Encyclopaedia of DesignTheory: MOLS

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Mutually orthogonal Latin squares

Two Latin squares L1 and L2 of order n are orthogonal if, for each pair (k,l) of symbols, there is a unique cell (i,j) in which L1 has symbol k and L2 has symbol l.

Here is an example, where we have used A,B,C as symbols in one square and a,b,c in the other, and then superimposed the two squares for convenience.

Aa Bb Cc
Bc Ca Ab
Cb Ac Ba

Here it is using colours rather than symbols:


In this situation, one could use different alphabets (e.g. Latin and Greek) for the symols in the two squares. For this reason, a pair of orthogonal Latin squares is sometimes called a Graeco-Latin square.

If B is a Latin square orthogonal to A, we call B an orthogonal mate of A.

If B is an orthogonal mate of A, then the cells containing each symbol of B form a transversal of A; that is, one in each row, one in each column, and one containing each symbol. So a necessary and sufficient condition for a Latin square to have an orthogonal mate is that its cells can be partitioned into transversals.

A family of s Latin squares of order n is mutually orthogonal if any two squares in the family are orthogonal. We use the abbreviation MOLS for "mutually orthogonal Latin squares", and speak of s MOLS of order n, for example.

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Peter J. Cameron
16 April 2002