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, jv 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:

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

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

1 2 3
3 1 2
2 3 1

while the second column produces the latin square,

1 3 2
3 2 1
2 1 3

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.

Read more about this topic:  Hyper-Graeco-Latin Square Design

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