Title: Existence of small families of t-wise independent permutations and t-designs via local limit theorems
Speaker: Shachar Lovett, Institute of Advanced Study
Date: Wednesday, September 21, 2011 11:00-12:00pm
Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ
Abstract:
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)}.
2. A t-(n,m,lambda) design is a family of sets of size m in a universe of size n such that each t elements belong to exactly lambda sets. Constructions of t-designs are known only for some specific settings of parameters. We show that there exist small t-designs for any n,m whose size (and lambda) are n^{O(t)}.
The main technical ingredients in both cases are local limit theorems used to study random walks on lattices.
Joint work with Greg Kuperberg and Ron Peled.