Reversible complement cyclic codes over Galois rings with application to DNA codes

Abstract

Let R be a Galois ring of characteristic pa and cardinality pam, where p is a prime and a,m are natural numbers. For a unit u and an element k∈R such that u2=1 and uk=k, a generalized notion of a complement of an element e in R with respect to u and k has been given in this paper. Using this notion of complement, reversible complement cyclic codes with respect to u and k have been defined. Further, a necessary and sufficient condition for a reversible cyclic code of arbitrary length over R to be a reversible complement cyclic code with respect to u and k has been obtained. This generalization is significant in view of the fact that a reversible cyclic code of length n over R may be reversible complement with respect to one choice of u and k but may not be reversible complement with respect to some other choice of u and k. Further, reversible complement cyclic codes with respect to u and k (for different values of u and k) over a special Galois ring, Z4, have been used to construct some DNA codes. It has been observed that codes over Z4 in some cases give better DNA codes than DNA codes over fields and some rings of equal cardinality. A tradeoff between the two DNA properties namely avoiding secondary structure formation and constant GC−content has also been observed.