We show that each point occurs in some cell of each row. In the $e$ new rows, the points in $U$ appear in the blocks from the ${\mathrm{GHD}}_k(s_m+e,g_m+u)$, and the points in each $G_i$ occur in the columns corresponding to the ${\mathrm{GHD}}_k(s_i+e,g_i+u)$ for $1 \leq i \leq m$. In a row which includes part of the $s_i \times s_i$ subsquare corresponding to the group $G_i$, the elements of $G_j$ ($j \neq i$) appear in the frame, while the elements of $G_i \cup U$ appear in the added ${\mathrm{GHD}}_k(s_i+e,g_i+u)$. In a similar way, each element occurs exactly once in every column.

We will generally use the Basic Frame Construction with $u=0$, so that our ingredient GHDs have an $e \times e$ empty subsquare. For this special case, we have the following.

\[FrameCorollary\] Suppose that: (1) there exists a ${\mathrm{GHF}}_k$ of type $\Pi_{i=1}^m (s_i,g_i)$; (2) for each $i\in \{1,\ldots,m-1\}$ there exists a ${\mathrm{GHD}}_k(s_i+e,g_i)$ containing a trivial subdesign ${\mathrm{GHD}}_k(e,0)$; and (3) there exists a ${\mathrm{GHD}}_k(s_m+e,g_m)$. Then there exists a ${\mathrm{GHD}}_k(s+e,v)$, where $s=\sum_{i=1}^m s_i$ and $v=\sum_{i=1}^m g_i$.

Starter-adder method {#SA section}
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In practice, to apply the Basic Frame Construction, Theorem \[Frame Construction\], described in Section \[FrameConstr\], we need to first obtain small GHDs 