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/ leftlangle u_{i}^{2}-u_{i},u_{i}u_{j}-u_{j}u_{i}rightrangle right. $ are given, where $i,j=1,2,ldots ,t$, $ineq j$, $Z_{4}={0,1,2,3}$. A map between the elements of the $A_{t}$ and the alphabet $left{ A,T,C,Gright} ^{2^{t}}$ 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 $i=1,2,ldots ,t$.