Construction of $(σ,δ)$-cyclic codes over a non-chain ring and their applications in DNA codes

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.