Generalized Group Divisible Designs And Generalized Group Divisible Coverings

Group divisible design is a very important combinatorial configuration,which can be used to construct other combinatorial designs such as balanced incomplete block design,and also to construct constant-weight code and constant-composition code.Therefore,group divisible design has important applications in combinatorial design theory and coding.In 2020,Han Yue proposed the concepts of generalized group divisible t-design and generalized group divisible t-packing as a generalization of group divisible design and group divisible packing.She studied the structural characteristics,gave some direct and recursive construction methods,and obtained the existence of generalized group divisible t-designs and generalized group divisible t-packings for some small parameters.Based on the research of Han Yue,we will continue to generalize the concept of generalized group divisible t-design in this paper.We will give the definitions of generalized group divisible t-design with variable index At and generalized group divisible t-covering,and discuss the structural characteristics and existence problems of them.The main contents of this paper are as follows:In the first chapter,we introduce some basic concepts and existence results of group divisible design and group divisible covering.The definitions of generalized group divisible t-design with variable index At and generalized group divisible t-covering are presented,and the research problems in this paper are proposed.In the second chapter,we study generalized group divisible t-designs with variable index ?_t,give some basic properties and general conclusions for t=2.In particular,we show the existence of generalized group divisible t-designs with variable index ?_t for t=2 with block size 3,4,and t=3 with block size 4.In the third chapter,we consider generalized group divisible t-coverings.Firstly,we introduce some auxiliary designs such as strong group divisible covering(SGDC).Then,we discuss the parameters and basic properties of generalized group divisible tcoverings.Specially,we give some lower bounds of the number of blocks for generalized group divisible t-coverings.Finally,we establish the existence of optimal generalized group divisible t-coverings with some small parameters.