and one ($v=351$) was solved in [@ACCLWW]. Abel, Chan, Colbourn, Lamken, Wang and Wang [@ACCLWW] have determined the existence of doubly resolvable nearly Kirkman triple systems, with 64 possible exceptions. We thus have the following result.

\[Kirkman squares\] Let $v \equiv 0 \pmod{3}$ be a positive integer. Then a ${\mathrm{GHD}}(\lfloor\frac{v-1}{2}\rfloor,v)$ exists if and only if either $v=3$ or $v\geq 18$, with 75 possible exceptions. The symbol $N$ will be used to denote this set of possible exceptions throughout the paper.

Recently, Du, Abel and Wang [@DuAbelWang] have proved the existence of ${\mathrm{GHD}}(\frac{v-4}{2},v)$ for $v \equiv 0 \pmod{12}$ with at most 15 possible exceptions and ${\mathrm{GHD}}(\frac{v-6}{2},v)$ for $v \equiv 0 \pmod{6}$ and $v>18$ with at most 31 possible exceptions. Two of these possible exceptions (${\mathrm{GHD}}(9,24)$ and ${\mathrm{GHD}}(12,30)$) will be addressed later in this paper.

At the other end of the spectrum, the case when $s=n$, $v=kn$, $t=2$ and $\lambda=1$ is known as a ${\mathrm{SOMA}}(k,n)$. SOMAs, or [*simple orthogonal multi-arrays*]{}, were introduced by Phillips and Wallis [@PhillipsWallis] and have been investigated by Soicher [@Soicher] and Arhin [@ArhinThesis; @Arhin]. We note that the existence of a $2$-${\mathrm{GHD}}_{k}(n,kn;1)$ is guaranteed by the existence of $k$ mutually orthogonal Latin squares (MOLS) of side $n$. (For 