On construction of k-wise independent random variables howard

A O-1 probability space is a probability space (Ω, 2Ω, P), where the sample space Ω ⊆ {O, 1}n for some n. A probability space is k-wise independent if, when Yi is defined to be the ith coordinate of the random n-vector, then any subset of k of the U's is (mutually) independent. We say it is a probability space for PI, p2, . . . . pn if P[Yi = 1] = pi. We study constructions of k-wise independent O-1 probability spaces in which the pi's are arbitrary. It was known that for any P1, pZ, . . . . pn, a k-wise independent probability space of size m(n, k) = (n/k) + (k-l) + (nk/2) +...+n/0 always exists. We prove that for some P1,P2) . ..1 Pn ϵ [0, 1], Ω(n, k) is a lower bound on the size of any k-wise independent O-1 probability space. For each fixed k, for each n we prove the existence of a p c [0, 1], depending on n, such that every k-wise independent O-1 probability space if all pi = p has size fl(n). (This is in contrast with the known construction of size O(n^(k/2) when all the probabilities are 1/2.) For a very large degree of independence = [an], for cs > l/2-And all pi = 1/2, we prove a lower bound on the size of 2n (1 -1/2α), in contrast with the previous lower bound of βn where β < 2 is a function only of a. We give explicit constructions of k-wise independent O-1 probability spaces with arbitrary pi, and tight constructions for pairwise spaces with identical pi's.