Construction of New Matrix-Product Codes and Their Applications

Hao Song,Jingjie Lv,Junli Wang,Ruihu Li

doi:10.1109/access.2019.2952193

Hao Song, Jingjie Lv + Show 2 more

Open Access

https://doi.org/10.1109/access.2019.2952193

Copy DOI

Abstract

In this paper, from special code chains $\mathcal {C}_{1} \supseteq \mathcal {C}_{2}~\supseteq \mathcal {C}_{3}$ such that $\mathcal {C}^{\bot _{h}}_{1}\subseteq \mathcal {C}_{3}$ and $\mathcal {C}^{\bot _{h}}_{2}\subseteq \mathcal {C}_{2}$ , some Hermitian dual-containing (HDC) matrix-product (MP) codes are presented, where $\mathcal {C}_{3}$ is not HDC. By studying some constacyclic codes of lengths $n=q^{2}\pm 1$ and $n=\frac {q^{2}-1}{2}$ , we construct many HDC MP codes of length $3n$ . Consequently, many $q$ -ary quantum codes with larger designed distance $d\geq q+1$ are obtained from these MP codes, where $4\leq q\leq 9$ .

Highlights

Matrix-product code was proposed first in [11] by Blackmore and Norton
Mankean and Jitman in [19], [20] discussed construction of Euclidean and Hermitian self-orthogonal MP codes, they gave some new codes of length 2n by employing some special matrices and code chains C1 ⊇ C2 of length n
In [22] they proved that entanglement-assisted quantum codes could be constructed via matrix-product codes

Summary

INTRODUCTION

Matrix-product code was proposed first in [11] by Blackmore and Norton. Soon afterwards it became more and more interesting, since it can be viewed as a generalization of the Plotkin’s (u|u + v)-construction etc., see [12]–[18]. In 2017, Liu et al [21] constructed quantum codes, whose minimum distances does not exceed q + 1, from matrix product codes. We will review relevant concepts on constacyclic codes and q2-cyclotomic cosets modulo rn. We can see that T is a union of some q2-cyclotomic cosets modulo rn and the dimension of C is n − |T | It is well-known that q2-ary constacyclic codes can be described by q2-cyclotomic cosets [10], [29], [30]. In order to construct QECCs, necessary and sufficient conditions for which HDC constacyclic codes exist were discussed, see [6]. VOLUME 7, 2019 using terminology of skew asymmetric cosets If C is a constacyclic code over Fq2 with defining set T , C⊥h ⊆ C if and only if one of the following holds:.

MATRIX-PRODUCT CODES

SOME HDC MP CODES AND RELATED QECCs

CONCLUSION