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So I have this question which I can't seem to find a solution to:

Prove that if $X = (X_1, ... X_n)$ is $k$-wise uniform* and each $X_i$ is Boolean then $\left|\operatorname{Supp}(X)\right| \geq \mathbf B(\frac{k}{2}, n)$, where $\mathbf B(r, n)$ is the number of words of weight at most $r$.

$*$ Where $k$-wise uniform means that, for each subset $S=\{i_1,\dots, i_s\}\subseteq [n]$ of size $s\leq k$, the distribution of $(X_{i_1},\dots,X_{i_s})$ is uniform.

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  • $\begingroup$ Isn't the constant distribution $\delta_{1^n}$ (i.e., $X_i = 1$ a.s. for every $i\in[n]$) $k$-wise independent for every $1\leq k\leq n$? $\endgroup$ – Clement C. Mar 23 '18 at 10:29
  • $\begingroup$ no, it is not even 1-wise independent, k-wise independent means that if we look at a subset (every subset) of length k it will look uniform. $\endgroup$ – Itaysason Mar 24 '18 at 10:47
  • $\begingroup$ That's only one of the possible definitions. One of the standard ones defines it as "every subset of at most k of the variables is mutually independent". (With yours, pairwise and 2-wise do not coincide.) $\endgroup$ – Clement C. Mar 24 '18 at 13:59
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    $\begingroup$ For this question, I recommend using the term "k-wise uniform" instead of "k-wise independent". $\endgroup$ – Ravi Boppana Mar 24 '18 at 16:23
  • $\begingroup$ @RaviBoppana and Itaysason: I have edited the OP that way, to make it clear the assumption implies uniformity. $\endgroup$ – Clement C. Mar 25 '18 at 16:51
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I believe this question was answered positively in three independent papers:

C. R. Rao (1947), "Factorial experiments derivable from combinatorial arrangements of arrays", Journal of the Royal Statistical Society 9:1, 128-139.

B. Chor, J. Friedman, O. Goldreich, J. Hastad, S. Rudich, and R. Smolensky (1985), "The bit extraction problem or t-resilient functions", 26th IEEE Symposium on Foundations of Computer Science, 396-407.

N. Alon, L. Babai, and A. Itai (1986), "A fast and simple randomized algorithm for the maximal independent set problem", Journal of Algorithms 7:4, 567-583.

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