Cyclic Codes over the Matrix Ring $M_2(F_p)$ and Their Isometric Images over $F_{p^2}+uF_{p^2}$

Abstract

Let $F_p$ be the prime field with $p$ elements. We derive the homogeneous weight on the Frobenius matrix ring $M_2(F_p)$ in terms of the generating character. We also give a generalization of the Lee weight on the finite chain ring $F_{p^2}+uF_{p^2}$ where $u^2=0$. A non-commutative ring, denoted by $\mathcal{F}_{p^2}+\mathbf{v}_p \mathcal{F}_{p^2}$, $\mathbf{v}_p$ an involution in $M_2(F_p)$, that is isomorphic to $M_2(F_p)$ and is a left $F_{p^2}$-vector space, is constructed through a unital embedding $\tau$ from $F_{p^2}$ to $M_2(F_p)$. The elements of $\mathcal{F}_{p^2}$ come from $M_2(F_p)$ such that $\tau(F_{p^2})=\mathcal{F}_{p^2}$. The irreducible polynomial $f(x)=x^2+x+(p-1) \in F_p[x]$ required in $\tau$ restricts our study of cyclic codes over $M_2(F_p)$ endowed with the Bachoc weight to the case $p\equiv$ $2$ or $3$ mod $5$. The images of these codes via a left $F_p$-module isometry are additive cyclic codes over $F_{p^2}+uF_{p^2}$ endowed with the Lee weight. New examples of such codes are given.