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 $\mathfrak{R}.$ 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}.

翻译：本文提出了一种从 $\mathbb{Z}_4$ 的非链扩张上的线性码构造 DNA 码的新方法。我们研究了环 $\mathfrak{R}=\mathbb{Z}_4+\omega\mathbb{Z}_4, \omega^2=\omega,$ 上的 $(\text{\textbaro},\mathfrak{d}, \gamma)$-常循环码，其中 $\text{\textbaro}$ 是 $\mathfrak{R}$ 的一个自同构，$\mathfrak{d}$ 是 $\mathfrak{R}$ 上的一个 $\text{\textbaro}$-导子。进一步，我们确定了任意长度下环 $\mathfrak{R}$ 上 $(\text{\textbaro},\mathfrak{d}, \gamma)$-常循环码的生成元，并为这些码建立了反向约束条件。除了推导反向互补码的充要条件外，我们还提出了一种从这些可逆码获得 DNA 码的构造方法。此外，我们利用 $(\text{\textbaro},\mathfrak{d},\gamma)$-常循环码的另一种构造来生成更多新的、最优的经典码。最后，我们给出了几个 $(\text{\textbaro},\mathfrak{d}, \gamma)$-常循环码的例子，并根据已建立的结果构造了 DNA 码。根据 \cite{z4codes} 中可用的码进行比较，这些在 $\mathbb{Z}_4$ 上的线性码的参数更优且达到最优。