Almost k-wise independence versus k-wise independence

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