New Families of Frequency-Hopping Sequences of Length mN Derived from the k-Fold Cyclotomy

Abstract

Let N = p ₁ ⋯ p _k where p _i , 1 ≤ i ≤ k, are odd primes such that p ₁ < ⋯ < p _k and p _i  = M _i f + 1 for some positive integers M _i and f. In this paper, we construct frequency-hopping sequence (FHS) sets by using the properties of the k-fold cycltomy. We give FHS sets with length 2N and frequency set size (N − 1)/f, which are optimal with respect to the Peng-Fan bound if k = 1, and near-optimal if k ≥ 2. We also present near-optimal FHS sets with length mN and frequency set size (N − 1)/f + 1 for any integer m with 2 ≤ m ≤ M ₁. The FHS sets constructed in this paper have new parameters not covered in the literature.

This research was supported in part by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the NIPA (National IT Industry Promotion Agency) (NIPA-2010-(C1090-1011-0011)), and by Mid-career Researcher Program through NRF grant funded by the MEST (No. 2010-0000170).