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Advances in Mathematics of Communications

New non-binary quantum codes from constacyclic codes over $ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $

Fanghui Ma, Jian Gao, Fang‐Wei Fu

Published Jan 1, 2019• 46 citations
Publisher

Abstract

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 $. In this paper, we construct new non-binary quantum codes from ($ \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. 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.

DOI
10.3934/amc.2019027

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Let R=Fq[u,v]/⟨u2−1,v2−v,uv−vu⟩R = \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangleR=Fq​[u,v]/⟨u2−1,v2−v,uv−vu⟩ be a finite non-chain ring, where qqq is an odd prime power and u2=1u^{2} = 1u2=1, v2=vv^2 = vv2=v, uv=vuuv = vuuv=vu. In this paper, we construct new non-binary quantum codes from (α+βu+γv+δuv\alpha+\beta u+\gamma v+\delta uvα+βu+γv+δuv)-constacyclic codes over RRR. We give the structure of (α+βu+γv+δuv\alpha+\beta u+\gamma v+\delta uvα+βu+γv+δuv)-constacyclic codes over RRR and obtain self-orthogonal codes over Fq\mathbb{F}_qFq​ by Gray map. By using Calderbank-Shor-Steane (CSS) construction and Hermitian construction from dual-containing (α+βu+γv+δuv\alpha+\beta u+\gamma v+\delta uvα+βu+γv+δuv)-constacyclic codes over RRR, some new non-binary quantum codes are obtained.
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​.
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​.
In this paper, we considerR=Fq[u,v]/<u2−u,v2−v,uv−vu>R=\mathbb{F}_{q}[u, v]/ < u^{2}-u, v^{2}-v, uv - vu >R=Fq​[u,v]/<u2−u,v2−v,uv−vu>which is a finite non-chain ring, whereqqqis an odd prime power andu2=u,v2=v,uv=vuu^{2}=u, v^{2}=v, uv=vuu2=u,v2=v,uv=vu. We construct new quantum codes from(α+βu+γv+δuv)(\alpha+\beta u+\gamma v+\delta uv)(α+βu+γv+δuv)- constacyclic codes overRRR. We give the structure of(α+βu+γv+δuv)(\alpha+\beta u+\gamma v+\delta uv)(α+βu+γv+δuv)- constacyclic codes overRRRand obtain self-orthogonal codes overFq\mathbb{F}_{q}Fq​by Gray map.
For an odd prime $ p $ and positive integers $ m $ and $ \ell $, let $ \mathbb{F}_{p^m} $ be the finite field with $ p^{m} $ elements and $ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $. Thus $ R_{\ell,m} $ is a finite commutative non-chain ring of order $ p^{2^{\ell} m} $ with characteristic $ p $. In this paper, we aim to construct quantum codes from skew constacyclic codes over $ R_{\ell,m} $. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.
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.
Let R= Fq+uFq+vFq+uvFq be a commutative ring with u2=u, v2=v, uv=vu, where q is a power of an odd prime. Ashraf and Mohammad constructed some new quantum codes from cyclic codes. Under this background, another Gray map from R to Fq4 is given. This map can be naturally extended to Rn. The problem on the ring turns to the field by this isomorphic map now. Therefore, This mapping is obviously a weight-preserving and distance-preserving map. The results show that the codes after mapping are self-orthogonal codes over Fq if they are self-orthogonal codes over R. Some computational examples show that some better non-binary quantum codes can be obtained under this Gray map. We discuss the structure of linear codes. On this basis, the structure of the generating matrix of linear codes is obtained. The structure of their dual codes is also obtained. The CSS construction guarantees the existence of quantum codes. Finally, with the help of the CSS construction, we get some good quantum codes. By comparison, our quantum codes have better parameters.