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''orthogonal table,  $   \mathop{\rm OA} ( N, k, n, t, \lambda ) $''

A  $  ( k \times N) $-
dimensional [[Matrix|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Dénes,   A.D. Keedwell,   "Latin squares and their applications" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Hall,   "Combinatorial theory" , Wiley  (1986)</TD></TR></table>

====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. [[#References|[a1]]]). For  $  \lambda = 1 $, 
see [[Orthogonal Latin squares|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. [[#References|[a1]]] for some fundamental results and [[#References|[a2]]] for a recent survey.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Beth,   D. Jungnickel,   H. Lenz,   "Design theory" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Jungnickel,   "Latin squares, their geometries and their groups. A survey" , ''Proc. IMA Workshops on Coding and Design Theory Minneapolis, 1988'' , Springer  (to appear)</TD></TR></table>