Improved Singleton bound on frequency hopping sequences and optimal constructions

Frequency hopping (FH) sequences play an important role in FH spread spectrum communication systems. In this paper, a new theoretical bound on the FH sequences with respect to the size of the frequency slot set, the sequence length, the family size, and the maximum periodic Hamming correlation is established. The new bound is tighter than the Singleton bounds on the FH sequences derived by Ding et al. (IEEE Trans Inf Theory 55:3297–3304, 2009) and Yang et al. (IEEE Trans Inf Theory 57:7605–7613, 2011) (first by Sarwate in: Glisic and Leppanen (eds) Code division multiple access communications, Springer, Boston, 1995). In addition, the new bound employes the M\(\ddot{\text {o}}\)bius function. Then, more optimal FH sequence sets are obtained from Reed–Solomon codes. By utilizing the properties of cyclic codes, a new class of optimal FH sequence sets is obtained whose parameters meet the new bound. Further, two new constructions of FH sequence sets are presented. More new FH sequence sets are obtained by choosing proper base sequence sets. Meanwhile, the FH sequence sets constructed are optimal with respect to the new bound.