Mutually orthogonal Latin hypercubes

Question

A $d$-dimensional Latin hypercube with side length $n$ is a $d$-dimensional array with $n$ symbols such that along any line parallel to an axis, each symbol appears exactly once. Let us call a $(n,d)$ mutually orthogonal Latin hypercube (MOLH) a set of $d$ $d$-dimensional Latin hypercubes of side length $n$ such that overlaying the $d$ hypercubes results in every element of the Cartesian product of the symbols appearing exactly once. For example, $(n,2)$ MOLH are the traditional Graeco-Latin squares.

I am interested in the case where $n=d+1$. $(3,2)$ MOLHs exist, because there are Graeco-Latin squares of side length 3. But are there $(d+1,d)$ MOLHs for $d\ge3$? I haven't been able to find this answered in the literature on the subject.

Thanks for any help.

Answer 1

0123  1032  2310  3201
1032  2310  3201  0123
2310  3201  0123  1032
3201  0123  1032  2310

----------------------

0231  3102  1320  2013
1320  2013  0231  3102
3102  1320  2013  0231
2013  0231  3102  1320

----------------------

3210  2301  1032  0123
1032  0123  3210  2301
0123  3210  2301  1032
2301  1032  0123  3210

answers the question in the affirmative for $d=3$. This is one of 493056 distinct (but not necessarily non-isomorphic) solutions where the first cube is one of the five representatives of the main classes and the other two are considered as an unordered pair of cubes derived from one not later than the first one in the list of representatives.

Since there are considerably more 4-dimensional Latin hypercubes of side length 5, I expect there to be correspondingly more $(5, 4)$ MOLHs, but the computational task of finding them is rather larger.