frames required come from the following special cases of Lemma \[n=38 lemma\].

\[Frame 7\^uv\] Let $h,m,v$ be integers with $m \geq 5,$ $m \notin \{6,10,22\}$. Then there exists a GHF of type $h^mv^1$ if either:

1.  $h=6$ and $m-1 \leq v \leq 2(m-1)$;

2.  $h\in \{7,8\}$ and $0 \leq v \leq 2(m-1)$;

3.  $h\in \{9,11\}$, $0 \leq v \leq 3(m-1)$ and $(h,v) \neq (9,1)$.

Apply Lemma \[n=38 lemma\], and let $h$ and $m$ be as given in that lemma. The admissible values of $m$ guarantee the existence of $4~{\mathrm{MOLS}}(m)$ (see Lemma \[n=14 lemma\]). The range for $v= \sum_{i=1}^{m-1} v_i$ is easily determined from the feasible values of $v_i$. We require that $3~{\mathrm{IMOLS}}(h+v_i, v_i)$ exist, so in light of Lemma \[n=10 lemma\], we may take $v_i \in \{1,2\}$ when $h=6$, $v_i \in \{0, 1,2\}$ when $h \in \{7,8\}$, $v_i \in \{0,2,3\}$ when $h=9$ and $v_i \in \{0,1,2,3\}$ when $h=11$.

We next discuss how to construct GHDs from frames. The following result is a generalization of the Basic Frame Construction for doubly resolvable designs; see [@ACCLWW; @CLLM; @Lamken95].

\[Basic Frame Construction\] \[Frame Construction\] Suppose there exists a ${\mathrm{GHF}}_k$ of type $\Pi_{i=1}^m (s_i,g_i)$. Moreover, suppose that for $1 \leq i \leq m$, there 