Probabilistic existence of rigid combinatorial structures

TL;DR

This paper proves the existence of certain rigid combinatorial structures like orthogonal arrays and t-designs using probabilistic methods, with sizes close to optimal, and introduces a new local central limit theorem for lattice random walks.

Contribution

It establishes the existence of various rigid combinatorial objects previously unknown to exist, using probabilistic techniques and a novel local central limit theorem.

Findings

Existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations.

Object sizes are optimal up to polynomial factors.

Probabilistic proof with positive probability for the properties.

Abstract

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.