Optimal Frequency-Hopping Sequences With New Parameters

A frequency-hopping sequence (FHS) of length v and frequency set size M is called a (v,M,¿)-FHS if its maximum out-of-phase Hamming autocorrelation is ¿. 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,p₁,p₁(p₁ + 2 ),2^m -1,or p₁ ^m-1, where p₁ and p₁+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 ¿ 2 N-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.