GDD$(n_1,n,n+1,4;{\lambda }_1,{\lambda }_2)$:$n_1=1or2$

Abstract

The subject matter for this paper is GDDs with three groups of sizes $n_1,n(n\ge n_1)$ and $n+1$, for $n_1=1or2$ and block size four. A block having Configuration $(1,1,2)$ means that the block contains 1 point from two different groups and 2 points from the remaining group. a block having Configuration $(2,2)$ means that the block has exactly two points from two of the three groups. First, we prove that a GDD$(n_1,n,n+1,4;{\lambda }_1,{\lambda }_2)$ for $n_1=1or2$ does not exist if we require that exactly halh of the blocks have the Configuration $(1,1,2)$ and the other half of the blocks have the configuration $(2,2)$ except possibly for n=7 when $n_1=2$. Then we provide necessary conditions for the existence of a GDD$(n_1,n,n+1,4;{\lambda }_1,{\lambda }_2)$ for $n_1=1and2$, and prove that these conditions are sufficient for several families of GDDs. For $n_1=2$, these usual necessary conditions are not sufficient in general as we provide a glimpse of challenges which exist even for the case of $n_1=2$. A general results that a GDD$(n_1,n_2,n_3,4;{\lambda }_1,{\lambda }_2)$ exists if $n_1+n_2+n_3=0,4$ $(mod12)$ is also given.