For a test T 0 1 n define k 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 T = x 0 1 n : \abs i x i − n 2 t we prove k = ( t 2 n + 1 ) .

For T = x 0 1 n : i x i c ( mod m ) we prove that k = ( n m 2 + 1 ) . 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 T . Finally, for any fixed odd m we show that there is an integer k = ( 1 − (1)) n such that any k -wise uniform distribution lands in T with probability exponentially close to T 2 n ; and this result is false for any even m .