Sets of frequency hopping sequences under aperiodic Hamming correlation: Upper bound and optimal constructions

Abstract

In order to evaluate the goodness of frequency hopping (FH) sequence design, the periodic Hamming correlation function is used as an important measure. Aperiodic Hamming correlation of FH sequences matters in real applications, while it received little attraction in the literature compared with periodic Hamming correlation. In this paper, an upper bound on the family size of FH sequences, with respect to the size of the frequency slot set, the sequence length, the maximum aperiodic Hamming correlation is established. Further, a construction of optimal FH sequence sets under aperiodic Hamming correlation from Reed-Solomon codes is presented, whose parameters meet the upper bound with equality. From generalized $m$ sequences (GM sequences) and generalized Gordon-Mills-Welch sequences (GGMW sequences), two classes of optimal FH sequence sets under aperiodic Hamming correlation are also presented, whose parameters meet the upper bound with equality.