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

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