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

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.