$\mathbb{F}_q\mathcal{R}$-skew cyclic codes and their application to quantum codes
(2305.10404)Abstract
Let $p$ be a prime and $\mathbb{F}q$ be the finite field of order $q=pm$. In this paper, we study $\mathbb{F}q\mathcal{R}$-skew cyclic codes where $\mathcal{R}=\mathbb{F}q+u\mathbb{F}q$ with $u2=u$. 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.
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