On reversibility problem in DNA bases over a class of rings

Let Rk=Z4u1,u2,…,uk/⟨ui2−ui,uiuj−ujui⟩ be a non-chain ring of characteristic 4, where 1≤i,j≤k and k≥1. In this article, we discuss reversible cyclic codes of odd lengths over the ring Rk. We construct bijections between the elements of the ring Rk and DNA-2k bases for k = 1, 2 in such a manner that the reversibility problem is solved. Employing these bijections, reversible complement cyclic codes of odd lengths are generated. Furthermore, we construct a Gray map Φ:Rkn→Z42kn and as an application of the Gray map Φ, we obtain the GC-content of cyclic codes of arbitrary odd length over the ring Rk. Meanwhile, we provide some examples of reversible cyclic codes of odd lengths over the ring Rk for different values of k, and also obtain the Lee distances of these codes.