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:
| 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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