Encyclopaedia of DesignTheory: Latin squares
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In this representation, the Latin square is an an orthogonal array of degree n, strength 1 and index 1, over an alphabet of size n. This means that
There is another important representation of a Latin square as an array. Assume that the symbols in the square are 1,2,...,n. Then let S be the set of n2 triples of the form (i,j,k), where the symbol in row i and column j of the square is k.
For example, given the square
we obtain the following nine triples:
This is an orthogonal array of degree 3, strength 2 and index 1, over an alphabet of size n. This means that
Conversely, from any orthogonal array of degree 3, strength 2 and index 1, we can reconstruct a Latin square, by putting symbol k in row i and column j if the row (i,j,k) occurs in the array. The order of the Latin square is the number of symbols in the array.
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Peter J. Cameron
27 November 2002