Abstract. In this paper, we extend the results given in [3] to a nonchain ring Rp = Fp + vFp + · · · + v p−1 Fp, where v p = v and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring Rp and study the structure of their duals. We classify cyclic codes containing their duals over Rp by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map π defined in [4], we give the parameters of the quantum codes of length pn over Fp which are obtained from cyclic codes over Rp. Finally, we illustrate the results by giving some examples.
The constructions of entanglement-assisted quantum codes have been studied intensively by researchers. Nevertheless, it is hard to determine the number of shared pairs required for constructing entanglement-assisted quantum codes from linear codes. In this paper, by making use of the notion of decomposition for defining sets of constacyclic codes, we construct several new families of entanglement-assisted quantum MDS codes from constacyclic codes, some of which are of minimum distances greater than q + 1. Moreover, we tabulate their parameters to illustrate what we find in this paper.
In this study, we introduce a new Gray map which preserves the orthogonality from the chain ring F_2 [u] / (u^s ) to F^s_2 where F_2 is the finite field with two elements. We also give a condition of the existence for cyclic codes of odd length containing its dual over the ring F_2 [u] / (u^s ) . By taking advantage of this Gray map and the structure of the ring, we obtain two classes of binary quantum error correcting (QEC) codes and we finally illustrate our results by presenting some examples with good parameters.
Linear codes with complementary duals (LCD) have a great deal of significance amongst linear codes. Maximum distance separable (MDS) codes are also an important class of linear codes since they achieve the greatest error correcting and detecting capabilities for fixed length and dimension. The construction of linear codes that are both LCD and MDS is a hard task in coding theory. In this paper, we study the constructions of LCD codes that are MDS from negacyclic codes over finite fields of odd prime power q elements. We construct four families of MDS negacyclic LCD codes of length n| q−1 2 , n| q+1 2 and a family of negacyclic LCD codes of length n = q − 1. Furthermore, we obtain five families of q 2 -ary Hermitian MDS negacyclic LCD codes of length n| (q − 1) and four families of Hermitian negacyclic LCD codes of length n = q 2 + 1. For both Euclidean and Hermitian cases the dimensions of these codes are determined and for some classes the minimum distances are settled. For the other cases, by studying q and q 2 -cyclotomic classes we give lower bounds on the minimum distance.
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