 
Summary: Almost kwise vs. kwise independent permutations,
and uniformity for general group actions
Noga Alon
TelAviv University and IAS, Princeton
nogaa@tau.ac.il
Shachar Lovett
IAS, Princeton
slovett@math.ias.edu
June 24, 2011
Abstract
A family of permutations in Sn is kwise independent if a uniform permutation
chosen from the family maps any distinct k elements to any distinct k elements equally
likely. Efficient constructions of kwise independent permutations are known for k = 2
and k = 3, but are unknown for k 4. In fact, it is known that there are no nontrivial
subgroups of Sn for n 25 which are 4wise independent. Faced with this adversity,
research has turned towards constructing almost kwise independent families, where
small errors are allowed. Optimal constructions of almost kwise independent families
of permutations were achieved by several authors.
Our first result is that any such family with small enough error is statistically close
to a distribution which is perfectly kwise. This allows for a simplified analysis of algo
