Skew generalized cyclic codes with derivations: Construction and DNA applications

This paper discusses the explicit form of the generators of skew cyclic codes of an arbitrary length with nontrivial automorphisms and derivations over a nonchain ring $ \mathcal{R} = \mathbb{Z}_4+\nu \mathbb{Z}_4+\omega \mathbb{Z}_4, $ where $ \nu^2 = \omega^2 = \nu \omega = \omega \nu = 0 $. We address the reversibility problem for these codes by presenting a novel approach to resolve it and establish reverse and reverse-complement constraints for these codes, ensuring their biological feasibility. In addition, we propose an unconventional way to construct DNA codes by extending reversible codes in a specific manner and obtain several DNA codes using this technique. In comparison to the known codes, we provide some new classical and DNA codes by employing various Gray maps under different automorphisms and derivations, which have better parameters.