# Orthogonal array

*orthogonal table, $ \mathop{\rm OA} ( N, k, n, t, \lambda ) $*

A $ ( k \times N) $- dimensional matrix whose entries are the numbers $ 1 \dots n $, and possessing the property that in each of its $ ( t \times N) $- dimensional submatrices any of the $ n ^ {t} $ possible $ t $- dimensional vector-columns with these numbers as coordinates is found in the columns of this submatrix precisely $ \lambda $ times. The definition of an orthogonal array implies that $ N = \lambda n ^ {t} $. One often considers the special case $ \mathop{\rm OA} ( N, k, n, t, \lambda ) $ with $ t = 2 $ and $ \lambda = 1 $, which is then denoted by $ \mathop{\rm OA} ( n, k) $. When $ k > 3 $, an orthogonal array $ \mathop{\rm OA} ( n, k) $ is equivalent to a set of $ k- 2 $ pairwise orthogonal Latin squares. For given $ n, t, \lambda $, the maximum value of the parameter $ k $ has been determined only in a number of specific cases, such as, for example, $ k \leq ( \lambda n ^ {2} - 1)/( n- 1) $ when $ t = 2 $, or $ k _ \max = t+ 1 $ when $ \lambda $ is odd and $ n = 2 $.

#### 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 $ t= 2 $ and $ \lambda \neq 1 $ states the existence of $ \mathop{\rm OA} ( \lambda n ^ {2} , 7 , n, 2, \lambda ) $ for all $ n \geq 2 $( H. Hanani, cf. [a1]). For $ \lambda = 1 $, see Orthogonal Latin squares. In geometric terms, an $ \mathop{\rm OA} ( \lambda n ^ {2} , k, n, 2, \lambda ) $ 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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_array&oldid=48073