On DNA codes from a family of chain rings

In this work, we focus on reversible cyclic codes which correspond to reversible DNA codes or reversible-complement DNA codes over a family of finite chain rings, in an effort to extend what was done by Yildiz and Siap in [20]. The ring family that we have considered are of size $2^{2^k}$, $k=1,2, \cdots$ and we match each ring element with a DNA $2^{k-1}$-mer. We use the so-called $u^2$-adic digit system to solve the reversibility problem and we characterize cyclic codes that correspond to reversible-complement DNA-codes. We then conclude our study with some examples

Reversible codes and applications to DNA codes over F42t[u]/(u2−1) F_{4^{2t}}[u]/(u^2-1)

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 $x^n-1$