Almost k-wise independence versus k-wise independence

We say that a distribution over {0,1}(n) is (epsilon,k)-wise independent if its restriction to every k coordinates results in a distribution that is epsilon-close to the uniform distribution. A natural question regarding (epsilon, k)-wise independent distributions is how close they are to some k-wise independent distribution. We show that there exist (epsilon,k)-wise independent distributions whose statistical distance is at least n(O(k).)epsilon from any k-wise independent distribution. In addition, we show that for any (epsilon,k)-wise independent distribution there exists some k-wise independent distribution, whose statistical distance is n(O(k).)epsilon. (C) 2003 Elsevier B.V. All rights reserved.