<s>[INST] De Bruijn tori in higher dimensions? 
Q. Do there exist De Bruijn tori in dimension $d > 2$?

A De Bruijn torus
is a two-dimensional generalization of a 
De Bruijn sequence.
A De Bruijn sequence is, for two symbols,
a cyclical bit-string that contains all bit strings of length $n$
as consecutive, left-to-right bits (with wrap-around).
For example, here is a sequence of $8$ bits that contains all $2^3$ bit
strings of length $3$:
$$
\begin{matrix}
\color{red}{0} & \color{red}{0} & \color{red}{0} & 1 & 1 & 1 & 0 & 1\\
0 & \color{red}{0}  & \color{red}{0}  & \color{red}{1} & 1 & 1 & 0 & 1\\
\color{red}{0} & 0 & 0 & 1 & 1 & 1 & \color{red}{0} & \color{red}{1}\\
0 & 0 & \color{red}{0} & \color{red}{1} & \color{red}{1} & 1 & 0 & 1\\
\color{red}{0} & \color{red}{0} & 0 & 1 & 1 & 1 & 0 & \color{red}{1}\\
0 & 0 & 0 & 1 & 1 & \color{red}{1} & \color{red}{0}  & \color{red}{1}\\
0 & 0 & 0 & 1 & \color{red}{1} & \color{red}{1} & \color{red}{0}  & 1\\
0 & 0 & 0 & \color{red}{1} & \color{red}{1} & \color{red}{1} & 0 & 1
\end{matrix}
$$
Here is a De Bruijn torus that includes all $2 \times 2$ bit-matrices 
exactly once (from here):
$$
\begin{matrix}
0 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 \\
1 & 1 & 1 & 0 \\
0 & 0 & 1 & 0
\end{matrix}
$$
A $4 \times 4$ De Bruijn torus has been explicitly constructed. 
My question is: Is it known that there exist 
De Bruijn tori in dimensions larger than $d=2$?
Perhaps for every dimension?
For example, a three-dimensional pattern of bits that includes
every $k \times k \times k$ bit-(hyper)matrix?
 [/INST] Yes. See "New constructions for de Bruijn tori" by Hurlbert and Isaak
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