On Self-Dual Cyclic Codes Over Finite Fields
Yan Jia, San Ling, Chaoping Xing · IEEE Transactions on Information Theory · 2011
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 lengthnover \BBFqexist if and only ifnis even andq= 2mwithma positive integer. The enumeration of such codes is also investigated. Whennandqare even, there is always a trivial self-dual cyclic code with generator polynomialxn/2+1. We, therefore, classify the existence of self-dual cyclic codes, for givennandq, into two cases: when only the trivial one exists and when two or more such codes exist. Givennandm, an easy criterion to determine which of these two cases occurs is given in terms of the prime factors ofn, for mostn. We also show that, over a fixed field, the latter case occurs more frequently as the length grows.