DNA Codes over the Ring ℤ_4 + wℤ_4
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In this present work, we generalize the study of construction of DNA codes over the rings ℛ_θ=ℤ_4+wℤ_4, w^2 = θ for θ∈ℤ_4+wℤ_4. Rigorous study along with characterization of the ring structures is presented. We extend the Gau map and Gau distance, defined in <cit.>, over all the 16 rings ℛ_θ. Furthermore, an isometry between the codes over the rings ℛ_θ and the analogous DNA codes is established in general. Brief study of dual and self dual codes over the rings is given including the construction of special class of self dual codes that satisfy reverse and reverse-complement constraints. The technical contributions of this paper are twofold. Considering the Generalized Gau distance, Sphere Packing-like bound, GV-like bound, Singleton like bound and Plotkin-like bound are established over the rings ℛ_θ. In addition to this, optimal class of codes are provided with respect to Singleton-like bound and Plotkin-like bound. Moreover, the construction of family of DNA codes is proposed that satisfies reverse and reverse-complement constraints using the Reed-Muller type codes over the rings ℛ_θ.
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