Constructions of Optimal Frequency-Hopping Sequences with Controlled Minimum Gaps

2025-06-01

IEEE Transactions on Information Theory

AI Summary

In frequency-hopping (FH) communication, achieving both wide minimum Hamming distance and low Hamming correlation simultaneously remains challenging. Method: This paper constructs frequency-hopping sequences (FHSs) that attain optimal Hamming correlation while enabling controllable minimum Hamming distance. First, it establishes the theoretical upper bound on the minimum Hamming distance for uniform FHSs. Second, it proposes explicit constructions of optimal wide-interval FHSs of lengths (2l) and (3l). Third, it designs a recursive framework based on sequence concatenation that preserves optimal Hamming correlation while achieving minimum distance control up to the theoretical bound. Results: The generated FHSs match state-of-the-art correlation performance, systematically generalizing and extending Li et al.’s (2022) work. This yields a new class of parameter-tunable, optimal FHSs, providing enhanced design flexibility for interference-resistant FH systems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

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 2l and 3l, 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 2l, 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.

Problem

• Constructs optimal frequency-hopping sequences with controlled gaps
• Investigates FHSs with optimal Hamming correlation and wide gaps
• Proposes recursive construction for FHSs with controlled minimum gaps

Innovation

• Optimal uniform wide-gap FHSs construction
• Recursive construction using shorter sequences
• Controlled minimum gaps for anti-interference