Some New Classes of Entanglement-Assisted Quantum MDS Codes Derived From Constacyclic Codes

Abstract

Although quantum maximal-distance-separable (MDS) codes that satisfy the quantum singleton bound have become an important research topic in the quantum coding theory, it is not an easy task to search for quantum MDS codes with the minimum distance that is larger than <inline-formula> <tex-math notation= LaTeX >$(q/2)+1$ </tex-math></inline-formula>. The pre-shared entanglement between the sender and the receiver can improve the minimum distance of quantum MDS codes such that the minimum distance of some constructed codes achieves <inline-formula> <tex-math notation= LaTeX >$(q/2)+1$ </tex-math></inline-formula> or exceeds <inline-formula> <tex-math notation= LaTeX >$(q/2)+1$ </tex-math></inline-formula>. Meanwhile, how to determine the required number of maximally entangled states to make the minimum distance of quantum MDS codes larger than <inline-formula> <tex-math notation= LaTeX >$(q/2)+1$ </tex-math></inline-formula> is an interesting problem in the quantum coding theory. In this paper, we utilize the decomposition of the defining set and <inline-formula> <tex-math notation= LaTeX >$q^{2}$ </tex-math></inline-formula>-cyclotomic cosets of constacyclic codes with the form <inline-formula> <tex-math notation= LaTeX >$q=\\alpha m+t$ </tex-math></inline-formula> or <inline-formula> <tex-math notation= LaTeX >$q=\\alpha m+\\alpha -t$ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$n= (q^{2}+1/\\alpha)$ </tex-math></inline-formula> to construct some new families of entanglement-assisted quantum MDS codes that satisfy the entanglement-assisted quantum singleton bound, where <inline-formula> <tex-math notation= LaTeX >$q$ </tex-math></inline-formula> is an odd prime power and <inline-formula> <tex-math notation= LaTeX >$m$ </tex-math></inline-formula> is a positive integer, while both <inline-formula> <tex-math notation= LaTeX >$\\alpha $ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$t$ </tex-math></inline-formula> are positive integers such that <inline-formula> <tex-math notation= LaTeX >$\\alpha =t^{2}+1$ </tex-math></inline-formula>. The parameters of these codes constructed in this paper are more general compared with the ones in the literature. Moreover, the minimum distance of some codes in this paper is larger than <inline-formula> <tex-math notation= LaTeX >$(q/2)+1$ </tex-math></inline-formula> or <inline-formula> <tex-math notation= LaTeX >$q+1$ </tex-math></inline-formula>.