The Design Of Optimal Sequence Pairs Based On Cyclotomy

Sequences and sequence pairs with ideal autocorrelation,balance properties and high linear complexity have many applications in the areas of engineering such as information enctyption,radar,sonar,navigation,synchronization,electronic countermeasure,telemetry,remote control,acoustic imaging and so on.Hence,the design of optimal sequences and sequence pairs has drawn the attention of the scholars in the fields of encoding theory,cryptology,spread spectrum communication,applied mathematics and so on.However,it is difficult to construct sequences or sequence pairs directly,so scholars often study the constructions of sequences and sequence pairs by the combinatorial design theories such as the difference set,almost difference set,difference set pairs（DSP）and almost difference set pairs（ADSP）.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 construction methods for optimal sequence pairs are studied based on cyclotomy,and new methods for constructing binary sequence pairs and quaternary sequences with good autocorrelation values are proposed by utilizing cyclotomic class and generalized cyclotomic classes.Firstly,the methods of constructing optimal binary sequence pairs with period N=p are studied and summed up based on cyclotmic classes of order 2,4 and 6,where p is an odd prime.It is found that optimal binary sequence pairs composed of two kinds of Legendre sequences can be constructed by cyclotmic classes of any even orders.Several new construction methods for binary sequence pairs with optimal three-level correlation are proposed by utilizing the relation between ADSP and binary sequence pair with three-level correlation based on cyclotmic classes of order 4 and 6 respectively.In addition,several new methods for constructing almost difference set pairs with high energy efficiency are also presented based on cyclotmic classes of order 4 and 6 in this thesis.Secondly,the constructions of optimal binary sequence pairs with period 3p are studied.Except for the binary sequence pairs composed of two kinds of Legendre sequences,the period of binary sequence pairs with optimal three-level correlation constructed by cyclotmic classes of order 4 and 6 directly are short.Hence,a generalized cyclotomy overZ_{3 p}is presented based on the Chinese remainder theorem and cyclotmic classes of order 2 in this thesis and it provides a new powerful mathematical tool for the constructions of sequences and sequence pairs with period of 3p.Many new methods for constructing binary sequence pairs with optimal three-level correlation are proposed by utilizing these new generalized cyclotomic classes in this thesis.Afterwards,in order to constructing optimal binary sequence pairs with period of pq,a new generalized cyclotomy overZ_pqis presented based on classic cyclotomy and the Chinese remainder theorem,where p and q are arbitrary two different odd primes.In contrast with the known Whiteman and Ding-Helleseth generalized cyclotomy over Z_pq,this new generalized cyclotomy is more flexible and can get many different kinds of generalized classes for a certain pair of p and q.Three new methods for constructing binary sequence pairs with optimal two-level correlation are proposed by utilizing the relation between DSP and binary sequence pair with two-level correlation based on these new generalized cyclotomic classes.All the binary sequence pairs obtained from these constructions have optimum balance property.Finally,the constructions of optimal quaternary sequences with period 2pare studied based on cyclotomic classes of order 4 and the Chinese remainder theorem.Two new construction methods for quaternary sequences with optimal three-level autocorrelation are presented.The quaternary sequences obtained from these constructions not only have optimum balance property but also haveR_max=2,which have come up to the standard of optimal.In addition,the constructions of quaternary sequences with low autocorrelation are also studied,and several new construction methods for quaternary sequences with low out-of-phase autocorrelation are presented.