Transitive. Projective Linear Group Under The 4 - Design

The construction of t-designs is an important problem with great theoretic significance and strong application background in the field of design theory. The theories and methods of t-designs play an important role in mathematical statistics, operations research, information science and computer science. Algebraic methods are efficient methods for the construction of simple t-designs, especially, the method of using the orbits of k-subets of projective line under action of PSL(2,q), the projective special linear group. This method is widely used recent 10 years. Experts at home and abroad have made great contribution to the study of the existence of 3-designs by using this method. This thesis is devoted to discussing the existence and construction of simple 4-designs and 5-designs with PGL(2,q) as an automorphism group.This thesis is devided into three chapters.The first chapter is devoted to the summarization of the dissertation. We disscuss the development of the subject, including the history and current situation of it. We also give the main results of this thesis.In the second chapter, we will introduce the elementary concepts and conclusions that will be used in this thesis. Then we can construct the basic theory system of this thesis.In the third chapter, we will introduce the existence of simple 3-designs, and 4-designs with PGL(2,q) as an automorphism group. And we get the main theorem as follows:Main Theorem 1:Let G=PGL(2,q),X=GF(q)∪{∞}, B∈X|7|, if(X,BG)is a 4-(q+1,7,λ)design,then the followings may happen: (1)q=16,λ∈{6,20,60}；(2)q=32,λ∈{14,28}；(3)q=17,λ∈{28,56}；(4)q=23,λ∈{20,40}；(5)q=37,λ∈{4,12,24}；(6)q=107,λ∈{4,8}.Main Theorem 2:Let X=GF(q)U{∞}, G=PGL(2,q), B∈X|6|,if (X,BG)is a 5-(q+1,6,λ)design,then q=11,λ=2.