Existence of Small Families of t-wise Independent Permutations and t-Designs Via Local Limit Theorems

Shachar Lovett
Institute for Advanced Study
September 20, 2011

We show existence of rigid combinatorial objects that previously were not known to exist. Specifically, we consider two families of objects:

1. A family of permutations on n elements is t-wise independent if it acts uniformly on tuples of t elements. Constructions of small families of t-wise independent permutations are known only for \( t=1,2,3 \) . We show that there exist small families of t-wise independent permutations for all t , whose size is \( n^{O(t)} \) .

On the Fourier Spectrum of Symmetric Boolean Functions

Amir Shpilka
Technion; on leave at Microsoft Research New England
March 14, 2011

It is well-known that any Boolean function f:{-1,+1}^n \to {-1,+1} can be written uniquely as a polynomial f(x) = \sum_{S subset [n]} f_s \prod_{i in S} x_i. The collection of coefficients (f_S's) this expression are referred to (with good reason) as the Fourier spectrum of f. The Fourier spectrum has played a central role in modern computer science by converting combinatorial and algorithmic questions about f into algebraic or analytic questions about the spectrum.