A 0-1 probability space is a probability space(\Omega ; 2\Omega ; P ), where the sample space\Omega ` f0; 1g n for some n. A probability space is k-wise independent if, when Y i is defined to be the ith coordinate of the random n-vector, then any subset of k of the Y i 's is (mutually) independent, and it is said to be a probability space for p1 ; p2 ; :::; pn if P [Y i = 1] = p i . We study constructions of k-wise independent 0-1 probability spaces in which the p i 's are arbitrary. It was known that for any p1 ; p2 ; :::; pn , a k-wise independent probability space of size m(n;k) = \Gamma n k \Delta + \Gamma n k\Gamma1 \Delta + \Gamma n k\Gamma2 \Delta + \Delta \Delta \Delta + \Gamma n 0 \Delta always exists. We prove that for some p1 ; p2 ; :::; pn 2 [0; 1], m(n;k) is a lower bound on the size of any k-wise independent 0-1 probability space. For each fixed k, we prove that every k-wise independent 0-1 probability space for all p i = k=n has size\Omega\Gamma n ..
