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We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings and ; using groups of order 4 and 8. Through these constructions and their extensions, we find binary self-dual codes of lengths 16, 32, 48, 64 and 68, many of which are extremal. In particular, we find forty new extremal binary self-dual codes of length 68, including twelve new codes with in , which is the first instance of such a value in the literature. 相似文献
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We characterize the structure of 2-quasi-cyclic codes over a finite field F by the so-called Goursat Lemma. With the characterization, we exhibit a necessary and sufficient condition for a 2-quasi-cyclic code being a dihedral code. And we obtain a necessary and sufficient condition for a self-dual 2-quasi-cyclic code being a dihedral code (if ), or a consta-dihedral code (if ). As a consequence, any self-dual 2-quasi-cyclic code generated by one element must be (consta-)dihedral. In particular, any self-dual double circulant code must be (consta-)dihedral. We also obtain necessary and sufficient conditions under which the three classes (the self-dual double circulant codes, the self-dual 2-quasi-cyclic codes, and the self-dual (consta-)dihedral codes) of codes coincide with each other. 相似文献
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《Discrete Mathematics》2022,345(4):112767
Let R be a finite commutative chain ring, be the dihedral group of size 2n and be the dihedral group ring. In this paper, we completely characterize left ideals of (called left -codes) when . In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over , where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where is a Galois field, we give a classification for left -codes over , for any positive integer N. In both cases we determine dual codes and identify self-dual ones. 相似文献
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《Discrete Mathematics》2024,347(1):113662
According to the well-known CSS construction, constructing quantum MDS codes are extensively investigated via Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. In this paper, given two Hermitian self-orthogonal GRS codes and , we propose a sufficient condition to ensure that is still a Hermitian self-orthogonal code. Consequently, we first present a new general construction of infinitely families of quantum MDS codes from known ones. Moreover, applying the trace function and norm function over finite fields, we give another two new constructions of quantum MDS codes with flexible parameters. It turns out that the forms of the lengths of our quantum MDS codes are quite different from previous known results in the literature. Meanwhile, the minimum distances of all the q-ary quantum MDS codes are bigger than . 相似文献
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In this paper, by Gauss sums, we determine Hamming weight distributions of generalized quasi-cyclic (GQC) codes over the finite chain ring , where . As applications, some new infinite families of minimal linear codes with and projective two-weight codes are constructed. 相似文献
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In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of -linear codes, where and that provides LCD codes over . 相似文献
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Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over are GRS codes (resp., extended GRS codes) when . In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs. 相似文献
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《Discrete Mathematics》2023,346(1):113126
New s-extremal extremal unimodular lattices in dimensions 38, 40, 42 and 44 are constructed from self-dual codes over by Construction A. In the process of constructing these codes, we obtain a self-dual code over . In addition, the code implies a code over . These codes have larger minimum weights than the previously known codes and codes, respectively. 相似文献
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We give methods for constructing many self-dual -codes and Type II -codes of length 2n starting from a given self-dual -code and Type II -code of length 2n, respectively. As an application, we construct extremal Type II -codes of length 24 for and extremal Type II -codes of length 32 for . We also construct new extremal Type II -codes of lengths 56 and 64. 相似文献
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In this paper, based on the structure of embedded fields, we investigate explicit construction of systematic mMDS sliding window codes with memory . First, over GF() with and , we propose an algorithm to construct mMDS codes with memory 2, which are optimal in the sense that is the maximum possible value of k for a sliding window code with memory 2 over GF() to be mMDS. When , every constructed code has the extra property that it contains a mMDS sliding window code with memory 2 as a subcode over the subfield GF(). Next, over GF() with and , we introduce a method to construct mMDS codes memory 3, and a few new codes have been obtained consequently. When , every code constructed by the new approach also has the property that it contains an mMDS subcode over the subfield GF(). The embedding subfield-subcode property enhances the flexibility and efficiency of the designed codes. 相似文献
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