Cyclic codes over F_2x(F_2+vF_2) and binary quantum codes

Abstract

In the present study, we define cyclic codes over the commutative principal ideal ring F_2 × (F_2 + vF_2) with v^2 = v and obtain some results on cyclic codes over F_2 × (F_2 + vF_2). We also investigate the dual of a cyclic code over F_2 × (F_2 + vF_2) depending on two inner products. We determine a generator polynomial of cyclic codes and give the calculation of the number of cyclic codes over F_2 × (F_2 + vF_2). Furthermore, we show that the Gray images of a cyclic code over F_2 × (F_2 + vF_2) of length n are binary quasi-cyclic codes of length 3n and of index 3. We find numerous binary codes as Gray images of cyclic codes over F_2 × (F_2 + vF_2) and tabulate the optimal ones. Moreover, we show that it is possible to obtain binary quantum error-correcting codes (QECCs) from cyclic codes over F_2 × (F_2 + vF_2).