On reversible codes over a non-chain ring

Abstract

Suppose $\mathbb {F}_{q}$ is a finite field with q elements and $q=p^{t},$ where p is a prime and $t\ge 1.$ Let $\mathfrak {R}_{q}={\mathbb {F}}_q+u_{1}{\mathbb {F}}_q+u_{2}\mathbb F_q+u_{1}u_{2}{\mathbb {F}}_q$, where $u_{1}^2=0$, $u_{2}^2=0,$ $u_{1}u_{2}=u_{2}u_{1}$ be a non-chain ring. A necessary and sufficient condition for a given cyclic code of length n over the ring $\mathfrak {R}_{q},$ where $(n, p)=1$ to be a reversible cyclic code is obtained and using this condition the reversible cyclic codes of length n over the ring $\mathfrak {R}_{q}$ are constructed. In addition, we determine the set of generators for the dual code of a cyclic code and obtain the reversibility condition for Moreover, we discuss reversible complement codes over the ring $\mathfrak {R}_{4}={\mathbb {F}}_4+u_{1}\mathbb F_4+u_{2}{\mathbb {F}}_4+u_{1}u_{2}{\mathbb {F}}_4,$ which are useful in DNA coding. Further, the reversibility problem that occur in DNA k-bases is solved using a bijection $\vartheta$ between the elements of the ring $\mathfrak {R}_{4}=\mathbb F_4+u_{1}{\mathbb {F}}_4+u_{2}{\mathbb {F}}_4+u_{1}u_{2}{\mathbb {F}}_4$ and $S_{D_{256}}$.