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 <i>n</i> over \BBF<i>q</i> exist if and only if <i>n</i> is even and <i>q</i> = 2<i>m</i> with <i>m</i> a positive integer. The enumeration of such codes is also investigated. When <i>n</i> and <i>q</i> are even, there is always a trivial self-dual cyclic code with generator polynomial <i>x</i>ⁿ/₂+1. We, therefore, classify the existence of self-dual cyclic codes, for given <i>n</i> and <i>q</i> , into two cases: when only the trivial one exists and when two or more such codes exist. Given <i>n</i> and <i>m</i> , an easy criterion to determine which of these two cases occurs is given in terms of the prime factors of <i>n</i>, for most <i>n</i> . We also show that, over a fixed field, the latter case occurs more frequently as the length grows.