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For a prime $p$ and a positive integer $m$, let $\mathbb{F}_{p^m}$ be thefinite 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 findingDNA codes. Towards this, we first define a Gray map to find classical codesover $\mathbb{F}_{p^m}$ using codes over the ring $\mathfrak{R}_l$. Later, wefind the conditions for a code to be reversible and a DNA code using $(\sigma,\delta)$-cyclic code. Finally, this algebraic method provides many classicaland DNA codes of better parameters.

Construction of $(\sigma,\delta)$-cyclic codes over a non-chain ring and their applications in DNA codes