Graeco-Latin cubes

Citation Author(s):
Sobolev Institute of Mathematics
Submitted by:
Denis Krotov
Last updated:
Mon, 10/02/2023 - 23:49
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A Latin square of order q is a q×q array with elements from {0,1,...,q-1} such that each value occurs exactly once in each row and column. Two Latin squares G and L are orthogonal if (G(x,y), L(x,y)) = (u,v) has exactly one solution for each (u,v) from {0,1,...,q-1}×{0,1,...,q-1}. The square obtained by superimposing two orthogonal Latin squares to get entries with pairs of elements is called a Graeco-Latin square. A Latin cube of order q and is a q×q×q array with elements from {0,1,...,q-1} such that when any two coordinates are fixed, each element of {0,1,...,q-1} occurs exactly once. Two Latin cubes are orthogonal if fixing any coordinate gives a Graeco-Latin square when the cubes are superimposed. The cube obtained by superimposing two orthogonal Latin cubes is called a Graeco-Latin cube. Graeco-Latin cubes are equivalent to orthogonal arrays OA(q^3,5,q,2) or to (5,q^3,3)q codes, q-ary unrestricted (not necessarily linear or additive) MDS codes of length 5 and minimum distance 3. Two standard methods to construct Graeco-Latin cubes are the following:

(i) if q is a prime power larger than 3, than a q×q×q Graeco-Latin cube can be constructed as a Reed-Solomon code over the finite field GF(q);

(ii) from a q'×q'×q' Graeco-Latin cube and a q''×q''×q'' Graeco-Latin cube, we can construct a q'q''×q'q''×q'q'' Graeco-Latin cube by Cartesian product.

It is easy to see that with only these two constructions one cannot construct a Graeco-Latin cube of order 2p, where p is odd, or 3p, where p is not divisible by 3.  In , there is a construction that allows to construct Graeco-Latin cubes of order 3p for some p not divisible by 3. This dataset contains examples of Graeco-Latin cubes of first few orders such that Graeco-Latin cubes of the same order cannot be constructed as a Reed-Solomon code or by Cartesian product from cubes of smaller orders.


The archives are *.zip, but the algorithm is LZMA. If unzip does not work, use 7z:

7z e

A cube of order q is saved in a archived text file with q*q*q lines. A line number q*q*x+q*y+z contains the two values in the cell (x,y,z) of the Graeco-Latin cube, in the hexadecimal format.


Final paper:
V. N. Potapov. Constructions of pairs of orthogonal latin cubes. Journal of Combinatorial Designs, 28(8), 2020, 604-613.
Funding information: Russian Science Foundation, Grant 18‐11‐00136

Submitted by Denis Krotov on Tue, 12/01/2020 - 14:57