Cyclic Codes over the Matrix Ring $M_2(\FFF_p)$ and Their Isometric Images over $\FFF_{p^2}+u\FFF_{p^2}$

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