  We show the existence of rigid combinatorial objects which previously were
not known to exist. Specifically, for a wide range of the underlying
parameters, we show the existence of non-trivial orthogonal arrays,
$t$-designs, and $t$-wise permutations. In all cases, the sizes of the objects
are optimal up to polynomial overhead. The proof of existence is probabilistic.
We show that a randomly chosen such object has the required properties with
positive yet tiny probability. The main technical ingredient is a special local
central limit theorem for suitable lattice random walks with finitely many
steps.
