On Self-Dual Cyclic Codes Over Finite Fields

In coding theory, self-dual codes and cyclic codes are important classes of codes which have been extensively studied. The main objects of study in this paper are self-dual cyclic codes over finite fields, i.e., the intersection of these two classes. We show that self-dual cyclic codes of length n over q exist if and only if n is even and q = 2 m with m a positive integer. The enumeration of such codes is also investigated. When n and q are even, there is always a trivial self-dual cyclic code with generator polynomial x n / 2 +1. We, therefore, classify the existence of self-dual cyclic codes, for given n and q , into two cases: when only the trivial one exists and when two or more such codes exist. Given n and m , an easy criterion to determine which of these two cases occurs is given in terms of the prime factors of n , for most n . We also show that, over a fixed field, the latter case occurs more frequently as the length grows.