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

Let $n\ge 1$ 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 $x^n-1$.