Constructions Of Optimal Group Divisible Packings With Block Size Four

Combinatorial design theory is an important branch of discrete mathematics,which is specialized in arranging things according to specific requirements and a knowledge of its nature.Group divisible design(GDD)is a very important concept in combinatorial design,and it is also the basis for many important problems in combinatorial design.In group divisible design,all elements from different groups have the same number of occurrences.In 1977,Brouwer gives a necessary and sufficient condition for the existence of 4-GDD.For a given order,When GDD does not exist,people focus on constructing a very close correlation structure,that is,group divisible packing(GDP).In 1968,Spencer determined the packing number(3,1~n).For general g,in 2001,Yin Jianxing determined the packing number D_?(3,g~n).In 1979,Brouwer determined the packing number(4,1~n).This paper studies the construction of optimal 4-GDP with g~n.This paper is divided into four chapters:In the first chapter,we give the definitions and existing results of group divisible packing and group divisible design,and give an upper bound of the block number of4-GDP.In the second chapter,the definitions of incomplete group divisible design,group divisible design with holes and other related aided designs are given,and the systematic method of constructing group divisible packing is established by using these aided designs.In the third chapter,by using the recursive construction established in the second chapter and combining with the direct construction of small parameter design,the packing numbers of some classes of optimal 4-GDP are determined.The fourth chapter summarizes the main conclusions of the article and further re-search issues.