2}, \ldots, \infty_{\ell, v_{\ell}}$.

It is straightforward to verify that the resulting array $A'$ is a GHF of type $h^mv^1$.

\[n=68 lemma\] Let $h$, $m$, $x$ and $t$ be positive integers. Suppose there exist $3+x~{\mathrm{MOLS}}(m)$ and $3~{\mathrm{MOLS}}(h)$. For each $1 \leq i \leq x-1$, let $w_i$ be a non-negative integer such that there exist $3~{\mathrm{IMOLS}}(h+w_i, w_i)$. Then if $w=w_1+ w_2+ \cdots+w_{x-1}$, there exists a GHF of type $h^{m-1}(m+w)^1$.

To apply Lemmas \[n=38 lemma\] and \[n=68 lemma\], we need some information on existence of $3+x~{\mathrm{MOLS}}(m)$, $4~{\mathrm{MOLS}}(m)$ and $3~{\mathrm{IMOLS}}(h+a,a)$. These existence results are given in the next two lemmas.

\[n=14 lemma\] There exist $4~{\mathrm{MOLS}}(m)$ for all integers $m \geq 5$ except for $m=6$ and possibly for $m \in \{10,22\}$. Also, if $m$ is a prime power, there exist $m-1~{\mathrm{MOLS}}(m)$.

See [@Todorov] for $n=14$, [@Abel] for $n=18$ and [@Handbook Section III.3.6] for other values.

\[n=10 lemma\] [@Handbook Section III.4.3] Suppose $y,a$ are positive integers with $y \geq 4a$. Then $3~{\mathrm{IMOLS}}(y,a)$ exist, except for $(y,a) = (6,1)$, and possibly for $(y,a) = (10,1)$.

Most of the frames that we require in Section \[TwoEmptySection\] are obtained using Lemma \[n=38 lemma\], but Theorem \[Uniform Frames\] and Lemma \[n=68 lemma\] will also be used. The majority of 