. 2. Aydin, N., Ray-Chaudhuri, D. K.: Quasi cyclic codes over Z4 and some new binary codes. IEEE Trans. Inform. Theory 48, 2065-2069(2009) 3. Bosma, W., Cannon, J., and Playoust: The Magma algebra system. J. Symb. Comput. 24, 235-265(1997) 4. Cao, Y.: Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration, Appl. Algebra Eng. Commun. Comput. 22, 219-233(2011) 5. Cao, Y.: Structural properties and enumeration of 1-generator generalized quasi-cyclic codes. Des. Codes Cryptogr. 60, 67-79(2011) 6. Esmaeili, M., Yari, S.: Generalized quasi-cyclic codes: Structural properties and codes construction. Appl. Algebra Eng. Commun. Comput. 20, 159-173(2009) 7. Gao, J., Shi, M., Wu, T., Fu, F.-W.: On double cyclic codes over Z4 . arXivpreprint, arXiv: 1501.01360v1(2015) 8. Gao, J., Fu, F.-W., Shen, L., Ren, W.: Some results on generalized quasi-cyclic codes over Fq + uFq , IEICE Trans. Fundamentals 97, 1005-1011(2014) 9. Grassl, M.: Table of Bounds on Linear Codes [Online]. Available: http://www.codetables.de/. accessed on 14.05.2015. 10. Hammons, J., Kumar, P. V., Calderbank, A. R., Sloane, N. J., Sol´ e, P.: The Z4 -linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40(2), 301-319(1994) 11. Van Lint, J. H.: Introduction to Coding Theory. Springer-Verlag, Berlin(1999) 12. Pless, V., Qian, Z.: Cyclic codes and quadratic residue codes over Z4 . IEEE Trans. Inform. Theory 42, 1594-1600(1996) 13. Siap, I., Abualrub, T., Aydin, N.: Quaternary quasi-cyclic codes with even length components. Ars Combinatoria 101, 425-434(2011)

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14. Siap, I., Kulhan, N.: The structure of generalized quasi-cyclic codes. Appl. Math. ENotes. 5, 24-30(2005) 15. Wan, Z.-X.: Quaternary Codes. World Scientific, Singapore(1997)