Optimal Holely Packing Designs

A holely packing with parameters(n,g,k,t)(HP(n,g,k,t))is a triple(X,g,B),where X is a set of ng elements;g ={G1,…,Gn} is a equipartition of X(called a group set);and B is a set of k-subsets of X(called blocks)satisfying:(1)|B ?G| ?1 for any B?G,G?g;(2)every t-subset of X from different groups is contained in at most one block.The packing number is the maximum number of blocks in an HP(n,g,k,t).The holely packing design is a generalization of group divisible t design.There exists a close relationship between an HP(n,g,k,t)and a(q+1)-ary constant weight code of length n,weight k,distance k-t+1.So determining the packing number is important.This thesis mainly investigates the existence of HP(n,g,4,3).In Chapter 2,we discuss the HP(n,g,4,3)with even group size.Firstly,by intro-ducing auxiliary designs:lattice designs,lattice HP(n,g,k,t),and by using 1-fan designs,we establish a construction of HPs.Secondly,starting with Steiner quadruple systems and applying the construction of HPs,we determine the existence of HP(6,g,4,3)for g?2,4(mod 6).Then starting with G-designs,we obtain an optimal HP(n,g,4,3)for n?0(mod 6),g?2,4(mod 6).Thirdly,starting with H(n,2,{4,6},3)and applying the construction,we almost determine the existence of HP(n,g,4,3)for n?3(mod 6),g?2,4(mod 6).Finally,it is proved that for any positive integers n,g,n?4,g?0(mod 2),and(n,g)(?){(a,b):a?3(mod 6),b? {26,38}},an optimal HP(n,g,4,3)exists.In Chapter 3,we discuss the HP(n,g,4,3)with n?1(mod 4)and g=3.First-ly,we prove that there exists an optimal HP(n,3,4,3)if and only if there exists a GDD(3,4,3n+1)of type 3n11.Secondly by using 2-fan designs and a pair of weak matching partial candelabra quadruple systems,a construction of GDD(3,4,3n+1)of type 3n11 is established.Finally,it is proved that an optimal HP(n,3,4,3)exists if n?0,1,2,4,6(mod 8)and n?17.