Group Divisible Designs with $\lambda_1=3$ and Large Second Index

Abstract

A group divisible design $\mbox{GDD}(m,n;\lambda_1,\lambda_2)$, is an ordered pair $(V, \cal{B})$ where $V$ is an $(m+n)$-set of symbols while $\cal{B}$ is a collection of $3$-subsets (called blocks) of $V$ satisfying the following properties: the $(m+n)$-set is divided into 2 groups of size $m$ and of size $n$: each pair of symbols from the same group occurs in exactly $\lambda_1$ blocks in $\cal{B}$, and each pair of symbols from different groups occurs in exactly $\lambda_2$ blocks in $\cal{B}$. $\lambda_1$ and $\lambda_2$ are referred to as first index and second index, respectively. Here, we focus on an existence problem of $\mbox{GDD}$s when $\lambda_1=3$ and $\lambda_2>3$. We obtain the necessary conditions and prove that these conditions are sufficient for most of the cases.