Minimal Enclosings of Group Divisible Designs with Block Size 3 and Group Size 2

Abstract

We obtain necessary conditions for the enclosing of a group divisible design with block size 3, $\text{GDD}(n,m;\lambda )$, into a group divisible design $\text{GDD}(\text{n},\text{m+1};\lambda +\text{x})$ with one extra group and minimal increase in $\lambda$. We prove that the necessary conditions are sufficient for the existence of all such enclosings for GDDs with group size 2 and $\lambda \le 6$, and for any $\lambda$ when $v$ is sufficiently large relative to $\lambda$.