On Double Cyclic Codes over Finite Chain Rings for DNA Computing.

Let e be a fixed positive integer and n1,n2 be odd positive integers. The main objective of this article is to investigate the algebraic structure of double cyclic codes of length (n1,n2) over the finite chain ring Re = F4e+vF4e, where v2=0. Building upon this structural framework, we further demonstrate the construction of DNA codes derived from these double cyclic codes over Re. In addition, we provide the necessary and sufficient criteria showing that these codes possess reversibility and reverse-complement properties over Re. Furthermore, we introduce a generalized Gray map that extends the classical Gray map from the ring F2+vF2 with v2=0 to the ring Re, showing a direct correspondence between elements of Re and DNA sequences over S={A,T,G,C} utilizing double cyclic codes. To illustrate the applicability of our results, we present some examples demonstrating the effectiveness of the mapping in generating reversible and reverse-complement DNA codes from algebraic structures over the ring Re.