Hyper-Graeco-Latin Square Design - Mutually Orthogonal Latin Squares

Mutually Orthogonal Latin Squares

A 2-(v,k,1) orthogonal array is equivalent to a set of k − 2 mutually orthogonal latin squares of order v.

Index one, strength 2 orthogonal arrays are also known as Hyper-Graeco-Latin square designs in the statistical literature.

Let A be a strength 2, index 1 orthogonal array on a v-set of elements, identified with the set of natural numbers {1,...,v}. Chose and fix, in order, two columns of A, called the indexing columns. All ordered pairs (i, j) with 1 ≤ i, j ≤ v appear exactly once in the rows of the indexing columns. Take any other column of A and create a square array whose entry in position (i,j) is the entry of A in this column in the row that contains (i, j) in the indexing columns of A. The resulting square is a latin square of order v. For example, consider the 2-(3,4,1) orthogonal array:

By chosing columns 3 and 4 (in that order) as the indexing columns, the first column produces the latin square,

while the second column produces the latin square,

The latin squares produced in this way from an orthogonal array will be orthogonal latin squares, so the k − 2 columns other than the indexing columns will produce a set of k − 2 mutually orthogonal latin squares.

This construction is completely reversible and so strength 2, index 1 orthogonal arrays can be constructed from sets of mutually orthogonal latin squares.