Enumeration of Self-Dual Cyclic Codes of some Specific Lengths over Finite Fields

Self-dual cyclic codes form an important class of linear codes. It has been shown that there exists a self-dual cyclic code of length n over a finite field if and only if n and the field characteristic are even. The enumeration of such codes has been given under both the Euclidean and Hermitian products. However, in each case, the formula for selfdual cyclic codes of length n over a finite field contains a characteristic function which is not easily computed. In this paper, we focus on more efficient ways to enumerate self-dual cyclic codes of lengths 2p and 2pq, where ν, r, and s are positive integers. Some number theoretical tools are established. Based on these results, alternative formulas and efficient algorithms to determine the number of self-dual cyclic codes of such lengths are provided.