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