A New Method to Construct Strictly Optimal Frequency Hopping Sequences With New Parameters

In order to evaluate the goodness of frequency hopping (FH) sequence design, the periodic Hamming correlation function is used as an important measure. Usually, the length of correlation window is shorter than the period of the chosen FH sequence, so the study of the partial Hamming correlation of FH sequence is particularly important. If an FH sequence or an FH sequence set has an optimal partial Hamming correlation with respect to the partial Hamming correlation bound for all length of correlation window, then the FH sequence or the FH sequence set is said to be strictly optimal. In this paper, we first prove that there is no strictly optimal FH sequence set of family size M and sequence length N over a frequency slot set of size q with respect to the partial Hamming correlation bound derived by Niu et al. when N > q
²
/M and q ≥ 2, and that by Cai et al. when N > q
²
/M and q ≥ 2N/(N - 2). Furthermore, we define a special partition-type difference packing (DP) called [N, ∇, H
_a
^l
] PDP and give several classes of [N, ∇, H
_a
^l
] PDPs. Then, we present a new construction of strictly optimal FH sequences. By choosing different PDPs, the FH sequences constructed can give new and flexible parameters. By utilizing this construction method recursively, we can obtain new [N, ∇, H
_a
^l
] PDPs, which lead to infinitely many classes of strictly optimal FH sequences with new parameters. Moreover, based upon an [N, ∇, H
_a
^l
] PDP, we present a construction of strictly optimal FH sequence sets. By preceding construction method and recursive construction, we can also obtain infinite classes of strictly optimal FH sequence sets which can give new and flexible parameters.