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Encyclopaedia of DesignTheory: Latin squares

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Latin squares as arrays

The definition of a Latin square is as a square array of symbols, with the property that each symbol occurs exactly once in each row and column.

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

Obviously the same applies with rows in place of columns. Furthermore, these two properties characterise Latin squares.

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

1 2 3
2 3 1
3 1 2

we obtain the following nine triples:

111
122
133
212
223
231
313
321
332

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