Support size lower bound for $k$-wise uniform distribution 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.
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.