On the structure of cyclic codes over M₂$({\mathbb {F}}_{\it{p}}$ + u${\mathbb {F}}_{\it{p}})$

Abstract

In this article, for a prime p such that $p\equiv 2$ or $3 \pmod {5}$, we identify cyclic codes of length N over $R=M_{2}({\mathbb {F}}_{p}+u{\mathbb {F}}_{p})$, $u^2=0$ as right R-submodules of $R[x]/\langle x^N-1\rangle$. Also, we define an isometry from $M_{2}({\mathbb {F}}_{p}+u{\mathbb {F}}_{p})$ to ${\mathbb {F}}_{p^2}+u{\mathbb {F}}_{p^2}+v{\mathbb {F}}_{p^2}+uv{\mathbb {F}}_{p^2}$, where $u^2=v^2=0,uv=vu$ and determine the structure of cyclic codes, in particular self-dual cyclic codes of length N where $\gcd (N,p)=1$. Moreover, several optimal and near to optimal codes are obtained as the Gray images of these codes over R.