
Encyclopaedia of DesignTheory: Mutually orthogonal Latin squares 
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A similar representation works for sets of mutually orthogonal Latin squares. Assume that the symbols in the squares are 1,2,...,n, and let L_{1},...,L_{r} be the squares. Then let S be the set of n^{2} (r+2)tuples of the form (i,j,k_{1},...,k_{r}), where the symbol in row i and column j of the square L_{t} is k_{t}.
For example, given the orthogonal Latin squares


we obtain the following nine quadruples:
1  1  1  1 
1  2  2  2 
1  3  3  3 
2  1  2  3 
2  2  3  1 
2  3  1  2 
3  1  3  2 
3  2  1  3 
3  3  2  1 
This is an orthogonal array of degree 4, strength 2 and index 1, over an alphabet of size n. This means that
Conversely, from any orthogonal array of degree r+2, strength 2 and index 1, we can reconstruct a set of r mutually orthogonal Latin squares, by putting symbol k in row i and column j of the tth square if the row (i,j,...,k,...) occurs in the array (where the k is in column t+2). The order of the Latin square is the number of symbols in the array.
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Peter J. Cameron
27 November 2002