Here we are interested in the case *t* = 2, λ = 1,
so omit these parameters from the notation, writing OA(*n*,*m*).

An orthogonal array OA(*n*,*m*) is equivalent to a
*transversal design* TD[*m*;*n*].
If we pick two rows (that is, two groups of the transversal design)
then the symbols found there can be regarded as row and column indices,
and we find a set of *m* – 2 mutually orthogonal Latin squares.
In particular, an orthogonal array OA(*n*,3) with designated pair
of rows is equivalent to a Latin square (and hence exists for all *n*).

If *n* is a prime power, then TD[*m*;*n*] exists
if and only if *m* is at most *n* + 1.
(The design can be obtained from a projective plane of order *n*
by removing a point and *n* – *m* + 1 lines on that point.)

The *block graph* of a transversal design TD[*m*;*n*]
is the graph with the transversals (blocks of size *m*) as vertices,
where blocks are adjacent when they have nonempty intersection.
Such a graph is strongly regular, with parameters
*v* = *n*^{2},
*k* = *m*(*n* – 1),
λ = (*m* – 1)(*m* – 2) + *n* – 2,
μ = *m*(*m* – 1).

For *m* = 2 this is just the Hamming graph
H(2,*n*), that is, the *n*×*n* grid.