New Non-Binary Quantum Codes from Cyclic Codes Over Product Rings

Abstract

For any odd prime $p$ , and a divisor $\ell $ of $p$ , we consider $I_\ell $ to be the set of all divisors of $p-1$ , which are less than or equal to $\ell $ . In this letter, we construct quantum codes from cyclic codes over $\mathbb F_{p}$ and $\mathbb F_{p} S_\ell $ , where $S_\ell =\prod _{i\in I_\ell }R_{i}$ , for $R_{i}=\frac {\mathbb F_{p}[u]}{\langle u^{i+1}-u\rangle }$ . For that, first we construct linear codes and a Gray map over $R_\ell $ . Using this construction, we study cyclic codes over $R_\ell $ , and then extend that over $\mathbb F_{p} S_\ell $ . We also give a Gray map over $\mathbb F_{p} S_\ell $ . Then, using necessary and sufficient condition of dual containing property for cyclic codes, we construct quantum MDS codes. It is observed that the quantum codes constructed are new in the literature.