Cyclic codes over $M_{2}(\mathbb {F}_{2}+u\mathbb {F}_{2})$

Abstract

Let $A=M_{2}(\mathbb {F}_{2}+u\mathbb {F}_{2})$, where u ² = 0, the ring of 2 × 2 matrices over the finite ring $\mathbb {F}_{2}+u\mathbb {F}_{2}$. The ring A is a non-commutative Frobenius ring but not a chain ring. In this paper, we derive the structure theorem of cyclic codes of odd length over the ring A and use them to construct some optimal cyclic codes over $\mathbb {F}_{4}$. Let v ² = 0 and u v = v u. We also give an isometric map from A to $\mathbb {F}_{4}+v\mathbb {F}_{4}+u\mathbb {F}_{4}+uv\mathbb {F}_{4}$ using their respective Bachoc weight and Lee weight.