Incomplete Group Divisible Designs With Block Size Four

Incomplete group divisible designs (IGDDs) have been studied by many researchers. This design is instrumental in the construction of various combinatorial design problems such as GDD, MOLS, etc..The main purpose of this paper is to investigate the existence spectrum of incomplete group divisible designs with block size four, group-type (g, h)~u. There are five parts in this paper. Firstly, we state the definition and background of IGDD. Secondly, we introduce some definitions and notations as well as preliminary results which will be used in the sequel. Thirdly, we show some recursive constructions for IGDD. Fourthly, we give the existence spectrum for the caseλ= 3. Finally, we shall show that the necessary conditions for the existence of incomplete group divisible designs with block size four, group-type (g, h)~u are also sufficient with 3 exceptions and 6 possible exceptions.