Strictly Optimal Frequency-Hopping Sequences And Related Codes

Frequency-hopping sequences(FHSs)are an integral part of frequency-hopping multiple access communication systems,where the partial Hamming correlation property dominates the performance of the system and also the synchronization and acquisition of the code sequence at a receiver.Apart from relevance to applications,in mathematics,the partial Hamming correlation of FHSs is also an important object of theoretical interest,since it contains the periodic and aperiodic Hamming correlations as special cases.Thus,in this thesis,we focus on FHSs with optimal partial Hamming correlation properties(strictly optimal).We are going to investigate the relationship between the partial Hamming correlation and other parameters of frequency-hopping sequences and to propose new constructions for strictly optimal frequency-hopping sequences and other related codes.Firstly,we improve the known lower bounds on the partial Hamming correlation of FHSs and FHS sets.Some known frequency-hopping sequences are proved to be optimal achieving the improved bound.Upper bounds on the family sizes of FHS sets with respect to partial Hamming correlation are also derived from classical bounds on error-correcting codes.These bounds are tight in some cases,where we can construct optimal frequency-hopping sequences with respect to those bounds.Secondly,we introduce constructions for FHSs with optimal partial Hamming correlations.Both individual FHSs and FHS sets having optimal partial Hamming correlation with respect to the improved bounds are constructed via generalized cyclotomy.By applying disjoint cyclic perfect Mendelsohn difference families(CPMDFs),a combinatorial construction of strictly optimal FHSs is introduced.This construction is generic in the sense that it works for any disjoint CPMDFs.In addition,based on the interleaving technology,a construction of FHSs with low partial Hamming correlation was also proposed.Employing this construction,FHS sets with new parameters having both optimal family sizes and optimal partial Hamming correlation were obtained.Next,a construction of zero-difference balanced(ZDB)functions is proposed via the generalized cyclotomy.This construction can generate ZDB functions with new parameters.It also yields optimal constant-composition codes,optimal perfect difference systems of sets,and optimal frequency-hopping sequences with new parameters,as applications of the ZDB functions.Finally,we discuss the bounds and constructions for two-dimensional optical orthogonal code(2-D OOC)satisfying both at most one-pulse per wavelength(AM-OPPW)and at most one-pulse per time slot(AM-OPPTS)perporties.On one hand,a Johnson type bound is derived for 2-D OOC satisfying both AM-OPPW and AM-OPPTS perporties.On the other hand,two constructions of optimal 2-D OOCs with both AM-OPPW and AM-OPPTS with respect to a new theoretic bound are introduced.By further adding some codewords,several optimal 2-D OOCs with AM-OPPW having new parameters can also be yielded.