# Orthogonal array

*orthogonal table, *

A -dimensional matrix whose entries are the numbers , and possessing the property that in each of its -dimensional submatrices any of the possible -dimensional vector-columns with these numbers as coordinates is found in the columns of this submatrix precisely times. The definition of an orthogonal array implies that . One often considers the special case with and , which is then denoted by . When , an orthogonal array is equivalent to a set of pairwise orthogonal Latin squares. For given , the maximum value of the parameter has been determined only in a number of specific cases, such as, for example, when , or when is odd and .

#### References

[1] | J. Dénes, A.D. Keedwell, "Latin squares and their applications" , Acad. Press (1974) |

[2] | M. Hall, "Combinatorial theory" , Wiley (1986) |

#### Comments

Regarding existence, the only general result for and states the existence of for all (H. Hanani, cf. [a1]). For , see Orthogonal Latin squares. In geometric terms, an is equivalent to a "transversal designtransversal design" , respectively a "netnet" ; cf. [a1] for some fundamental results and [a2] for a recent survey.

#### References

[a1] | T. Beth, D. Jungnickel, H. Lenz, "Design theory" , Cambridge Univ. Press (1986) |

[a2] | D. Jungnickel, "Latin squares, their geometries and their groups. A survey" , Proc. IMA Workshops on Coding and Design Theory Minneapolis, 1988 , Springer (to appear) |

**How to Cite This Entry:**

Orthogonal array. V.M. Mikheev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Orthogonal_array&oldid=17901