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

This work introduces a novel approach to constructing DNA codes from linear codes over a non-chain extension of ${\mathbb{Z}}_4$. We study $(\text{textbaro},\mathfrak{d},\gamma )$-constacyclic codes over the ring $\mathfrak{R}={\mathbb{Z}}_4+\omega {\mathbb{Z}}_4,{\omega }^2=\omega ,$ with an $\mathfrak{R}$-automorphism $\text{textbaro}$ and a $\text{textbaro}$-derivation $\mathfrak{d}$ over $\mathfrak{R}\mathrm{.}$ Further, we determine the generators of the $(\text{textbaro},\mathfrak{d},\gamma )$-constacyclic codes over the ring $\mathfrak{R}$ of any arbitrary length and establish the reverse constraint for these codes. Besides the necessary and sufficient criterion to derive reverse-complement codes, we present a construction to obtain DNA codes from these reversible codes. Moreover, we use another construction on the $(\text{textbaro},\mathfrak{d},\gamma )$-constacyclic codes to generate additional optimal and new classical codes. Finally, we provide several examples of $(\text{textbaro},\mathfrak{d},\gamma )$ constacyclic codes and construct DNA codes from established results. The parameters of these linear codes over ${\mathbb{Z}}_4$ are better and optimal according to the codes available at \cite{z4codes}.