DNA cyclic codes over the ring F2[u,v]/⟨u2−1,v3−v,uv−vu⟩

In this paper, our main objective is to find out the necessary and sufficient conditions for a cyclic code of arbitrary length over the ring of four elements R₁ = F₂ + uF₂ (u² = 1) to be a reversible cyclic code. We also obtain the structure of cyclic DNA codes of odd length over the ring R = F₂[u,v]/⟨u² − 1, v³ − v, uv − vu⟩, which plays an important role in Computational Biology. Furthermore, we establish a direct link between the elements of ring R and 64 codons used in the amino acids of living organisms by introducing a Gray map from R to R₁. Among others, binary images of cyclic codes over R are also investigated. As applications, some cyclic DNA codes over R using the Gray map are provided.