A study of FqR-cyclic codes and their applications in constructing quantum codes

Let $R=\mathbb F_{q}+u\mathbb F_{q}+v\mathbb F_{q}+uv\mathbb F_{q}$ , with $u^{2}=u,v^{2}=v,uv=vu$ , where $q=p^{m}$ for a positive integer $m$ and an odd prime $p$ . We study the algebraic structure of $\mathbb F_{q}R$ -cyclic codes of block length $(r,s)$ . These codes can be viewed as $R[x]$ -submodules of $\mathbb F_{q}[x]/\langle x^{r}-1\rangle \times R[x]/\langle x^{s}-1\rangle $ . For this family of codes we discuss the generator polynomials and minimal generating sets. We study the algebraic structure of separable codes. Further, we discuss the duality of this family of codes and determine their generator polynomials. We obtain several optimal and near-optimal codes from this study. As applications, we discuss a construction of quantum error-correcting codes (QECCs) from $\mathbb F_{q}R$ -cyclic codes and construct some good QECCs.