Given a cyclic word $w$ of length $N$ over a $q$-ary alphabet and $k \in \mathbb{Z}_+$, consider the directed multigraph $G_k(w) = (V,E)$ with $V \subset$ {$1,\dots,q$}$^k$ given by the $k$-lets (i.e., subwords of $k$ symbols) that appear in $w$ (without multiplicity) and $E$ given by the $(k+1)$-lets in $w$ that appear with multiplicity. If $w$ is a de Bruijn sequence, $G_k(w)$ is a de Bruijn graph. So call $G_k(w)$ a generalized de Bruijn graph corresponding to $w$ (and $k$). It is not hard to compute the number of words $w'$ having $G_k(w)$ as their generalized de Bruijn graph, using the matrix-tree and BEST theorems.

In two dimensions, the picture is much less clear. De Bruijn tori are basically periodic rectangular arrays of symbols in which all possible subarrays of a certain size occur with multiplicity 1. There is a hypergraph--the "generalized de Bruijn hypergraph"--corresponding to a generic rectangular array of symbols in a generalization of the sketch above, so by analogy call a rectangular array of symbols over a finite alphabet a generalized de Bruijn torus in this context.

How can the number of arrays corresponding to a generalized de Bruijn hypergraph be enumerated?

(Note that even the existence of de Bruijn tori for nonsquare subarrays is uncertain, which is why I'm working in the "generalized" context.)