Construction of good quantum codes via classical codes is an important task for quantum information and quantum computing. In this work, by virtue of a decomposition of the defining set of constacyclic codes we have constructed eight new classes of entanglement-assisted quantum maximum distance separable codes. Mathematics Subject Classification IntroductionQuantum error-correcting (QEC for brevity) codes were introduced for security of quantum information. Construction of good quantum codes via classical codes is a crucial task for quantum information and quantum computing (see Refs. [1,4,5,7,17,26,27,30,32] for example). A q-ary quantum code Q, denoted by parameters n, k, d q , is a q k dimensional subspace of the Hilbert space C q n . A quantum code C with parameters n, k, d q satisfy the quantum Singleton bound: k ≤ n − 2d + 2 (see [17]). If k = n − 2d + 2, then C is called a quantum maximum-distance-separable (MDS) code. In recent years, many researchers have been working to find quantum MDS codes via constacyclic codes (for instance, see [6,19,16,18,32,33]).Entanglement-assisted quantum error correcting (EAQEC for short) codes use pre-existing entanglement between the sender and receiver to improve
The reversibility problem for linear cellular automata with null boundary defined by a rule matrix in the form of a pentadiagonal matrix was studied over the binary field ℤ2 by Martín del Rey et al. [Appl. Math. Comput.217, 8360 (2011)]. Recently, the reversibility problem of 1D Cellular automata with periodic boundary has been extended to ternary fields and further to finite primitive fields ℤp by Cinkir et al. [J. Stat. Phys.143, 807 (2011)]. In this work, we restudy some of the work done in Cinkir et al. [J. Stat. Phys.143, 807 (2011)] by using a different approach which is based on the theory of error correcting codes. While we reestablish some of the theorems already presented in Cinkir et al. [J. Stat. Phys.143, 807 (2011)], we further extend the results to more general cases. Also, a conjecture that is left open in Cinkir et al. [J. Stat. Phys.143, 807 (2011)] is also solved here. We conclude by presenting an application to Error Correcting Codes (ECCs) where reversibility of cellular automata is crucial.
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.
Constacyclic codes are preferred in engineering applications due to their efficient encoding process via shift registers. The class of constacyclic codes contains cyclic and negacyclic codes. The relation and presentation of cyclic codes as group algebras has been considered. Here for the first time, we establish a relation between constacyclic codes and group algebras and study their algebraic structures. Further, we give a method for constructing constacyclic codes by using zero-divisors in group algebras. Some good parameters for constacyclic codes which are derived from the proposed construction are also listed.
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