Which Graeco-Latin hypercubes are impossible? Define a Graeco-Latin hypercube of dimension $n$ and order $k$ as an $n$-dimensional grid, with $k$ cells in each direction (for a total of $k^n$ cells), where:


*

*Each cell contains an ordered tuple $(x_1, x_2, x_3, ..., x_n)$ where each $x_i$ is a number from $1$ to $k$.

*For each row in any direction, no number is repeated in the same position on any two ordered tuples.

*Each possible ordered tuple is represented exactly once in the hypercube.
The case of $n = 2$ and $k = 6$ is the 36 officers problem, which Euler proved was impossible. Are there any other cases known to be impossible? Has there been any research done on this topic?
 A: Here is a selection of recent papers on orthogonal Latin hypercube designs. I'm not sure these are the same as Graeco-Latin hypercubes, but surely these papers give some idea of what designs are, and are not, possible. 
MR2659850 (2011k:62223) 
Sun, Fasheng; Liu, Min-Qian; Lin, Dennis K. J.
Construction of orthogonal Latin hypercube designs with flexible run sizes. 
J. Statist. Plann. Inference 140 (2010), no. 11, 3236–3242. 
MR3183676 
Georgiou, Stelios D.; Efthimiou, Ifigenia
Some classes of orthogonal Latin hypercube designs. 
Statist. Sinica 24 (2014), no. 1, 101–120. 
MR3377513 
Cao, Rui-Yuan; Liu, Min-Qian
Construction of second-order orthogonal sliced Latin hypercube designs. 
J. Complexity 31 (2015), no. 5, 762–772. 
MR3254915 
Georgiou, S. D.; Stylianou, S.; Drosou, K.; Koukouvinos, C.
Construction of orthogonal and nearly orthogonal designs for computer experiments. 
Biometrika 101 (2014), no. 3, 741–747. 
MR3183339 
Huang, Hengzhen; Yang, Jian-Feng; Liu, Min-Qian
Construction of sliced (nearly) orthogonal Latin hypercube designs. 
J. Complexity 30 (2014), no. 3, 355–365. 
MR3183681 
Yang, Jinyu; Liu, Min-Qian; Lin, Dennis K. J.
Construction of nested orthogonal Latin hypercube designs. 
Statist. Sinica 24 (2014), no. 1, 211–219. 
MR2933184 
Yang, Jinyu; Liu, Min-Qian
Construction of orthogonal and nearly orthogonal Latin hypercube designs from orthogonal designs. 
Statist. Sinica 22 (2012), no. 1, 433–442. 
MR2861300 (2012j:05072)
Sun, FaSheng; Pang, Fang; Liu, MinQian
Construction of column-orthogonal designs for computer experiments. 
Sci. China Math. 54 (2011), no. 12, 2683–2692. 
