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

TL;DR

This paper investigates the existence of group divisible designs with a fixed first index of 3 and a larger second index, establishing necessary and sufficient conditions for most cases.

Contribution

It provides a comprehensive set of necessary and sufficient conditions for the existence of GDDs with λ₁=3 and large λ₂, advancing the understanding of these combinatorial structures.

Findings

Necessary conditions for existence are established.

Most cases of the conditions are proven to be sufficient.

The results extend the theory of group divisible designs with specific index parameters.

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.