Research On The Construction Method Of Almost Difference Set Pairs And Sequence Pairs Based On Cyclotomy

Sequences and sequence pairs with ideal autocorrelation have many applications in the areas of engineering such as radar,navigation,synchronization,electronic countermeasure,remote control and so on.Therefore,The design of optimal sequences and sequence pairs is closely related to many fields,such as mathematics,communication,computer and so on,and has very important significance in theory and application.The design of optimal sequences and sequence pairs has drawn the attention of the scholars in the these fields.Combinatorial design theories such as the almost difference sets,difference sets,almost difference set pairs（ADSP）and difference set pairs（DSP）are often used by scholars to study the construction methods of sequences and sequence pairs.Cyclotomy is a favourite mathematical tool used in combinatorial design theory,which is widely used in the design of sequence and sequence pairs and the constructions of difference sets,almost difference set,difference set pairs and almost difference set pairs.In this thesis,the algorithm for cyclotomic classes,the computer checking algorithm for ADSPs and the computer search algorithm for ADSPs are designed based on the Chinese residue theorem,cyclotomic number,cyclotomic class and the properties of ADSPs in finite field.The construction methods for ADSPs with new parameter forms are studied based on a large number of examples of ADSPs obtained by searching.The binary sequence pairs with optimal three-level autocorrelation function values are further obtained by using the equivalence relation between ADSPs and binary sequence pairs with three-level autocorrelation.Firstly,the methods of constructing ADSPs are studied based on cyclotmic classes of order 3 and 5.In the past,scholars mainly studied the construction method of ADSPs based on cyclotmic classes of even order,and few based on cyclotmic classes of odd order.In this thesis,several new construction methods of ADSPs are proposed based on cyclotmic classes of order 3 and 5,and the ADSPs constructed by these methods have new parameter forms.In addition,according to the equivalent relations between ADSPs and binary sequence pairs with three-level autocorrelation,the corresponding binary sequence pairs of these ADSPs have optimal three-level autocorrelation function values.Secondly,methods for constructing ADSPs of order 2n are studied.In this thesis,the construction methods for ADSPs and quaternary sequences are studied by utilizing generalized cyclotomic classes of order e overZ_2nbased on the Chinese remainder theorem and cyclotomy of order e.At first,several new construction methods of ADSPs are proposed based on the generalized cyclotomic classes of order 2 overZ_2n.Then,binary sequence pairs with optimal three-level autocorrelation whose sidelobe values are{-4,0}are obtained based on the generalized cyclotomic classes of order 4 overZ_2n.In addition,several new classes of balanced quaternary sequences with low autocorrelation function values are proposed based on the generalized cyclotomic classes of order 4 over Z_2n.Afterwards,methods for constructing ADSPs of order pq are studied.Several kinds of construction methods for ADSPs are proposed by utilizing generalized cyclotomic classes of order 2-2 over Z_pq,and these methods are divided into three categories according to the remainders of p module 4 and q module 4 respectively.The binary sequence pairs which are equivalent to these ADSPs have optimal three-level autocorrelation function values whose sidelobe values are{-3,1}or{-1,3}.In addition,the binary sequence pairs constructed by these methods have not only optimal three-level autocorrelation function values but also optimum balance property.Finally,methods for constructing ADSPs of order 5q are studied.Because the cyclotomic numbers of order 2 over Z₅ have only two values 0 and 1,the generalized cyclotomic classes of order 2-2 over Z_5qq have more combinatorial characteristics.Furthermore,it is found that the methods of constructing ADSPs over Z_5qq have their own unique features.In this thesis,four new methods for constructing ADSPs are proposed based on the cyclotomic classes of order 2-2 over Z_5q.All the characteristic sequence pairs of these ADSPs have optimal autocorrelation function values{-1,3}.