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Entanglement-assisted quantum error-correcting (EAQEC, for short) codes use pre-existing entanglements between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQEC code from any classical linear code, unlike standard quantum error-correcting codes, they can only be constructed from classical linear codes which contain their Hermitian dual codes. However, how to determine the parameters of ebits c in EAQEC codes is not an easy task. In this paper, let p be prime and e, k be integers, we construct six classes of EAQEC codes based on k-Galois dual codes over finite fields , where . The parameter of ebits c of these EAQEC codes can be easily generated algebraically. Furthermore, the six classes of EAQEC codes are of maximal entanglement, most of which have better parameters than current EAQEC codes available. 相似文献
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Minimal linear codes have important applications in secret sharing schemes and secure multi-party computation, etc. In this paper, we study the minimality of a kind of linear codes over from Maiorana-McFarland functions. We first obtain a new sufficient condition for this kind of linear codes to be minimal without analyzing the weights of its codewords, which is a generalization of some works given by Ding et al. in 2015. Using this condition, it is easy to verify that such minimal linear codes satisfy for any prime p, where and denote the minimum and maximum nonzero weights in a code, respectively. Then, by selecting the subsets of , we present two new infinite families of minimal linear codes with for any prime p. In addition, the weight distributions of the presented linear codes are determined in terms of Krawtchouk polynomials or partial spreads. 相似文献
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Hai Q. Dinh Xiaoqiang Wang Hongwei Liu Songsak Sriboonchitta 《Discrete Mathematics》2019,342(5):1456-1470
Let be an odd prime, , be positive integers, be nonzero elements of the finite field such that . In this paper, we show that, for any positive integer , the Hamming distances of all repeated-root -constacyclic codes of length can be determined by those of certain simple-root -constacyclic codes of length . Using this result, Hamming distances of all constacyclic codes of length are obtained. As an application, we identify all MDS -constacyclic codes of length . 相似文献
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Hai Q. Dinh Xiaoqiang Wang Hongwei Liu Songsak Sriboonchitta 《Discrete Mathematics》2019,342(11):3062-3078
Let be an odd prime, and be a nonzero element of the finite field . The -constacyclic codes of length over are classified as the ideals of quotient ring in terms of their generator polynomials. Based on these generator polynomials, the symbol-pair distances of all such -constacyclic codes of length are obtained in this paper. As an application, all MDS symbol-pair constacyclic codes of length over are established, which produce many new MDS symbol-pair codes with good parameters. 相似文献
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In this paper, we explore some properties of hulls of cyclic serial codes over a finite chain ring and we provide an algorithm for computing all the possible parameters of the Euclidean hulls of that codes. We also establish the average -dimension of the Euclidean hull, where is the residue field of R, as well as we give some results of its relative growth. 相似文献
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Huffman (2013) [12] studied -linear codes over and he proved the MacWilliams identity for these codes with respect to ordinary and Hermitian trace inner products. Let S be a finite commutative -algebra. An -linear code over S of length n is an -submodule of . In this paper, we study -linear codes over S. We obtain some bounds on minimum distance of these codes, and some large classes of MDR codes are introduced. We generalize the ordinary and Hermitian trace products over -algebras and we prove the MacWilliams identity with respect to the generalized form. In particular, we obtain Huffman's results on the MacWilliams identity. Among other results, we give a theory to construct a class of quantum codes and the structure of -linear codes over finite commutative graded -algebras. 相似文献
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The -additive codes are subgroups of , and can be seen as linear codes over when , -additive codes when , or -additive codes when . A -linear generalized Hadamard (GH) code is a GH code over which is the Gray map image of a -additive code. Recursive constructions of -additive GH codes of type with are known. In this paper, we generalize some known results for -linear GH codes with to any prime when , and then we compare them with the ones obtained when . First, we show for which types the corresponding -linear GH codes are nonlinear over . Then, for these codes, we compute the kernel and its dimension, which allow us to classify them completely. Moreover, by computing the rank of some of these codes, we show that, unlike -linear Hadamard codes, the -linear GH codes are not included in the family of -linear GH codes with when prime. Indeed, there are some families with infinite nonlinear -linear GH codes, where the codes are not equivalent to any -linear GH code with . 相似文献
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《Discrete Mathematics》2022,345(11):113059
Let be the finite field of q elements and let be the dihedral group of 2n elements. Left ideals of the group algebra are known as left dihedral codes over of length 2n, and abbreviated as left -codes. Let . In this paper, we give an explicit representation for the Euclidean hull of every left -code over . On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left -codes over . In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left -codes and self-dual left -codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left -code over , and present several numerical examples to illustrative our applications. 相似文献