Construction of cyclic DNA codes over $$\varvec{\mathrm{{Z}}_4R}$$ Z 4 R

This study is devoted to the construction of cyclic DNA codes of odd length over $$\mathbb {Z}_4\mathbb {R}$$ Z 4 R . The generator polynomial of cyclic codes of this ring and the structure of its separable codes are studied in detail. A map $$\psi _\beta $$ ψ β from $$\mathbb {R}^\beta $$ R β to $$\lbrace A, T, G, C \rbrace ^{2\beta }$$ { A , T , G , C } 2 β is defined and then constructed a one-to-one correspondence between DNA codons of alphabets and the elements of $$\mathbb {R}$$ R . Furthermore, necessary and sufficient circumstances of cyclic codes over $$\mathbb {Z}_4$$ Z 4 , $$\mathbb {R}$$ R and $$\mathbb {Z}_4\mathbb {R}$$ Z 4 R to be reversible codes and reverse-complement constraints codes are determined. To support the theories, some important examples are illustrated.