Bounded Independence versus Symmetric Tests.

For a test T ⊆ {0, 1}n, define k*(T) to be the maximum k such that there exists a k-wise uniform distribution over {0, 1}n whose support is a subset of T. For Ht = {x ∈ {0, 1}n : | ∑ixi − n/2| ≤ t}, we prove k*(Ht) = Θ (t2/n + 1). For Sm, c = {x ∈ {0, 1}n : ∑ixi ≡ c (mod m)}, we prove that k*(Sm, c) = Θ (n/m2). For some k = O(n/m) we also show that any k-wise uniform distribution puts probability mass at most 1/m + 1/100 over Sm, c. Finally, for any fixed odd m we show that there is an integer k = (1 − Ω(1))n such that any k-wise uniform distribution lands in T with probability exponentially close to |Sm, c|/2n; and this result is false for any even m.