Almost k -wise independence versus k -wise independence Noga Alon ∗ Sackler Faculty of Exact Sciences Tel Aviv University Ramat-Aviv, Israel. nogaa@post.tau.ac.il Oded Goldreich † Department of Computer Science Weizmann Institute of Science Rehovot, Israel. oded@wisdom.weizmann.ac.il Yishay Mansour ‡ School of Computer Science Tel Aviv University Ramat-Aviv, Israel. mansour@post.tau.ac.il December 12, 2003 Abstract We say that a distribution over {0, 1} n is (ǫ, k)-wise independent if its restriction to every k coordinates results in a distribution that is ǫ-close to the uniform distribution. A natural question regarding (ǫ, k)-wise independent distributions is how close they are to some k-wise independent distribution. We show that there exists (ǫ, k)-wise independent distribution whose statistical distance is at least n O(k) · ǫ from any k-wise independent distribution. In addition, we show that for any (ǫ, k)-wise independent distribution there exists some k-wise independent distribution, whose statistical distance is n O(k) · ǫ. Keywords: Small probability spaces, k-wise independent distributions, almost k-wise indepen- dent distributions, small bias probability spaces. * Research supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. † Supported by the MINERVA Foundation, Germany. ‡ Research supported in part by a grant from the Israel Science Foundation.