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

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 $F_{p^2}+u{F}_{p^2}$ 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 $F_{p^2}+u{F}_{p^2}$ endowed with the Lee weight. New examples of such codes are given.

arXiv comment14 pages, presented at the International Zurich Seminar on Communications, Zurich, Switzerland, February 2014

References in corpus (1)

• Cyclic codes over $M_2(\F_2)$