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 $R_{ \theta}$.
Categories: cs.IT math.IT math.RA