Improved Bound on Low-Hit-Zone Frequency Hopping Sequence Sets and Their Linear Programming Bound

There are two types of bounds on low-hit-zone (LHZ) frequency hopping (FH) sequence sets. One is the lower bound on the maximum Hamming correlation of LHZ FH sequence sets derived by Peng et al. in 2006 (Peng-Fan-Lee bound), and the other is the upper bound on the family size of LHZ FH sequence sets derived by Liu et al. in 2019. Those bounds are not tight in some cases although they can be met in most cases. In this paper, by introducing the Möbius function we derive a new bound on LHZ FH sequence sets which is tighter than that derived by Liu et al.. The new bound deduces the bound on conventional FH sequence sets for Z = N −1, which was derived by Liu et al. in 2014. Further, by establishing an inequality relationship between LHZ FH sequence sets and Krawtchouk polynomials, we derive a linear programming bound on LHZ FH sequence sets, which is sometimes tighter than the Peng-Fan-Lee bound and the upper bound derived by Liu et al.. It is actually a nonlinear programming bound for gcd(H_m + 1;N) ≠ 1 and Z ≥ N/gcd(H_m+1;N). By setting Z = N −1, a linear programming bound on conventional FH sequence sets can be deduced which is tighter than that derived by Liu in 2025 for some cases.