Cyclic and Quasi-Cyclic DNA Codes

Abstract

In this paper, we discuss DNA codes that are cyclic or quasi-cyclic over ${\mathbb{Z}}_4+\omega {\mathbb{Z}}_4$, where ${\omega }^2=2+2\omega$ along with methods to construct these with combinatorial constraints. We also generalize results obtained for the ring ${\mathbb{Z}}_4+\omega {\mathbb{Z}}_4$, where ${\omega }^2=2+2\omega$, and some other rings to the sixteen rings $R_{\theta }={\mathbb{Z}}_4+\omega {\mathbb{Z}}_4$, where ${\omega }^2=\theta \in {\mathbb{Z}}_4+\omega {\mathbb{Z}}_4$, 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 $R_{\theta }$.

draft, 16 pages

References in corpus (1)

• The Art of DNA Strings: Sixteen Years of DNA Coding Theory