Define a Latin hypercube of order n and dimension d to be a set of nd tuples (x1,x2,...,xd+1) over {0,1,...,n-1} such that each of the nd possible d-tuples appears exactly once in each set of d coordinate positions.
Latin hypercubes of dimension 1, 2 and 3 are commonly called
permutations, Latin squares, and Latin cubes,
respectively. The hypercube can be represented
as a d-dimensional array C with
C[x1,x2,...,xd]
= xd+1 for each (d+1)-tuple.
In the example below, we use this array representation to show a
Latin hypercube with n=4 and d=3.
| 0123 1032 2301 3210 |
1032 0123 3210 2301 |
2301 3210 0123 1032 |
3210 2301 1032 0123 |
| C[i,j,0] | C[i,j,1] | C[i,j,2] | C[i,j,3] |
A Latin hypercube is reduced if, whenever d-1 entries of a tuple are 0, the other two entries are equal. In the case of Latin squares, this is equivalent to the first row and column being in natural order. The example above is reduced.
Given a Latin hypercube A={(x1,...,xd+1)} and permutations (p1,...,pd+1) of {0,1,...,n-1}, then B={(p1(x1),...,pd+1(xd+1))} is also a Latin hypercube, said to be isotopic to the first. This is an equivalence relation, and the equivalence classes are called isotopy classes. If in fact A=B, then (p1,...,pd+1) is called an autotopism of A. The autotopisms of A form a group under composition, called the autotopy group Is(A).
Given a Latin hypercube A={(x1,...,xd+1)} and a permutation q of {1,2,...,d+1}, then B={(y1,...,yd+1)} is also a Latin hypercube, where yi=xq(i) for each i, is also a Latin hypercube, called a conjugate of A.
The operation of combining a conjugacy and an isotopism is called a paratopism. Paratopism is an equivalence operation whose equivalence classes are called main classes. A paratopism which takes a Latin hypercube A onto itself is an autoparatopism of A. The autoparatopisms of A form a group under composition, called the autoparatopy group Par(A). The autotopy group is a subgroup of the autoparatopy group.
The relative counts of Latin hypercubes and their isotopy or main classes depends on the group sizes:
For Latin squares, see the Latin squares page. Here we give one representative from each main class of Latin hypercubes for some small orders and dimensions. The files have one line for each main class, containing a hypercube, then |Is(A)| and |Par(A)|/|Is(A)|. The hypercube is given as a list of the values C[x1,x2,...,xd] for subscripts in lexicographic order [0,0,...,0], [0,0,...,1], ..., [n-1,n-1,...,n-1]. (See above for the definition of C.)
This data was computed by Brendan McKay and Ian Wanless.
| n | d | reduced hypercubes |
isotopy classes |
main classes |
| 4 | 3 | 64 | 12 | 5 |
| 4 | 4 | 7132 | 328 | 26 |
| 4 | 5 | 201538000 | 2133586 | 4785 |
| 5 | 3 | 40246 | 59 | 15 |
| 5 | 4 | 31503556 | 5466 | 86 |
| 5 | 5 | 50490811256 | 1501786 | 3102 (gzipped) |
| 6 | 3 | 95909896152 | 5678334 | 264248 (gzipped) |
B. D. McKay and I. M. Wanless, A census of small Latin hypercubes, SIAM Journal on Discrete Mathematics 22 (2008) 719-736. preprint in PDF
Page Master: Brendan McKay, brendan.mckay@anu.edu.au