Optimal Frequency-Hopping Sequences With New Parameters

Abstract

A frequency-hopping sequence (FHS) of length and frequency set size M is called a (v, M, lambda)-FHS if its maximum out-of-phase Hamming autocorrelation is lambda. Three new classes of optimal FHSs with respect to the Lempel-Greenberger bound are presented in this paper. First, new optimal (p, M, f)-FHSs are constructed when p = Mf + 1 is an odd prime such that f is even and p 3 mod 4. And then, a construction for optimal (kp, p, k)-FHSs is given for any odd prime p and a positive integer k < p such that k = 2, 4, p(1), p(1) (p(1) + 2), 2(m) - 1, or p(1)(m) - 1, where p(1) and p(1) + 2 are odd primes. Finally, several new optimal FHSs with maximum out-of-phase Hamming autocorrelation 1 or 2 are also presented. In particular, the existence of optimal (v, N, 1)-FHSs is proven for any integer N >= 3 and any integer v with N + 1 <= v <= 2N - 1, as well as the existence of optimal (2N + 1,N,2)-FHSs is shown for any integer N >= 3. These classes of optimal FHSs have new parameters which are not covered in the literature.