Generalized Group Divisible T-designs And T-packings

Group divisible t-design is a very important combinatorial configuration in combina-torial design theory,which has been widely used in constructing various kinds of designs,such as t-design.In 2009,Peter J Cameron gave the concept of generalized t-design as a generalization of t-design.In 2013,Robert F Bailey and Andrea C Burgess gave the concept of generalized t-packing.In this paper,we present the concepts of generalized group divisible t-design and generalized group divisible t-packing based on the relation-ship between t-design and generalized t-design.In particular,the generalized t-design(t-packing)is a generalized group divisible t-design(t-packing)with group size 1.In this paper,we mainly study the constructions and existence of generalized group divisible t-design and generalized group divisible t-packing.The full text is divided into four chapters:In Chapter 1,we first introduce some basic concepts and related results,then we give the definitions of generalized group divisible t-design and generalized group divisible t-packing.In Chapter 2,we introduce the definition of strong group divisible packing(SGDP)and establish some recursive constructions about it.Chapter 3 is divided into two sections.In the first section,we give some results on parameter relations and the existence of generalized group divisible t-designs for small parameters;In the second section,we propose the upper bound of the number of blocks for generalized group divisible t-packing and construct some optimal generalized group divisible t-packings.Chapter 4 summarizes the conclusions of this paper and lists some problems for further study.