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

Abstract

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