with a hole. The notation $t$ ${\mathrm{IMOLS}}(s,a)$ is used for $t~{\mathrm{MOLS}}(s)$, each of which has a hole of side length $a$ on the same set of positions. Also, the notation $t~{\mathrm{IMOLS}}(s,a,b)$ is used for a set of $t~{\mathrm{MOLS}}(s)$ with two disjoint holes of orders $a$ and $b$, with two extra properties: (1) the two Latin holes have no symbols in common; (2) no common row or column within the larger square intersects both array holes.

Latin holes and array holes also both feature in the concept of frames, which are described in Section \[FrameConstr\].

Construction methods {#ConstrSection}
====================

Frames {#FrameConstr}
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In this section, we discuss a useful method, that of frames, which will allow us to construct infinite families of GHDs. The central idea has been used to construct GHDs on both ends of the spectrum. However, the terminology in the literature varies: for MOLS and GHDs with few empty cells, authors often refer to HMOLS [@AbelBennettGe; @BennettColbournZhu; @WangDu], while for doubly resolvable designs authors often speak of frames [@CLLM; @blue; @book]. We use the latter terminology, as it is more easily adaptable to more general kinds of GHDs. First we begin with a definition.

\[Frame Definition\] Let $s$ 