  In this paper, we discuss DNA codes that are cyclic or quasi-cyclic over
$\Z_{4}+\omega \Z_{4}$, where $\omega^{2}=2+2\omega$ along with methods to
construct these with combinatorial constraints. We also generalize results
obtained for the ring $\Z_{4}+\omega \Z_{4}$, where $\omega^{2}=2+2\omega$, and
some other rings to the sixteen rings $R_{\theta}=\Z_{4}+\omega \Z_{4}$, where
$\omega^{2}=\theta\in \Z_{4}+\omega \Z_{4}$, 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 $R_{\theta}$ which
enables us to attain an upper boundary for the Gau distance of free codes that
are self-dual over the rings $R_{\theta}$.
