# Graeco-Latin cubes

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- Denis Krotov
- Last updated:
- Sun, 03/15/2020 - 04:44
- DOI:
- 10.21227/xh3k-pv95
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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 https://arxiv.org/abs/1911.12960 , 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 filename.zip

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.

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doi = {10.21227/xh3k-pv95},

url = {http://dx.doi.org/10.21227/xh3k-pv95},

author = {Vladimir Potapov },

publisher = {IEEE Dataport},

title = {Graeco-Latin cubes},

year = {2020} }

T1 - Graeco-Latin cubes

AU - Vladimir Potapov

PY - 2020

PB - IEEE Dataport

UR - 10.21227/xh3k-pv95

ER -