For a prime $p$ and a positive integer $m$, let $\mathbb{F}_{p^m}$ be the finite field of characteristic $p$, and $\mathfrak{R}_l:=\mathbb{F}_{p^m}[v]/\langle v^l-v\rangle$ be a non-chain ring. In this paper, we study the $(\sigma,\delta)$-cyclic codes over $\mathfrak{R}_l$. Further, we study the application of these codes in finding DNA codes. Towards this, we first define a Gray map to find classical codes over $\mathbb{F}_{p^m}$ using codes over the ring $\mathfrak{R}_l$. Later, we find the conditions for a code to be reversible and a DNA code using $(\sigma, \delta)$-cyclic code. Finally, this algebraic method provides many classical and DNA codes of better parameters.

翻译：设$p$为素数，$m$为正整数，令$\mathbb{F}_{p^m}$表示特征为$p$的有限域，$\mathfrak{R}_l:=\mathbb{F}_{p^m}[v]/\langle v^l-v\rangle$为某非链环。本文研究$\mathfrak{R}_l$上的$(\sigma,\delta)$-循环码，并进一步探讨此类码在DNA编码中的应用。为此，首先定义一种Gray映射，利用$\mathfrak{R}_l$上的码构造$\mathbb{F}_{p^m}$上的经典码；随后，通过$(\sigma,\delta)$-循环码导出码字可逆及成为DNA码的条件。最终，该代数方法生成了大量参数更优的经典码与DNA码。