  We show the existence of regular 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 structure has the required properties with positive yet tiny
probability. Our method allows also to give rather precise estimates on the
number of objects of a given size and this is applied to count the number of
orthogonal arrays, t-designs and regular hypergraphs. The main technical
ingredient is a special local central limit theorem for suitable lattice random
walks with finitely many steps.
