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arXiv · 1111.0492

Probabilistic existence of rigid combinatorial structures

Greg Kuperberg·Shachar Lovett·Ron Peled
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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.

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Greg Kuperberg, Shachar Lovett, Ron Peled. 2011-11-02. Probabilistic existence of rigid combinatorial structures. https://arxiv.org/abs/1111.0492

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