k-fold cyclotomic numbers and their applications to frequency-hopping sequences

For an integer k ≥ 1, k-fold cyclotomic numbers of F_q1× ... × F_qk are introduced, where F_q is the finite field with q elements and q_i´s are powers of distinct primes. They are a generalization of the conventional cyclotomic numbers (k = 1 case). Some of their basic properties including k-fold diagonal sums are derived. As an application of the k-fold cyclotomy, frequency-hopping sequences (FHSs) of length p₁ ... p_k are constructed for distinct odd primes p₁, ..., p_k, which are optimal with respect to the Lempel-Greenberger bound and the Peng-Fan bound.