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

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.