Cyclic and Quasi-Cyclic DNA Codes

Abstract

In this paper, we discuss DNA codes that are cyclic or quasi-cyclic over Z 4 + ωZ 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 + ωZ 4 , where ω 2 = 2 + 2ω, and some other rings to the sixteen rings R θ = Z 4 + ωZ 4 , where ω 2 = θ ∈ Z 4 + ωZ 4 , using the generalized Gau map and Gau distance in [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 .