Constacyclic codes over $F_q[u_{1}, u_{2}, \cdots , u_{k}]/\langle u_{i}^3-u_{i},u_{i}u_{j}-u_{j}u_{i}\rangle$ and their applications of constructing quantum codes

Abstract

In this paper, we study constacyclic codes over the ring $R_{k} = F_q[u_{1}, u_{2}, \cdots , u_{k}]/\langle u_{i}^3-u_{i}, u_{i}u_{j}-u_{j}u_{i}\rangle$, for all $i,j = 1, \ldots ,k$, $q = p^m$ for any odd prime p and positive integers m, k. We study the structure of the ring $R_{k}$ and define a Gray map by a matrix and decompose constacyclic codes over the ring $R_{k}$ as a direct sum of constacyclic codes over $F_q$. We give the relevant properties of constacyclic codes over the ring $R_{k}$ and give necessary and sufficient conditions for constacyclic codes to be dual-containing. Taking $R_{2}$ as an example, we give some corresponding conclusions. As an application, we construct some new quantum error-correcting codes over $F_{q}$ from constacyclic codes over $R_k$.