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### Paper:

TR16-102 | 4th July 2016 12:55

#### Bounded independence vs. moduli

TR16-102
Authors: Ravi Boppana, Johan Håstad, Chin Ho Lee, Emanuele Viola
Publication: 4th July 2016 12:55
Keywords:

Abstract:

Let $k=k(n)$ be the largest integer such that there
exists a $k$-wise uniform distribution over $\zo^n$ that
is supported on the set $S_m := \{x \in \zo^n : \sum_i x_i \equiv 0 \bmod m\}$, where $m$ is any integer. We
show that $\Omega(n/m^2 \log m) \le k \le 2n/m + 2$. For
$k = O(n/m)$ we also show that any $k$-wise uniform
distribution puts probability mass at most $1/m + 1/100$
over $S_m$. Finally, for any fixed odd $m$ we show that
there is $k = (1-\Omega(1))n$ 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 $m$.

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