# 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)

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.