On the Codes over a Family of Rings and Their Applications to DNA Codes

Abstract

In this paper, the structures of the linear codes over a family of the rings $A_{t}=Z_{4}\left[ u_{1},\ldots ,u_{t}\right] \left/ \left\langle u_{i}^{2}-u_{i},u_{i}u_{j}-u_{j}u_{i}\right\rangle \right. $ are given, where $i,j=1,2,\ldots ,t$, $i\neq j$, $Z_{4}=\{0,1,2,3\}$. A map between the elements of the $A_{t}$ and the alphabet $\left\{ A,T,C,G\right\} ^{2^{t}}$ is constructed. The DNA codes are obtained with three different methods, by using the cyclic, skew cyclic codes over a family of the rings $A_{t}$ and $\theta _{i}$-set, where $\theta _{i}$ is a non trivial automorphism on $A_{i}$, for $i=1,2,\ldots ,t$.