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Procedia Computer Science

New Quantum Codes From Matrix-Product Codes

Hao Song, Luo Bin Guo, Liang Dong Lv, Gang Chen

Published Jan 1, 2019• 2 citations
Publisher

Abstract

In this paper, the construction of matrix-product codes with Euclidean dual-containing are studied. The critical issue is to determine the parameters of the matrix-product code and its dual code. Therefore, the necessary conditions for which the matrix-product code is Euclidean dual-containing are given. Consequently, new non-binary quantum codes are derived from those matrix-product codes by q-ary Steane’s construction, and some examples are given for q =4,5.

DOI
10.1016/j.procs.2019.06.107

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In this paper, the construction of matrix-product codes with Euclidean dual-containing are studied. The critical issue is to determine the parameters of the matrix-product code and its dual code. Therefore, the necessary conditions for which the matrix-product code is Euclidean dual-containing are given. Consequently, new non-binary quantum codes are derived from those matrix-product codes by q-ary Steane’s construction, and some examples are given for q =4,5.
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good qqq-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and qqq-ary quantum codes derived from these matrix-product codes.
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good qqq-ary quantum codes of large length. Specifically, it follows from CSS construction (resp. Hermitian construction) that a matrix-product code over Fq\mathbb{F}_{q}Fq​ (resp. Fq2\mathbb{F}_{q^{2}}Fq2​) which is Euclidean dual-containing (resp. Hermitian dual-containing) can produce a qqq-ary quantum code. In order to obtain such matrix-product codes, a common way is to construct quasi-orthogonal matrices (resp. quasi-unitary matrices) as the defining matrices of matrix-product codes over Fq\mathbb{F}_{q}Fq​ (resp. Fq2\mathbb{F}_{q^{2}}Fq2​). The usage of NSC quasi-orthogonal matrices or NSC quasi-unitary matrices in this process enables the minimum distance lower bound of the corresponding quantum codes to reach its optimum. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in \cite{Cao2020Constructioncaowang,Cao2020New,Cao2020Constructionof}; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and qqq-ary quantum codes derived from these newly constructed matrix-product codes.
In this paper, from special code chainsC1⊇C2 ⊇C3\mathcal {C}_{1} \supseteq \mathcal {C}_{2}~\supseteq \mathcal {C}_{3}C1​⊇C2​ ⊇C3​such thatC1⊥h⊆C3\mathcal {C}^{\bot _{h}}_{1}\subseteq \mathcal {C}_{3}C1⊥h​​⊆C3​andC2⊥h⊆C2\mathcal {C}^{\bot _{h}}_{2}\subseteq \mathcal {C}_{2}C2⊥h​​⊆C2​, some Hermitian dual-containing (HDC) matrix-product (MP) codes are presented, whereC3\mathcal {C}_{3}C3​is not HDC. By studying some constacyclic codes of lengthsn=q2±1n=q^{2}\pm 1n=q2±1andn=q2−12n=\frac {q^{2}-1}{2}n=2q2−1​, we construct many HDC MP codes of length3n3n3n. Consequently, manyqqq-ary quantum codes with larger designed distanced≥q+1d\geq q+1d≥q+1are obtained from these MP codes, where4≤q≤94\leq q\leq 94≤q≤9.
For an odd prime p and m ≥ 1, let R = Fpm[v]/{v3} = v}. Here our aim is to construct new non-binary quantum codes from skew constacyclic codes over this non-chain ring ℛ. In order to achieve our goal, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring ℛ. Then we determine a necessary and sufficient condition for these codes to contain their Euclidean duals. Finally, by applying CSS construction on these dual containing skew constacyclic codes and using Gray map, we obtain several new quantum codes.
Quantum maximum-distance-separable (MDS) codes form an important class of quantum codes. To get q-ary quantum MDS codes, one of the effective ways is to find linear MDS codes C over Fq2satisfying C⊥H⊆ C, where C⊥Hdenotes the Hermitian dual code of C. For a linear code C of length n over Fq2, we say that C is a dual-containing code if C⊥H⊆ C and C≠ Fq2n. Several classes of new quantum MDS codes with relatively large minimum distance have been produced through dual-containing constacyclic MDS codes. These works motivate us to make a careful study on the existence conditions for dual-containing constacyclic codes. We obtain necessary and sufficient conditions for the existence of dual-containing constacyclic codes. Four classes of dual-containing constacyclic MDS codes are constructed and their parameters are computed. Consequently, the quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have minimum distance bigger than the ones available in the literature.
Three code constructions generating new families of good nonbinary quantum codes are presented in this paper. The first two ones are derived from Hermitian self-orthogonal non-narrow-sense Bose-Chaudhuri-Hocquenghem (BCH) codes. The third one is derived from qqq-ary ($q\ensuremath{\ne}2$ is a prime power) Steane's enlargement of Calderbank-Shor-Steane codes applied to Euclidean self-orthogonal non-narrow-sense BCH codes. The quantum nonbinary BCH codes presented here have parameters better than the ones available in the literature.