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In this paper, we study the structure of cyclic DNA codes of arbitrary length over the ring \(R=\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2\) , \(u^{2}=0, v^{2}=v, uv=vu\) . By defining a Gray map, we establish a relation between R and \(R^{2}_{1}\) , where \(R_{1}=\mathbb {F}_2+u\mathbb {F}_2\) 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.
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