The DNA codes from $(\text{\textbaro}, \mathfrak{d}, \gamma)$-constacyclic codes over $\mathbb{Z}_4+\omega\mathbb{Z}_4$

DNA Codes from Skew-Constacyclic Codes over $\mathbb{Z}_4$

Abstract

The paper introduces a novel technique for constructing DNA codes from skew cyclic codes over a non-chain extension of $\mathbb{Z}_4.$ We discuss $(\text{\textbaro },\mathfrak{d}, \gamma)$-constacyclic codes over the ring $\textfrak{R}=\mathbb{Z}_4+\omega\mathbb{Z}_4, \omega^2=\omega,$ with an $\textfrak{R}$-automorphism $\text{\textbaro }$ and a $\text{\textbaro }$-derivation $\mathfrak{d}$ over $\textfrak{R}$. By determining the generators of these codes of any arbitrary length over $\R$, we propose a construction on the $(\text{\textbaro },\mathfrak{d},\gamma)$-constacyclic codes to generate additional classical codes with improved and new parameters. Further, we demonstrate the reversibility of these codes and investigate the necessary and sufficient criterion to derive reverse-complement codes. Moreover, we present another construction to generate DNA codes from these reversible codes. Finally, we provide numerous $(\text{\textbaro },\mathfrak{d}, \gamma)$ constacyclic codes and applying the established results, we construct reversible and DNA codes. The parameters of these linear codes over $\mathbb{Z}_4$ are better and optimal according to the available online database of the $\mathbb{Z}_4$ codes.