${\mathbb{F}}_q\mathcal{R}$-skew cyclic codes and their application to quantum codes

Abstract

Let $p$ be a prime and ${\mathbb{F}}_q$ be the finite field of order $q=p^m$. In this paper, we study ${\mathbb{F}}_q\mathcal{R}$-skew cyclic codes where $\mathcal{R}={\mathbb{F}}_q+u{\mathbb{F}}_q$ with $u^2=u$. To characterize ${\mathbb{F}}_q\mathcal{R}$-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 ${\mathbb{F}}_q\mathcal{R}$ and obtain their ${\mathbb{F}}_q$-Gray images. As an application, we apply the CSS (Calderbank-Shor-Steane) construction on Gray images of dual containing ${\mathbb{F}}_q\mathcal{R}$-skew cyclic codes and obtain many quantum codes with better parameters than the best-known codes available in the literature.

17 pages. This paper is under modification in the Quantum Information Processing