DNA cyclic codes over rings

In this paper we construct new DNA cyclic codes over rings. Firstly, we introduce a new family of DNA cyclic codes over the ring $R=\mathbb{F}_2[u]/(u.6)$. A direct link between the elements of such a ring and the $64$ codons used in the amino acids of the living organisms is established. Using this correspondence we study the reverse-complement properties of our codes. We use the edit distance between the codewords which is an important combinatorial notion for the DNA strands. Next, we define the Lee weight, the Gray map over the ring $R$ as well as the binary image of the DNA cyclic codes allowing the transfer of studying DNA codes into studying binary codes. Secondly, we introduce another new family of DNA skew cyclic codes constructed over the ring $\tilde {R}=\mathbb{F}_2+v\mathbb{F}_2=\{0, 1, v, v+1\}, $ where $v^2=v$. The codes obtained are cyclic reverse-complement over the ring $\tilde {R}$. Further we find their binary images and construct some explicit examples of such codes.