On reversible DNA codes over the ring \({\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle\) based on the deletion distance

Let \({\mathfrak {R}}= {\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle\) $\mathfrak{R}={\mathbb{Z}}_4[u,v]/⟨u^2-2,uv-2,v^2,2u,2v⟩$ be a ring, where \({\mathbb {Z}}_{4}\) ${\mathbb{Z}}_4$ is a ring of integers modulo 4. This ring \({\mathfrak {R}}\) $\mathfrak{R}$ is a local non-chain ring of characteristic 4. The main objective of this article is to construct reversible cyclic codes of odd length n over the ring \({\mathfrak {R}}.\) $\mathfrak{R}.$ Employing these reversible cyclic codes, we obtain reversible cyclic DNA codes of length n,  based on the deletion distance over the ring \({\mathfrak {R}}.\) $\mathfrak{R}.$ We also construct a bijection \(\Gamma\) $\mathrm{\Gamma }$ between the elements of the ring \({\mathfrak {R}}\) $\mathfrak{R}$ and \(S_{D_{16}}.\) $S_{D_{16}}.$ As an application of \(\Gamma ,\) $\mathrm{\Gamma },$ the reversibility problem which occurs in DNA k-bases has been solved. Moreover, we introduce a Gray map \(\Psi _{\hom }:{\mathfrak {R}}^{n}\rightarrow {\mathbb {F}}_{2}^{8n}\) ${\mathrm{\Psi }}_{hom}:{\mathfrak{R}}^n\to {\mathbb{F}}_2^{8n}$ with respect to homogeneous weight \(w_{\hom }\) $w_{hom}$ over the ring \({\mathfrak {R}}\) $\mathfrak{R}$ . Further, we discuss the GC-content of DNA cyclic codes and their deletion distance. Moreover, we provide some examples of reversible DNA cyclic codes.