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.

翻译：令$p$为素数，$\mathbb{F}_q$为阶为$q=p^m$的有限域。本文研究$\mathbb{F}_q\mathcal{R}$-斜循环码，其中$\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_q$且$u^2=u$。为刻画$\mathbb{F}_q\mathcal{R}$-斜循环码，我们首先建立其代数结构，进而通过考虑非退化内积讨论其对偶包含性质。进一步地，我们在$\mathbb{F}_q\mathcal{R}$上定义Gray映射并获得其$\mathbb{F}_q$-Gray像。作为应用，我们将CSS（Calderbank-Shor-Steane）构造应用于包含其对偶的$\mathbb{F}_q\mathcal{R}$-斜循环码的Gray像，得到了一批参数优于已知最优码的量子码。