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International Journal of Information and Coding Theory

Constacyclic codes over the ring Fp + vFp and their applications of constructing new non-binary quantum codes

Jian Hui Gao, Yongkang Wang

Published Jan 1, 2018• 2 citations
Publisher

Abstract

In this paper, we study some results on constacyclic codes over Fp + vFp, where p is an odd prime and v2 = v. Under some special Gray map, the image of the (α + βv)-constacyclic code over Fp + vFp is a (–α(α + β))-constacyclic code over Fp, where α and β are non-zero elements of Fp satisfying β = –2α. As an application, some new non-binary quantum codes are obtained.

DOI
10.1504/ijicot.2018.10016416

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In this paper, we study some results on constacyclic codes over Fp + vFp, where p is an odd prime and v2 = v. Under some special Gray map, the image of the (α + βv)-constacyclic code over Fp + vFp is a (–α(α + β))-constacyclic code over Fp, where α and β are non-zero elements of Fp satisfying β = –2α. As an application, some new non-binary quantum codes are obtained.
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In this paper, we study the structural properties of ( α + u 1 β + u 2 γ + u 1 u 2 δ ) -constacyclic codes over R = F q [ u 1 , u 2 ] / ⟨ u 1 2 − u 1 , u 2 2 − u 2 , u 1 u 2 − u 2 u 1 ⟩ where q = p m for odd prime p and m ≥ 1 . We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4 n . Also we have classified all possible self dual linear codes over this ring R . We have given the applications by computing non-binary quantum codes over this ring R .
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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​.
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.