Bounded Independence versus Symmetric Tests

Abstract

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 H t = { x ∈ {0, 1} n : | ∑ i x i − n /2| ≤ t }, we prove k * ( H t ) = Θ ( t 2 / n + 1). For S m, c = { x ∈ {0, 1} n : ∑ i x i ≡ c (mod m )}, we prove that k * ( S m, c ) = Θ ( n / m 2 ). 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 S m, 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 | S m, c |/2 n ; and this result is false for any even m .