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International Conference on Advanced Communication Technologies and Networking (CommNet)

Construction of some new quantum codes

Mohammed Sabiri

Published Dec 12, 2022• 4 citations
Publisher

Abstract

In this paper, we consider$R=\mathbb{F}_{q}[u, v]/ < u^{2}-u, v^{2}-v, uv - vu >$which is a finite non-chain ring, where$q$is an odd prime power and$u^{2}=u, v^{2}=v, uv=vu$. We construct new quantum codes from$(\alpha+\beta u+\gamma v+\delta uv)$- constacyclic codes over$R$. We give the structure of$(\alpha+\beta u+\gamma v+\delta uv)$- constacyclic codes over$R$and obtain self-orthogonal codes over$\mathbb{F}_{q}$by Gray map.

DOI
10.1109/commnet56067.2022.9993864

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Let $ R = \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $ be a finite non-chain ring, where $ q $ is an odd prime power and $ u^{2} = 1 $, $ v^2 = v $, $ uv = vu .Inthispaper,weconstructnewnon−binaryquantumcodesfrom(. In this paper, we construct new non-binary quantum codes from (.Inthispaper,weconstructnewnon−binaryquantumcodesfrom( \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R .Wegivethestructureof(. We give the structure of (.Wegivethestructureof( \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $ and obtain self-orthogonal codes over $ \mathbb{F}_q $ by Gray map. By using Calderbank-Shor-Steane (CSS) construction and Hermitian construction from dual-containing ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $, some new non-binary quantum codes are obtained.
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New non-binary quantum codes from skew constacyclic codes over the ring Fpm+vFpm+v2Fpm\mathbb{F}_{p^m}+v\mathbb{F}_{p^m}+v^2 \mathbb{F}_{p^m}Fpm​+vFpm​+v2Fpm​
In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring R=Fpm[v]/⟨v3=v⟩\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangleR=Fpm​[v]/⟨v3=v⟩ where ppp is an odd prime and m≥1m \geq 1m≥1. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring R\mathcal{R}R. Then a necessary and sufficient condition for skew constacyclic codes over R\mathcal{R}R to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over Fpm\mathbb{F}_{p^m}Fpm​.
This paper focuses on studying the properties of constacyclic codes, quantum error-correcting codes. The code is studied over a specific mathematical structure called the ring S\mathfrak{S}S, which is defined as S=Iq[u,v]/⟨u2−α2, v2−α2, uv−vu⟩\mathfrak{S}=\mathfrak{I}_q[\mathfrak{u},\mathfrak{v}]/\langle \mathfrak{u}^2-\alpha^2,~ \mathfrak{v}^2-\alpha^2,~\mathfrak{u}\mathfrak{v}-\mathfrak{v}\mathfrak{u} \rangleS=Iq​[u,v]/⟨u2−α2, v2−α2, uv−vu⟩, where Iq\mathfrak{I}_qIq​ is a finite field of qqq elements, α\alphaα be the nonzero elements of the field Iq\mathfrak{I}_qIq​ and qqq is a power of an odd prime ppp such that q=pm, for m≥1q=p^m, ~\textup{for}~ m \ge 1q=pm, for m≥1. The paper also introduces a Gray map and use it to decompose constacyclic codes over the ring S\mathfrak{S}S into a direct sum of constacyclic codes over Iq\mathfrak{I}_qIq​. We construct new and better quantum error-correcting codes over the ring S\mathfrak{S}S (cf.; Table 1 and Table 2). Moreover, we also obtain best known linear codes as well as best dimension linear codes (cf.; Table 4).
In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring R=Fpm[v]/⟨v3=v⟩\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangleR=Fpm​[v]/⟨v3=v⟩ where ppp is an odd prime and m≥1m \geq 1m≥1. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring R\mathcal{R}R. Then a necessary and sufficient condition for skew constacyclic codes over R\mathcal{R}R to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over Fpm\mathbb{F}_{p^m}Fpm​.