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

Jianzhang Chen,Chunhui Feng,Yuanyuan Huang,Riqing Chen,Youqin Chen

doi:10.1109/access.2019.2927294

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 (q/2) + 1. 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 (q/2) + 1 or exceeds (q/2) + 1. Meanwhile, how to determine the required number of maximally entangled states to make the minimum distance of quantum MDS codes larger than (q/2) + 1 is an interesting problem in the quantum coding theory. In this paper, we utilize the decomposition of the defining set and q 2 -cyclotomic cosets of constacyclic codes with the form q = αm + t or q = αm + α - t and n = (q 2 + 1/α) to construct some new families of entanglement-assisted quantum MDS codes that satisfy the entanglement-assisted quantum singleton bound, where q is an odd prime power and m is a positive integer, while both α and t are positive integers such that α = t 2 + 1. 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 (q/2) + 1 or q + 1.

Highlights

In the quantum information and quantum computing, an important subject is to constuct some good quantum error-correcting codes [3], [5], [7],[8], [15], [18], [30], [31], [35]–[37], [42]
We utilize the decomposition of the defining set and q2-cyclotomic cosets of constacyclic codes with the form q = αm + t or q = αm + α − t and n = (q2 + 1/α) to construct some new families of entanglement-assisted quantum MDS codes that satisfy the entanglement-assisted quantum singleton bound, where q is an odd prime power and m is a positive integer, while both α and t are positive integers such that α = t2 + 1
Based on the results of [22], [25], we proposed a decomposition of the defining set of negacyclic codes and utilized this method to construct some families of entanglement-assisted quantum MDS codes with different lengths in [6]

Summary

INTRODUCTION

In the quantum information and quantum computing, an important subject is to constuct some good quantum error-correcting codes (quantum codes for short) [3], [5], [7],. Based on the results of [22], [25], we proposed a decomposition of the defining set of negacyclic codes and utilized this method to construct some families of entanglement-assisted quantum MDS codes with different lengths in [6]. The method that is the decomposition of the defining set of constacyclic codes with length q2+1 α is used to determine the number of pre-shared maximally entangled states, and to construct some new families of entanglement-assisted quantum MDS codes with length. We can use the same method of the decomposition of the defining set of constacyclic codes to obtain other entanglement-assisted quantum MDS codes with the number of pre-shared maximally entangled states that exceeds 9 in the Hermitian construction.

PREMILINARIES

AND DISCUSSION