Further Constructions Of Almost Difference Set Pairs Based On Cyclotomic Classes Of Order Eight

As we all know,difference sets and almost difference sets are equivalent to binary and ternary autocorrelation sequences,so they can be used to construct binary and ternary autocorrelation sequences,respectively.In order to get more extensive and high-quality signals,scholars put forward the theory of almost optimal autocorrelation sequence pairs.When a certain parameter condition is satisfied,the best autocorrelation binary sequence pairs is equivalent to difference set pairs.Therefore,difference set pairs,an important tool to study the best autocorrelation binary sequence pairs,is proposed.In 2014,Luliang Zheng put forward the concept and some properties of almost difference set pairs,and constructed several kinds of almost difference set pairs by using cyclotomic classes of order 4 and order 6,which provided a new possibility for constructing the optimal binary sequence pairs.At present,the main method of constructing almost difference set pairs is cyclotomic classes,which mostly based on cyclotomic classes of order 2,order 4 or order 6.Only a few almost difference set pairs are constructed by cyclotomic classes of order 8.This paper is concerned with the further construction of almost difference set pairs by using cyclotomic classes of order 8.It is similar to the previous method of constructing almost difference set pairs by cyclotomic classes.First of all,we sort out some possible combinations of cyclotomic classes,and then calculate the list of difference functions in eachcombination by using cyclotomic numbers.Using this method,we can construct several new classes of almost difference set pairs.The corresponding value of is obtained by program,and then some examples and explicit expressions of are given.The first chapter summarizes the research status and development trend of almost difference set pairs,the main content and structure arrangement of this paper are briefly introduced.The second chapter introduces the basic theory and the main conclusions about the structure of cyclotomic classes and almost difference set pairs are introduced.We introduces some basic knowledge of cyclotomic classes,and some basic concepts,such as cyclotomic number,difference function,almost difference set pairs,and some basic properties of them,which provide the basis for constructing almost difference set pairs based on cyclotomic classes.The third chapter constructs some almost difference set pairs by means of cyclotomic classes.Based on the theory of almost difference set pairs,we put forward several kinds ofnew almost difference set pairs based on cyclotomic classes of order 8.We check our results by programing and give several examples.