  If the number of runs in a (mixed-level) orthogonal array of strength 2 is
specified, what numbers of levels and factors are possible? The collection of
possible sets of parameters for orthogonal arrays with N runs has a natural
lattice structure, induced by the ``expansive replacement'' construction
method. In particular the dual atoms in this lattice are the most important
parameter sets, since any other parameter set for an N-run orthogonal array can
be constructed from them. To get a sense for the number of dual atoms, and to
begin to understand the lattice as a function of N, we investigate the height
and the size of the lattice. It is shown that the height is at most [c(N-1)],
where c= 1.4039... and that there is an infinite sequence of values of N for
which this bound is attained. On the other hand, the number of nodes in the
lattice is bounded above by a superpolynomial function of N (and
superpolynomial growth does occur for certain sequences of values of N). Using
a new construction based on ``mixed spreads'', all parameter sets with 64 runs
are determined. Four of these 64-run orthogonal arrays appear to be new.
