Bounded Independence vs. Moduli
Let k = k(n) be the largest integer such that there exists a k-wise uniform distribution over {0,1}^n that is supported on the set S_m := {x in {0,1}^n: sum_i x_i equiv 0 mod m}, where m is any integer. We show that Omega(n/m^2 log m) <= k <= 2n/m + 2. For 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. For any fixed odd m there is k \ge (1 - Omega(1))n such that any k-wise uniform distribution lands in S_m with probability exponentially close to |S_m|/2^n; and this result is false for any even m.
Bounded independence
Modulus
24:1-24:9
Regular Paper
Ravi
Boppana
Ravi Boppana
Johan
Håstad
Johan Håstad
Chin Ho
Lee
Chin Ho Lee
Emanuele
Viola
Emanuele Viola
10.4230/LIPIcs.APPROX-RANDOM.2016.24
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