T—Design And Multiple Delivery Groups

The study of t-designs(t? 3)is a very difficult problem in the field of combinatorial designs,especially the study of existence and construction of simple t-designs.The theory of permutation groups helps much in study of simple t-designs which have an automorphism group multiply transitive.Since the subgroup structure of the projective special linear group PSL(2,q)(q=pn)is well-studied,so it is quite popular in the study of the existence and construction of t-designs with an automorphism group.So far,the existence of simple 3-designs with block size k with k?0 orl(mod p)has been completely finished.Note that when p-2,i.e.q-2n,it always follows that k?0 or1(mod p).This dissertation focuses on the existence of simple 3-designs with PSL(2,2n)as an automorphism group.In the first chapter,we present a summary of the background of t-designs,the main work of this thesis and some related theoretical knowledge.In the second chapter,for any positive integer d where d|(2n-1)and d?3 and any odd l,we give an infinite family of simple 3-designs with block size 2ld+1 by determining the length of the orbits containing a 2ld+1-subset which is fixed by an element of order d under the action of PSL(2,2n).Then by calculating the number of orbits with this length,more simple t-designs are obtained.In the third chapter,by determining the length of the orbits containing a 2d-subset and 2d+2(d?5)-subset each of which is fixed by an element of order d under the action of PSL(2,2n),two infinite families of simple 3-designs with block size 2d and 2d+2(d?5)are given,respectively.