Quantum Codes Obtained from Skew ϱ - λ -Constacyclic Codes over R k

Abstract

Let ${\mathfrak{R}}_k={\mathbb{F}}_q\left[u_1,u_2,\cdots ,u_k\right]/<u_i^2={\alpha }_i{u}_i,u_i{u}_j=u_j{u}_i=0>$ , where $q=p^m$ , $p$ is an odd prime, ${\alpha }_i$ is a unit over ${\mathbb{F}}_q$ , and $i,j=\mathrm{1,2},\cdots ,k\text{.}$ In this article, we define a Gray map from ${\mathfrak{R}}_k^n$ to ${\mathbb{F}}_q^{\left(k+1\right)n}$ , we study the structure of skew $\varrho$ - $\lambda$ -constacyclic codes over ${\mathfrak{R}}_k$ , and then we give the necessary and sufficient conditions for skew $\varrho$ - $\lambda$ -constacyclic codes over ${\mathfrak{R}}_k$ to satisfy dual containing. Further, we have obtained some new nonbinary quantum codes from skew $\varrho$ - $\lambda$ -constacyclic over ${\mathfrak{R}}_k$ by using the CSS construction.