Group divisible designs of four groups and block size five with configuration (1; 1; 1; 2)

Abstract

We present constructions and results about GDDs with four groups and block size five in which each block has Configuration (1; 1; 1; 2), that is, each block has exactly one point from three of the four groups and two points from the fourth group. We provide the necessary conditions of the existence of a GDD(n; 4; 5; _1; _2) with Configuration (1; 1; 1; 2), and show that the necessary conditions are sufficient for a GDD(n; 4; 5; _1; _2) with Configuration (1; 1; 1; 2) if n 6_ 0(mod 6), respectively. We also show that a GDD(n; 4; 5; 2n; 6(n 􀀀 1)) with Configuration (1; 1; 1; 2) exists, and provide constructions for a GDD(n = 2t; 4; 5; n; 3(n 􀀀 1)) with Configuration (1; 1; 1; 2) where n 6= 12, and a GDD(n = 6t; 4; 5; 4t; 2(6t 􀀀 1)) with Configuration (1; 1; 1; 2) where n 6= 6 and 18, respectively.