analogous to the HMOLS used in [@AbelBennettGe; @BennettColbournZhu; @WangDu].

[@AbelBennett; @AbelBennettGe] \[Uniform Frames\] Let $h\geq 1$ and $u\geq 5$ be integers. Then there exists a GHF of type $h^u$ except for $(h,u)=(1,6)$, and possibly for $(h,u) \in \{(1,10), (3,28), (6,18)\}$.

The following two lemmas are similar to Lemmas 2.2 and 2.3 in [@BennettColbournZhu] where they were written in the language of transversal designs with holes and holey MOLS. More information on the construction methods used to establish them can be found in [@BrouwerVanRees]. For the sake of clarity, we give a brief proof here of Lemma \[n=38 lemma\]; the proof of Lemma \[n=68 lemma\] is similar.

\[n=38 lemma\] Let $h,m$ be positive integers such that $4~{\mathrm{MOLS}}(m)$ exist. Let $v_1, v_2, \ldots, v_{m-1}$ be non-negative integers such that for each $i=1,2,\ldots,m-1$, there exist $3~{\mathrm{IMOLS}}(h+v_i, v_i)$. Then for $v=v_1+v_2+\cdots+v_{m-1}$, there exists a GHF of type $h^mv^1$.

Take the first three ${\mathrm{MOLS}}(m)$ on disjoint symbol sets and superimpose them to form a ${\mathrm{GHD}}(m,3m)$, $A$. These MOLS possess $m$ disjoint transversals, $T_0, T_1, \ldots, T_{m-1}$, since the cells occupied by any symbol in the fourth square form such a transversal. Permuting rows and columns if necessary, we can assume that $T_0$ is on the 