A study of reversible DNA cyclic codes over a non-chain ring based on the deletion distance

Abstract

Let $\mathcal{R}=\mathbb{Z}_8[u]/\langle u^2-4,2u\rangle$ be a non-chain  ring of characteristic $8$. In  this article,  DNA codes of odd lengths over the ring $\mathcal{R}$ based on the deletion distance are discussed. For this purpose, we study cyclic codes of  any odd length over the ring $\mathcal{R}$ satisfying the reversible and the reversible complement constraints. Also, a bijection $\vartheta$ between the elements of the ring $\mathcal{R}$ and $S_{D_{16}}$  is constructed in such a way that the reversibility problem is solved. Moreover, we  introduce a  homogeneous weight $w_{\hom}$ over the ring $\mathcal{R}$ and by utilizing $w_{\hom},$  a new Gray map $\theta_{\hom}:\mathcal{R}^{n}\rightarrow\mathbb{F}^{8n}_{2}$ is obtained. Furthermore, we study the $GC$-content of DNA codes and provide some examples of DNA codes with their respective deletion distance.