# Optimal Constructions of Low-Hit Zone Frequency-Hopping Sequence Set Based on _m_-Sequence
## 1 Introduction  
### Background  
Frequency hopping (FH) is a spread-spectrum technology that achieves spectral broadening through carrier frequency hopping [1, 2, 3, 4]. In an FH system, the transmitting end employs pseudorandom sequences to control the central frequency of the data-modulated carrier wave, causing it to randomly hop over frequency sets according to a specific pattern. Correspondingly, the receiving end follows the same pattern for synchronization, demodulation, and signal recovery. The pseudorandom sequence is termed the FH sequence (FHS). When multiple users share the same spectrum resources, during certain time intervals, different users may simultaneously occupy the same frequency channel, and interference occurs due to the signals from different users overlapping. This type of interference is what is referred to as multiple-access interference (MAI). Studies have shown that MAI is the primary factor affecting the bit error rate (BER) performance of FH systems [5]. The severity of MAI between users is determined by the number of frequency collisions between their used FHSs. As one type of FH system, Quasi-Synchronous FH Multiple Access (QS-FHMA) systems seamlessly integrate FH technology with multiple-access (MA) technology. In thesesystems, multiple users communicate over different frequencies according to predetermined FHSs (FH patterns) to achieve spectrum sharing and interference avoidance. QS-FHMA systems exhibit relatively relaxed requirements for time synchronization, allowing for a certain degree of time delay or frequency offset. For QS-FHMA systems, the concept of low-hit zone (LHZ) FHS set was introduced [6; 7]. The asynchronicity of QS-FHMA systems is constrained near zero delay, and the MAI between users is determined by the Hamming correlation values of different FHSs within the synchronization error range. An LHZ FHS set possesses a delay zone Z near zero delay within which both the Hamming autocorrelation sidelobes of each sequence and the Hamming cross-correlation values between distinct sequences remain low, where \(Z\) is termed \"LHZ\". For an LHZ FHS set, when the delay does not exceed \(Z\), the Hamming correlation values are small and the number of sequences is large. QS-FHMA systems, supported by well-designed LHZ FHS sets, are integral to a variety of modern applications. For instance, in satellite communication, due to the long signal transmission distances and complex environments, interference issues are particularly prominent; the LHZ FHS sets support MA communication, optimize spectrum utilization, and significantly enhance the anti-interference capability and security of satellite communication. In military communications, the LHZ FHS sets are employed to prevent interception and jamming, thereby ensuring the confidentiality and integrity of transmitted information [8; 9]. Additionally, radar systems utilize the LHZ FHS sets to evade detection and countermeasures, enhancing the reliability of signals in complex operational environments. Current research on LHZ FHS sets focuses primarily on two aspects. (1) The first is exploring theoretical limits on the relationships between different parameters of LHZ FHS sets. Studies have shown that parameters such as the Hamming correlation values, sequence counts, frequency slot numbers, and sequence lengths of LHZ FHS sets are subject to certain theoretical constraints. These mathematical relationships are often referred to as \"theoretical bounds\" [10; 11; 12], serving as an important method for evaluating LHZ FHS set performance and guiding their design. (2) The second aspect is constructing LHZ FHS sets that achieve these theoretical bounds. An LHZ FHS set is termed \"optimal\" if its parameters satisfy the equality of the corresponding theoretical bound. The goal of LHZ FHS set design is to construct optimal LHZ FHS sets. In recent years, domestic and international scholars have actively engaged in research regarding the construction of LHZ FHS sets.