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In this paper, quantum codes from cyclic codes over A 2 ¼ F 2 þ uF 2 þ vF 2 þ uvF 2 ; u 2 ¼ u; v 2 ¼ v; uv ¼ vu, for arbitrary length n have been constructed. It is shown that if C is self orthogonal over A 2 ; then so is ÉðCÞ, where É is a Gray map. A necessary and su±cient condition for cyclic codes over A 2 that contains its dual has also been given. Finally, the parameters of quantum error correcting codes are obtained from cyclic codes over A 2 .

<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>

Abstract-The structures of the cyclic DNA codes of odd length over the finite rings R = Z 4 + wZ 4 , w 2 = 2 and S = Z 4 + wZ 4 + vZ 4 + wvZ 4 , w 2 = 2, v 2 = v, wv = vw are studied. The links between the elements of the rings R, S and 16 and 256 codons are established, respectively. The cyclic codes of odd length over the finite ring R satisfy reverse complement constraint and the cyclic codes of odd length over the finite ring S satisfy reverse constraint and reverse complement constraint are studied. The binary images of the cyclic DNA codes over the finite rings R and S are determined. Moreover, a family of DNA skew cyclic codes over R is constructed, its property of being reverse complement is studied.

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