In this paper, we discuss DNA codes that are cyclic or quasi-cyclic over $\Z_{4}+ω\Z_{4}$, where $ω^{2}=2+2ω$ along with methods to construct these with combinatorial constraints. We also generalize results obtained for the ring $\Z_{4}+ω\Z_{4}$, where $ω^{2}=2+2ω$, and some other rings to the sixteen rings $R_θ=\Z_{4}+ω\Z_{4}$, where $ω^{2}=θ\in \Z_{4}+ω\Z_{4}$, using the generalized Gau map and Gau distance in \cite{3}. We determine a relationship between the Gau distance and Hamming distance for linear codes over the sixteen rings $R_θ$ which enables us to attain an upper boundary for the Gau distance of free codes that are self-dual over the rings $R_θ$.