On skew cyclic codes over a mixed alphabet and their applications to DNA codes

In this paper, we introduce skew cyclic codes over the mixed alphabet [Formula: see text], where [Formula: see text] is the finite field with 4 elements and [Formula: see text]. Our results include a description of the generator polynomials of such codes and a necessary and sufficient condition for an [Formula: see text]-skew cyclic code to be reversible complement.

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

<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>

Skew cyclic codes over 𝔽4R

This paper considers a new alphabet set, which is a ring that we call [Formula: see text], to construct linear error-control codes. Skew cyclic codes over this ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize [Formula: see text]-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over [Formula: see text] are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of [Formula: see text]-skew cyclic codes which are reversible complement.