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International Journal of Quantum Information

Some results on the linear codes over the finite ring F2+v1F2+⋯+vrF2

Abdullah Dertli, Yasemin Çengellenmiş, Şenol Eren

Published Feb 1, 2016• 19 citations
Publisher

Abstract

In this paper, we study the structure of cyclic, quasi-cyclic codes and their skew codes over the finite ring [Formula: see text], [Formula: see text] for [Formula: see text]. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasi-cyclic codes over [Formula: see text] are obtained. A necessary and sufficient condition for cyclic code over [Formula: see text] that contains its dual has been given. The parameters of quantum error correcting codes are obtained from cyclic codes over [Formula: see text].

DOI
10.1142/s021974991650012x

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In the paper, we consider a finite non-chain commutative ring [Formula: see text], where [Formula: see text]. We mainly study the construction of quantum codes from cyclic codes over [Formula: see text]. For this, we obtained self-orthogonal codes over [Formula: see text] as gray images of linear and cyclic codes over [Formula: see text], then these codes over [Formula: see text] correspond to a cyclic code over [Formula: see text] of odd length [Formula: see text] used to determine the parameters of the quantum codes.
FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes and their application to quantum codes
Let ppp be a prime and Fq\mathbb{F}_qFq​ be the finite field of order q=pmq=p^mq=pm. In this paper, we study FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes where R=Fq+uFq\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_qR=Fq​+uFq​ with u2=uu^2=uu2=u. To characterize FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes, we first establish their algebraic structure and then discuss the dual-containing properties by considering a non-degenerate inner product. Further, we define a Gray map over FqR\mathbb{F}_q\mathcal{R}Fq​R and obtain their Fq\mathbb{F}_qFq​-Gray images. As an application, we apply the CSS (Calderbank-Shor-Steane) construction on Gray images of dual containing FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes and obtain many quantum codes with better parameters than the best-known codes available in the literature.