$k$ -Fold Cyclotomy and Its Application to Frequency-Hopping Sequences

For an integer k ≥ 1, let , 1≤ qi ≤ k,be prime powers such that qi - M_if + 1 for some integers M_i and f. In this paper, the k-fold cyclotomy of F_qk × ⋯ × F_qk as a nontrivial generalization of the conventional cyclotomy (k = 1 case) and its application to frequency-hopping sequences (FHSs) are presented, where F_q is the finite field with q elements. First, the definitions of k-fold cyclotomic classes and k-fold cyclotomic numbers are given. And then, their basic properties including k-fold diagonal sums are derived. Based on them, new optimal FHS sets of length N and frequency set size M or M + 1 with respect to the Peng-Fan bound are constructed for a product N of distinct odd primes and a di visor M of N - 1. Furthermore, new optimal FHSs of length N and frequency set size M with respect to the Lempel-Greenberger bound are constructed when N has at least one prime factor which is 3 modulo 4 and (N - 1)/M is an even integer. Our constructions give several new optimal parameters not covered in the literature, which are summarized in Table I.