and $v$ be integers, $X$ be a set of $v$ points, $\{G_1, \ldots, G_n\}$ be a partition of $X$, and $g_i=|G_i|$. Also let $s_1, \ldots, s_n$ be non-negative integers with $\sum_{i=1}^{n} s_i = s$. A [*generalized Howell frame*]{}, ${\mathrm{GHF}}_k$, of type $(s_1,g_1)\ldots (s_n, g_n)$ is a square array of side $s$, $A$, that has the following properties:

1.  Every cell of $A$ is either empty or contains a $k$-subset of elements of $X$. The filled cells are called [*blocks*]{}.

2.  No pair of points from $X$ appear together in more than one block, and no pair of points in any $G_i$ appear together in any block.

3.  The main diagonal of $A$ consists of empty $s_i\times s_i$ subsquares, $A_i$.

4.  Each row and column of the array with empty diagonal entry $A_i$ is a resolution of $X\setminus G_i$.

We will use an exponential notation $(s_1,g_1)^{n_1}\ldots (s_t,g_t)^{n_t}$ to indicate that there are $n_i$ occurrences of $(s_i, g_i)$ in the partition. In the case that $k=3$ and $g_i=3s_i$, we will refer to a GHF of type $s_1s_2\cdots s_n$, and will use exponential notation $s_1^{\alpha_1} \cdots s_r^{\alpha_r}$ to indicate that there are $\alpha_i$ occurrences of $s_i$. Note that this concept of frame is 