Binary Sequences With Optimal Autocorrelation Magnitude

The construction of binary sequences become an important problem with great theoretic significance and strong application backgroud in the field of design theory in recent years. There are some good results of the construction of the binary sequences, but it is not perfect. T.W. Cusick, C. Ding, A. Renvall proved that a binary sequence with optimal autocorrelation is equal to a difference set or an almost difference set with special parameters. Therefore, the combinatorial and algebraic methods are efficient for the construction of binary sequences with optimal autocorrelation and optimal autocorrelation magnitude. Many reseachers construct the binary sequences with optimal autocorrelation and optimal autocorrelation magnitude by cyclotomic classes and cyclotomic numbers of a finite field. This dessertation investigates the structure, the property and the construction method of binary sequences with optimal autocorrelation and optimal autocorrelation magnitude. We also investigate binary sequences with another definition.This thesis is devided into five chapters.The first chapter is devoted to the summarization of the disserta- tion. We disscuss the development of the subject, including the history and current situations of it, methods used and difficulties facing us. We also give the main results of this thesis.In the second chapter, we construct some classes of binary sequences with optimal autocorrelation maginatude by the difference function of the set and the cyclotomic classes and the cyclotomic numbers of order 4 over finite fields.In the third chapter and the fourth chapter, we analyze the polynomial properties and linear complexity by the polynomial representation of the sequences.In the fifth chapter, using a special type of cyclotomic classes over finite fields, we construction some classes of almost difference families.