Reversible DNA codes over a family of non-chain rings $R_{k,s}$

Abstract

In this paper, we solve the reversibility problem for DNA codes  over the non-chain ring $R_{k,s}=\mathbb{F}_{4^{2k}}[u_1,\ldots,u_{s}]/\langle u_1^2-u_1,\ldots, u_s^2-u_s\rangle$. We define an automorphism $\theta$ over $R_{k,s}$ which help us both find the idempotent decomposition of $R_{k,s}$ and solve the reversibility problem via skew cyclic codes. Moreover, we introduce a  generalized Gray map that preserves DNA reversibility.