k-fold cyclotomy and its application to frequency-hopping sequences

Abstract

For an integer k ≥ 1, let qi, 1 ≤ I ≤ k, be prime powers such that qi=Mi f + 1 for some integers M i and f. In this paper, the k -fold cyclotomy of double-struck F signq1 × ⋯ × double-struck F signqk as a nontrivial generalization of the conventional cyclotomy ( k=1 case) and its application to frequency-hopping sequences (FHSs) are presented, where double-struck F signq 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 divisor M of N-1. Furthermore, new optimal FHSs of length N and frequency set size M with respect to the LempelGreenberger 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.