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

Let Fp be the prime field with p elements. We derive the homogeneous weight on the Frobenius matrix ring M2(Fp) in terms of the generating character. We also give a generalization of the Lee weight on the finite chain ring Fp2+uFp2 where u 2 = 0. A non-commutative ring, denoted by Fp2+vpFp2 , vp an involution in M2(Fp), that is isomorphic to M2(Fp) and is a left Fp2vector space, is constructed through a unital embedding τ from Fp2 to M2(Fp). The elements of Fp2 come from M2(Fp) such that τ(Fp2) = Fp2 . The irreducible polynomial f(x) = x2+x+(p−1) ∈ Fp[x] required in τ restricts our study of cyclic codes over M2(Fp) endowed with the Bachoc weight to the case p ≡ 2 or 3 mod 5. The images of these codes via a left Fp-module isometry are additive cyclic codes over Fp2 + uFp2 endowed with the Lee weight. New examples of such codes are given.