Simple Permutations Mix Even Better

We study the composition of random permutations drawn from a small family of O(n(3)) simple permutations on {0, 1}(n). Specifically, we ask how many randomly selected simple permutations need be composed to yield a permutation that is close to k-wise independent. We improve on the results of Gowers (Combin Probab Comput 5 (1996) 119-130) and Hoory et al. (Presented at 3 1 st ICALP 2004) and show that it suffices to compose min(O(n(3)k(2)), (O) over tilde (n(2)k(2))) random permutations from this family for any n >= 3 and k <= 2(n) - 2. The 6 (O) over tilde notation suppresses a polylogarithmic factor in k and n. (c) 2007 Wiley Periodicals, Inc.