The Existence Of Optimal Group Divisible Packings With Block Size Four

Combinatorial design theory is an important branch of combinatorial mathematics,which has a wide range of connections with the fields of coding theory and cryptography.Group divisible t-design(t-GDD)is a classic design,and it is an important tool to construct other combinatorial designs.In this paper,we study the existence of group divisible t-packing(t-GDP)on the basis of group divisible t-design.It is known that a GDP(t,k,gn)of type gn.can be used to construct a(g+1)-ary constant-weight code with weight k,code length n and minimum Hamming distance k-t+1.Therefore,the group divisible t-packing has attracted much attention.Due to limit of difficulty,the uniform group divisible t-packing is the main topic.When t=2,Yin Jianxing determined the packing number D(2,3,gn)of GDP(2,3,gn)of type gn.In recent two years,Feng Qi and Zhou Pinpin discussed the packing number D(2,4,gn)of GDP(2,4,gn)of type gn and gave some existence results.When t=3,Zhou Wenchang studied the existence of GDP(3,4,gn)of type gn in 2019.He gave some recursive construction methods and partial of the existence results of packing number D(3,4,gn).In this paper,we will continue to study the existence of uniform optimal GDP with strength t=2,3 and block length 4.The main content of this paper is as follows:In the first chapter,some auxiliary designs are given,such as incomplete group divisible design(IGDD),holey group divisible design(HGDD),and double group divisible design(DGDD),and some recursive construction methods of group divisible t-packing are given by using these auxiliary designs.In the second chapter,we study the case of t=2,using the recursive construction methods given in the first chapter and combining with the method of computer search,we give some small parameter designs,and further perfect the results of optimal GDP(2,4,gn)of type gn,the exact values of several packing numbers D(2,4,gn)are determined.In the third chapter,we study the case of t=3 by using HGDDs,DGDDs and the recursive constructions.Combining with the small parameter designs,we determine the exact values of the packing number D(3,4,gn)for GDP(3,4,gn)of type gn with n≡0(mod 2)and g is a positive integer,or n≡1(mod 2)and g is even.The fourth chapter summarizes the main results of this paper and points out the problems that need further research.