set $H$ of points such that no pair of elements of $H$ appears in the blocks, while a hole in a Latin square means an empty subsquare, so that certain rows and columns do not contain all symbols. In the case of GHDs, three types of “hole” may occur, and each will be of importance later in the paper. We thus develop appropriate terminology to differentiate the uses of the term “hole”.

First, a [*pairwise hole*]{} in a GHD is a set of points $H \subseteq X$ with the property that no pair of points of $H$ appear in a block of the GHD. Thus, a pairwise hole corresponds with the notion of a hole in a BIBD.

A [*${\mathrm{GHD}}_k^*(s,v)$*]{}, $\cal{G}$, is a $2$-${\mathrm{GHD}}_k(s,v;1)$ with a pairwise hole of size $v-s(k-1)$. Thus the symbol set $X$ of a ${\mathrm{GHD}}_k^*(s,v)$ can be written as $$X=Y \cup (S \times \{1,2,\ldots,k-1\}),$$ where $|S|=s$, $|Y|=v-s(k-1)$, and $Y$ is a pairwise hole of $\cal{G}$.

Note that this definition extends in a natural way to higher $\lambda$; however, for our purposes it will be enough to only consider the case that $\lambda=1$. Also note that in the case that $k=2$, a ${\mathrm{GHD}}_2^*(s,v)$ is precisely the $\mathrm{H}^*(s,v)$ 