|
Encyclopaedia of DesignTheory: Mutually orthogonal Latin squares |
General | Partition | Incidence | Array |
Experimental | Other designs | Math properties | Stat properties |
Server | External | Bibliography | Miscellanea |
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
|
|
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
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.
Table of contents | Glossary | Topics | Bibliography | History
Peter J. Cameron
27 November 2002