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Bounded Independence versus Symmetric Tests

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Published:11 July 2019Publication History
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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 Ht = {x ∈ {0, 1}n : | ∑ixin/2| ≤ t}, we prove k*(Ht) = Θ (t2/n + 1).

For Sm, c = {x ∈ {0, 1}n : ∑ixic (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.

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