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# Encyclopaedia of DesignTheory: Mutually orthogonal Latin squares

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# MOLS as arrays

A Latin square can be represented as an orthogonal array of degree 3, strength 2, and index 1. This is described under the entry for Latin squares.

A similar representation works for sets of mutually orthogonal Latin squares. Assume that the symbols in the squares are 1,2,...,n, and let L1,...,Lr be the squares. Then let S be the set of n2 (r+2)-tuples of the form (i,j,k1,...,kr), where the symbol in row i and column j of the square Lt is kt.

For example, given the orthogonal Latin squares

 1 2 3 2 3 1 3 1 2
 1 2 3 3 1 2 2 3 1

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

• there are four columns;
• if you slide your fingers down any two columns of the array, you will see each ordered pair of symbols precisely once.

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.

Peter J. Cameron
27 November 2002