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Bounded Independence vs. Moduli

Authors Ravi Boppana, Johan Håstad, Chin Ho Lee, Emanuele Viola

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Ravi Boppana
Johan Håstad
Chin Ho Lee
Emanuele Viola

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Ravi Boppana, Johan Håstad, Chin Ho Lee, and Emanuele Viola. Bounded Independence vs. Moduli. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 24:1-24:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.24

Abstract

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.
Keywords
  • Bounded independence
  • Modulus

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