On the Application of a Hybrid Incomplete Exponential Sum to Aperiodic Hamming Correlation of Some Frequency-Hopping Sequences

Frequency-hopping sequences are essential in frequency-hopping spread spectrum communication systems due to their strong anti-interference capabilities, low probability of interception, and high confidentiality. Existing research has predominantly focused on the periodic Hamming correlation properties of sequences, whereas the aperiodic Hamming correlation performance more accurately reflects the actual system performance. Owing to the complexity of its application scenarios and considerable research challenges, results in this area remain scarce. In this paper, we utilize exponential sums over finite fields to derive an upper bound on a hybrid incomplete exponential sum. Then, based on this upper bound, we derive bounds on the aperiodic Hamming correlation of some frequency-hopping sequence sets constructed by trace functions. Finally, by analyzing the maximum estimation error between the average and actual frequency collision numbers of such sequence sets, the validity of the derived bound is demonstrated.