Design On Optimal Frequency-hopping Sequences For Spread-Spectrum Communication

Frequency-hopping spread spectrum(FHSS)systems,with properties of anti-jamming,anti-intercept,code division multiple access(CDMA),channel sharing,etc,are usually applied in military radio communication,mobile communication,modern radar and sonar echolocation systems.Frequency-hopping sequence(FHS)is an integral part of FHSS systems,where the correlation properties of the FHSs employed are the most important criterion of the performance of FHSS systems.In this dissertation,we give a deep study on FHSs and construct some FHSs and sets of FHSs with good performance.Details are as follows:Firstly,a kind of generalized cyclotomy with respect to a prime power is introduced and properties of the corresponding generalized cyclotomic numbers are investigated.Based on the generalized cyclotomy,three classes of sets of FHSs with prime-power period are presented.Meanwhile,we derive the Hamming correlation distribution of the new sets of FHSs by virtue of the properties of the generalized cyclotomic numbers.The results show that the proposed FHSs and sets of FHSs have optimal or near-optimal maximum Hamming correlation.The last two classes of near-optimal sets of FHSs have new parameters which are not covered in the literature.Secondly,based on the classical cyclotomy and the Chinese Remainder Theorem(CRT),a class of set of FHSs with a multiple of prime number period is constructed and the Hamming correlations of the new set are derived by some basic properties of the cyclotomic numbers.The results show that the proposed set is optimal with respect to the Peng-Fan bound and each FHS of the set is optimal or near-optimal with respect to the Lempel-Greenberger bound.This set has more large size,long period of these sequences which generalizes the early results given by Chu and Zhang based on the similar technique.Next,two classes of FHSs are proposed by means of two partitions of _v and the CRT,where v>3 is an odd positive integer.It is shown that all the constructed FHSs are optimal with respect to the Lempel-Greenberger bound.By choosing appropriate injective functions,infinitely many optimal FHSs can be recursively obtained.Above all,these FHSs have new parameters which are not covered in the former literature.Zero-difference balanced(ZDB)functions were introduced by Ding in connection with constructions of optimal constant composition codes.Based on such functions,some scholars have constructed optimal constant weight codes,optimal FHSs,optimal and perfect difference systems of sets.In order to obtain more optimal cryptographic objects,the ZDB function is generalized to the near zero-difference balanced(N-ZDB)function,whose characterizations are partially given.Furthermore,we prove that N-ZDB functions are equivalent to partitioned almost difference families(PADFs)in design theory.Most importantly,three classes of the N-ZDB functions are proposed by means of the partition of _n,where n>3 is an odd positive integer.Employing these N-ZDB functions,we obtain at the same time optimal FHSs and optimal difference systems of sets with flexible parameters.Afterwards,we present a simplified representation of the Peng-Fan bounds on the periodic Hamming correlation of sets of FHSs,which may make it convenient to check the optimality of an FHS set.Then we propose a recursive construction of FHS sets from the known ones using some injective functions and the CRT.It generalizes the previous construction of optimal FHSs and FHS sets with composite lengths.Without the limit of the specific function,our construction can produce new optimal FHSs and FHS sets that cannot be produced by the earlier construction.By choosing appropriate injective functions and known optimal FHSs and FHS sets,infinitely many new optimal FHSs and FHS sets can be recursively obtained.Finally,we propose a direct construction of disjoint cyclic perfect Mendelsohn difference family(CPMDF)from the Zeng-Cai-Tang-Yang generalized cyclotomy.As we all know,strictly optimal FHSs are a kind of optimal FHSs which has optimal Hamming auto-correlation for any correlation window.As an application of our disjoint CPMDFs,we present more flexible combinatorial constructions of strictly optimal FHSs,which can produce new strictly optimal FHSs.