described by Stinson [@Stinson].

We will refer to any ${\mathrm{GHD}}_k^*(s,v)$ as having the [*$*$-property*]{}. As our primary focus is on the case that $k=3$, we will omit the subscript $k$ in this case. Note that trivially any ${\mathrm{GHD}}(s,2s+1)$ (i.e. Kirkman square) has the $*$-property. It is also clear that any ${\mathrm{GHD}}(s,2s+2)$ (i.e. DRNKTS) has the as the values of the parameters force there to be an unused pair of points. In the case of a ${\mathrm{GHD}}(s,3s)$ formed by superimposing three ${\mathrm{MOLS}}(s)$, the unused pairs of points are those which occur within the three groups of size $s$ in the corresponding transversal design; one of these groups may be taken as $Y$, so any ${\mathrm{GHD}}(s,3s)$ formed in this manner is a ${\mathrm{GHD}}^*(s,3s)$. In addition, the existence of a decomposable ${\mathrm{SOMA}}(3,6)$ [@PhillipsWallis] and ${\mathrm{SOMA}}(3,10)$ [@Soicher] yield the existence of ${\mathrm{GHD}}^*(6,18)$ and ${\mathrm{GHD}}^*(10,30)$. Thus, we have the following.

\[GHD\*\]

1.  There exists a ${\mathrm{GHD}}^*\left(\left\lfloor \frac{v-1}{2} \right\rfloor,v \right)$ for any $v\equiv 0 \pmod 3$, whenever $v=3$ or $v \geq 18$ and $v \notin N$.

2.  There exists a ${\mathrm{GHD}}^*(n,3n)$ if and only if $n \neq 2$ or $3$.

The second type of hole in a GHD is an [*array hole*]{}. To define it, we 