Research On The Constructions Of Difference Set Pairs And Almost Difference Set Pairs

Optimal sequences and sequence pairs are widely used in many signals design fields due to their good autocorrelation properties.Among methods of constructing the best sequence pairs,the difference set is a relatively simple and direct method.However,is often difficult to construct new difference sets due to its strong requirements and many restrictions.Difference set pair and almost difference set pair have many similar properties with difference set.The difference set pair and almost difference set pair not only can be used to construct sequence pair with good properties,but also has more flexibility in construction,so it has attracted many scholars' attention.The difference set pairs are commonly constructed by the following three methods:multiplier theorem and multiplier conjecture method,characteristic polynomial method and cyclotomic class method of cyclotomic theory.Firstly,this thesis introduces the research background and significance.The definitions of classical cyclotomic classes,cyclotomic numbers and generalized cyclotomic classes with different forms are introduced.The research status of difference set pairs and almost difference set pairs and the latest research results of scholars are introduced.Secondly,this thesis introduces difference set pair and almost difference set pair constructed by classical cyclotomic classes of order 2 and 4 over some finite fields,which provides a theoretical basis for the proof of the difference set pair and almost difference set pair constructed by classical cyclotomic classes of order 8 over finite fields.Then,this thesis introduces the cyclotomic number of order 8 and proposes the theorem of difference set pairs with parameter(8f+1,2f,2f,0,f/2)and almost difference set pairs with parameter(8f+1,2f,2f,0,f/2,(f+2)/2).It enriches the research content of constructing difference set pairs and almost difference set pairs by classical cyclotomic classification.Finally,this thesis introduces the generalized cyclotomic class of order 4 based on Z2p2,and gives the generalized cyclotomic number on Z2p2.The difference set pairs with parameter of(2p2,3,3p2+1/2,p2,3p2+1.4,3p+1/4)and the almost difference set pairs with parameter of(2p2,p2-1/2,p2,p2-1/4,p2-1/4,p2-5/4)are constructed by using the generalized cyclotomic class of order 4 based on Z2p2,and the constructed difference set pairs and almost difference set pairs are theoretically proved.This is of great significance to the study of sequence pairs.