Simple permutations mix even better

We study the random composition of a small family of O(n) 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 [12] and Hoory et al. [13] and show that up to a polylogarithmic factor, nk compositions of random permutations from this family suffice. In addition, our results give an explicit construction of a degree O(n) Cayley graph of the alternating group of 2n objects with a spectral gap Ω(2/n), which is a substantial improvement over previous constructions.