Reversible $ G $-codes over the ring $ {\mathcal{F}}_{j,k} $ with applications to DNA codes

Abstract

<p style='text-indent:20px;'>In this paper, we show that one can construct a <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula>-codes are reversible of index <inline-formula><tex-math id="M5">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>. Additionally, we introduce a new family of rings, <inline-formula><tex-math id="M6">\begin{document}$ {\mathcal{F}}_{j,k} $\end{document}</tex-math></inline-formula>, whose base is the finite field of order <inline-formula><tex-math id="M7">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> and study reversible <inline-formula><tex-math id="M8">\begin{document}$ G $\end{document}</tex-math></inline-formula>-codes over this family of rings. Moreover, we present some possible applications of reversible <inline-formula><tex-math id="M9">\begin{document}$ G $\end{document}</tex-math></inline-formula>-codes over <inline-formula><tex-math id="M10">\begin{document}$ {\mathcal{F}}_{j,k} $\end{document}</tex-math></inline-formula> to reversible DNA codes. We construct many reversible <inline-formula><tex-math id="M11">\begin{document}$ G $\end{document}</tex-math></inline-formula>-codes over <inline-formula><tex-math id="M12">\begin{document}$ {\mathbb{F}}_4 $\end{document}</tex-math></inline-formula> of which some are optimal. These codes can be used to obtain reversible DNA codes.</p>