The Bound And Design Of Frequency Hopping Sequences With Low-Hit-Zone

Frequency hopping sequences (FHSs) play a very important role in frequency hopping communication systems. Generally speaking, the extent of the multiple-access interference of frequency hopping communication systems is determined by the performance of the FHSs largely. Hence FHSs influence the performance of frequency hopping communication systems directly. The theory of FHSs has two main research fields:the theoretical bound and sequence design. In this paper, the theoretical bounds on maximum aperiodic Hamming correlation of the FHSs with low-hit-zone (LHZ), maximum partial Hamming correlation of the FHSs with LHZ and the FHSs with no-hit-zone (NHZ) are investigated; the constructions and analysis of the FHSs with optimal average Hamming correlation, LHZ FHSs with optimal maximum periodic Hamming correlation, LHZ FHSs with optimal maximum partial Hamming correlation and optimal NHZ FHSs are also studied.First of all, the new lower bounds on maximum aperiodic Hamming correlation of the LHZ FHSs, with respect to the family size, the alphabet size, the sequence length, the LHZ, the maximum aperiodic Hamming autocorrelation and the maximum aperiodic Hamming crosscorrelation within the LHZ are established. Specially, the new bounds include the second powers of the maximum aperiodic Hamming autocorrelation and the maximum aperiodic Hamming crosscorrelation within the LHZ, but the previous bounds do not include them. The new lower bound on maximum aperiodic Hamming correlation of the conventional FHSs is also derived. And the new lower bound on maximum partial Hamming correlation of the LHZ FHSs, with respect to the family size, the alphabet size, the sequence length, the LHZ, the maximum partial Hamming autocorrelation and the maximum partial Hamming crosscorrelation within the LHZ are established. The new bounds are tighter than the corresponding previous bounds. In addition, some essential properties of NHZ FHSs are pointed out. The theoretical bounds on the NHZ FHSs are discussed. The relationships between these different theoretical bounds are disclosed and the equivalent conditions for these theoretical bounds are presented.Thereafter, the average Hamming correlation properties of FHSs are studied deeply. A sufficient and necessary condition for a set of FHSs with optimal average Hamming correlation is derived. Based on the polynomial theory over the finite field, a new set of FHSs with optimal average Hamming correlation is presented and studied. It is shown that the new FHS set includes the known cubic and constant FHS set as a special case. By interleaved techniques, a new construction for optimal average Hamming correlation FHSs is proposed. Some classes of optimal average Hamming correlation FHSs with new parameters are presented based on the new design. It is shown that the new construction includes the known Chung and Yang's construction as a special case.Next, based on the m-sequences and its decimated sequences, two classes of LHZ FHS sets with optimal maximum periodic Hamming correlation are presented, respectively. It is shown that the presented LHZ FHS sets have new parameters and include the known FHS set constructed by Ding and Yin as a special case. In addition, a general construction method of FHSs with NHZ is proposed. By this construction method, optimal FHS sets with NHZ of any length and any optimal FHS sets with NHZ can be obtained.Finally, we prove the nonexistence of an LHZ FHS set achieving the theoretical bound for all correlation windows in some conditions. A sufficient condition for an LHZ FHS set with strictly optimal average partial Hamming correlation is also given. A new design of LHZ FHS sets achieving the theoretical bound for all correlation windows is presented via interleaving techniques. By the new design, new classes of LHZ FHSs achieving the theoretical bound for all correlation windows are obtained. Based on the m-sequences, a class of LHZ FHS sets achieving the theoretical bound for some specific correlation windows is presented by interleaving techniques. In addition, a concatenated construction method is presented. The LHZ FHS sets achieving the theoretical bound for some specific correlation windows and the LHZ FHS sets with strictly optimal average partial Hamming correlation are constructed by the new design, respectively.