Constructions Of External Difference Families And Disjoint Difference Families

The concept of difference family is a natural extension of the concept of difference set.The difference family method is also one of the most commonly used and most effective methods to construct various types of designs.The concept of external differ-ence family is first proposed by Ogata in 2004,and applied to authentication code and secret sharing.Then,Chang and Ding thoroughly studied the constructions of external difference family and disjoint difference in 2006,and get a series of new parameters of external difference family and disjoint difference by means of cyclotomic construction and recursive construction,and establish the relationship between external difference family and special classes of difference sets and almost difference sets.In the study of the external difference family,cyclotomic construction is a very im-portant means.After Chang and Ding,Huang and Wu continued to construct a new type of external difference family and disjoint difference family with the four and six order cyclotomic classes.They found more external difference family under the condition of ku<v-1 and solved the problem 1 and the problem 4 which Chang and Ding left in the article.In 2016,Chen,Lin and Ling obtained the equivalent conditions of a class of external difference family and disjoint difference by dividing all square elements in the GF(q).For the construction of external difference family with q = 2ku+1,Touchev studied the existence of a class of external difference family which have the conditon of q?3(mod 4)and q-1 = 2ku in 2003.On the base of this,Chang and Ding continued to study the existence of(q,k,A;u)-EDF when q?1(mod 4)and q?1(mod 8),and gave the solution to the existence problem of k=2,4 in 2006.This paper mainly studies the general constructions of external difference family and disjoint difference family.In the first chapter,we introduce some basic concepts and symbols related to this article.In the second chapter,we use the eight order cyclotomic class to construct new external difference family and disjoint difference family and ex-tend it.We get several new types of external difference family.In the third chapter,we generalizes the problem of the existence of external difference family with parameters of(q,k,?;u)when q = 2ku + 1,and discuss the the existence problem of k=6,8.In the fourth chapter,we summarize some new parameters of the external difference family and disjoint difference derived from chapter two and three,and summarize the work of this paper.