Reversible codes and applications to DNA codes over $ F_{4^{2t}}[u]/(u^2-1) $

Abstract

<abstract><p>Let $ n \geq 1 $ be a fixed integer. Within this study, we present a novel approach for discovering reversible codes over rings, leveraging the concept of $ r $-glifted polynomials. This technique allows us to achieve optimal reversible codes. As we extend our methodology to the domain of DNA codes, we establish a correspondence between $ 4t $-bases of DNA and elements within the ring $ R_{2t} = F_{4^{2t}}[u]/(u^{2}-1) $. By employing a variant of $ r $-glifted polynomials, we successfully address the challenges of reversibility and complementarity in DNA codes over this specific ring. Moreover, we are able to generate reversible and reversible-complement DNA codes that transcend the limitations of being linear cyclic codes generated by a factor of $ x^n-1 $.</p></abstract>