LAUSR

In this paper, we study the structure of cyclic DNA codes of arbitrary length over the ring $$R=\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2$$R=F2+uF2+vF2+uvF2, $$u^{2}=0, v^{2}=v, uv=vu$$u2=0,v2=v,uv=vu. By defining a Gray map, we establish a relation between R and $$R^{2}_{1}$$R12, where $$R_{1}=\mathbb {F}_2+u\mathbb {F}_2$$R1=F2+uF2 is a ring with four elements. Cyclic codes of arbitrary length over R satisfying the reverse constraint and the reverse-complement constraint are studied in this paper. Furthermore, we introduce reversible codes which provide a rich source for DNA codes. The GC content constraint is also considered. We give some examples to support our study in the last.