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 (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. INDEX TERMS Entanglement-assisted quantum codes, constacyclic codes, maximal-distance-separable (MDS) codes.

Some Progress on Quantum Error Correction for Discrete and Continuous Error Models

“…Another direction to construct entanglement-assisted quantum codes with good error correction abilities is to find entanglement-assisted quantum MDS codes that achieve the equality of entanglement-assisted quantum Singleton bound, i.e., K ≤ 2 n+c−2 (d−1) . Entanglement-assisted quantum MDS codes can be constructed directly from classical MDS codes [26], constacyclic codes [8] and so on. The up-to-date families of entanglement-assisted quantum MDS codes were summarized in [27].…”

“…It was then presented in [7] that the minimal number of qubits to protect one-qubit information against one arbitrary error is five. Various advanced quantum codes with lower error correction cost were further constructed in the 21st century, such as entanglement-assisted quantum codes [8] and nonadditive quantum codes [9].…”