Design Of Optimal Frequency Hopping Sequences And Study Of Theoretical Bounds

For frequency hopping communication systems(FHCS),the frequency hopping sequences(FHSs)have a very important significance.The properties of FHSs have a great significance on the performences of FHCS,and directly affect the performences of systems,such as anti-jamming,anti-interception and networking capability.Currently,the studies of FHSs are very active,the researches of FHSs include the following two aspects:Search for the theoretical bounds which are used to evaluate the performance of FHSs;Design the FHSs which meet or almost meet the condition of equal in the theoretical bounds.In this paper,the theoretical bounds,the constructions of Low-hit-zone FHS set(LHZ FHS set)and the FHSs with the maximum aperiodic Hamming correlation are studied,and achieve some important results.First of all,the theoretical bounds of the frequency hopping sequence set(FHS set)are studied.For the FHS set,we study clearly the exact relation between the bounds of the maximum periodic partial correlation,and study clearly the exact relation between the bounds of the maximum aperiodic Hamming correlation.For the LHZ FHS set,we study clearly the exact relation between the bounds of the maximum periodic Hamming correlation,between the bounds of the average periodic Hamming correlation and between the bounds of the average partial correlation.Secondly,we study the LHZ FHS sets with optimal Hamming correlation.Using the traditional cyclotomy technology over the finite field and the Chinese Remainder Theorem,we propose generic constructions of several kinds of LHZ FHS set.The results show that the new FHS sets possess many advantages.For example,the number of FHS is big,cyclic shift is not equivalent,the parameters are flexible and FHS sets are optimal according to the maximum periodic Hamming correlation.Select any optimal FHS set(basic sequence)with respect to the Peng-Fan bounds and any number of positive integers which meet certain conditions,using the Chinese Remainder Theorem,we propose the generalized designs of severy classes of LHZ FHS set.Analysis results show that the new FHS sets have excellent properties.For example,cyclic shift is not equivalent,and compared with the base sequence,the value of periodic Hamming correlation within the LHZ doesn't increase.Furthermore,the length of LHZ can take any value.And select different basic sequence,we can construct any different classes of optimal LHZ FHS set.Then the designs of LHZ FHS set with optimal maximum periodic partial Hamming correlation are studied.We study the properties of the maximum periodic partial Hamming correlation of FHS set constructed by the Cartesian Product.Based on this,select any k FHS sets which are optimal according to the periodic partial Hamming correlation bounds and the maximum periodic Hamming correlation is 0 at the time delay 0,two classes of optimal LHZ FHS set was constructed.And select any two FHS sets which are optimal with respect to the periodic partial Hamming correlation bounds,and one of which satisfies that the maximum periodic Hamming correlation is 0 at the time delay 0,we construct another class of optimal LHZ FHZ set.All new FHS sets have the following excellent properties:cyclic shift is not equivalent and the number of FHS is big.Whereafter,for any prime or prime power q(29)2,and positive integer l(29)1,the l-decimated sequence of q ary m-sequence is investigated,whered-gcd(l,q~n-1)?1,d(q-1),2?d(27)q-1 and gcd(l,(q~n-1)(q-1))-1.And give some new properties of l-decimated sequence,such as array structure and cyclic array structure.We construct a class of LHZ FHS set.The results show that the new FHS set is optimal according to the maximum periodic partial Hamming correlation for any length of correlation window within the LHZ.Finally,the FHS sets with optimal maximum aperiodic Hamming correlation are studied.Basing on GM/GGMW sequence over polynomial residue class ring,we use the decimation theory of sequence to construct two classes of FHS set,which are optimal on the number of FHSs with respect to the maximum aperiodic Hamming correlation.The early constructions are the special cases of the new methods.