3-Group Divisible Designs with 3 Groups and Block Size 5

Abstract

A $3$ -GDD ( $n$ , $2$ , $k$ , ${\lambda }_1$ , ${\lambda }_2$ ) was defined by combining the definitions of a group divisible design and a t-design. In this paper, we extend the definitions to 3 groups and block size 5, and we denote such GDD by $3$ -GDD ( $n$ , $3$ , $5$ , ${\mu }_1$ , ${\mu }_2$ ). Some necessary conditions for the existence of these GDDs are developed, and several new constructions and specific instances of nonexistence are given.