Constructions of Optimal Frequency-Hopping Sequences With Controlled Minimum Gaps

Frequency-hopping sequences (FHSs) with low Hamming correlation and wide gaps significantly contribute to the anti-interference performance in FH communication systems. This paper investigates FHSs with optimal Hamming correlation and controlled minimum gaps. We start with the discussion of the upper bounds on the minimum gaps of uniform FHSs and then propose a general construction of optimal uniform wide-gap FHSs with length 2<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> and 3<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i>, which includes the work by Li et al. in IEEE Trans. Inf. Theory, vol. 68, no. 1, 2022 as a special case. Furthermore, we present a recursive construction of FHSs with length 2<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i>, which concatenate shorter sequences of known minimum gaps. It is shown that the resulting FHSs have the same Hamming correlation as the concatenation-ordering sequences. As applications, several known optimal FHSs are used to produce optimal FHSs with controlled minimum gaps.