Non-Binary Quantum Codes from Cyclic Codes over $\mathbb {F}_{p} \times (\mathbb {F}_{p}+v\mathbb {F}_{p})$

In this paper, we study cyclic codes over the ring $\mathbb {F}_{p} \times (\mathbb {F}_{p}+v\mathbb {F}_{p})$ , where p is an odd prime and v2 = v. We first investigate the properties of the ring $\mathbb {F}_{p} \times (\mathbb {F}_{p}+v\mathbb {F}_{p})$ and the linear codes over this ring. We also define a distance-preserving Gray map from $\mathbb {F}_{p} \times (\mathbb {F}_{p}+v\mathbb {F}_{p})$ to $\mathbb {F}_{p}^{3}$ . We discuss cyclic codes and their dual codes over the ring. Also, we define a set of generators for these codes. As an implementation, we show that quantum error-correcting codes can be obtained from dual containing cyclic codes over the ring by using the Calderbank-Shor-Steane (CSS) construction. Furthermore, we give some illustrative examples. Finally, we tabulate the non-binary quantum error-correcting codes obtained from cyclic codes over the ring.