Constructions Of Difference Set Pairs And Difference Families Based On Cyclotomic Classes

With the rapid development of mobile communication technology,ideal sequences and sequence pairs with good correlation function characteristics have attracted extensive attention in many engineering fields,such as satellite communication,radio positioning,underwater acoustic detection and so on.In order to break the limitation of the difficulty of constructing ideal sequences and sequence pairs,and meet their urgent needs in modern communication systems,scholars put forward the related concepts such as difference sets,difference families,difference set pairs and almost difference set pairs,and so on.These concepts not only enrich construction methods of ideal sequences,but also provide an effective mathematical tool for constructing high-quality signals.Because the number of sequence pairs and optical orthogonal codes with better performance is relatively small,and the difference set pairs and difference families are used as the design tools to transform the optimal autocorrelation sequence pairs and construct optical orthogonal codes respectively,the construction of difference set pairs and difference families has become the key breakthrough content in the research of combinatorial mathematics.In this thesis,based on the examples of constructing difference set pairs and difference families by using even order cyclotomic classes in the existing literature,new difference families,difference set pairs and almost difference set pairs are constructed by cyclotomic classes of order 8,which provide new mathematical support for the research of coding theory and cryptography.Firstly,the thesis summarizes the research background and development trend of difference families and difference set pairs,summarizes the research results of scholars in difference families and difference set pairs,and briefly explains their important application value in the field of communication.Secondly,the thesis introduces the basic concepts and related properties from two aspects.On the one hand,the thesis introduces the definition of classical cyclotomic classes,cyclotomic number and difference function,and completes the calculation of difference function with the help of basic cyclotomic number formula and operation law of cyclotomic classes.On the other hand,the thesis introduces the definitions of difference families,almost difference families,difference set pairs and almost difference set pairs,and illustrates the theoretical derivation process of constructing difference family and difference set pair by examples.Finally,the thesis uses computer programming to screen and refine a large number of related data which may form difference set pairs and difference family,and proposes new difference set pairs,difference set pairs and almost difference set pairs based on the construction of 8-order circle class,and completes the theoretical proof of the experimental results combined with the basic knowledge content of chapter 2.The examples of difference families,difference set pairs and almost difference set pairs constructed by the cyclotomic classes in this thesis can provide a new choice of mathematical tools for the research of the optical orthogonal code in code division multiple access communication system.The conclusion part of the article highly summarizes the research results of the subject,and also makes a detailed analysis of the problems in the research process,so as to provide reference experience for further research.