Randomly supported independence and resistance

We prove that for any positive integer k, there is a constant c_k such that a randomly selected set of c_k n^k log n Boolean vectors with high probability supports a balanced k-wise independent distribution. In the case of k ≤ 2 a more elaborate argument gives the stronger bound c_k n^k. Using a recent result by Austrin and Mossel this shows that a predicate on t bits, chosen at random among predicates accepting c₂ t² input vectors, is, assuming the Unique Games Conjecture, likely to be approximation resistant. These results are close to tight: we show that there are other constants, c_k', such that a randomly selected set of cardinality c_k' n^k points is unlikely to support a balanced k-wise independent distribution and, for some c>0, a random predicate accepting ct²/log t input vectors is non-trivially approximable with high probability. In a different application of the result of Austrin and Mossel we prove that, again assuming the Unique Games Conjecture, any predicate on t bits accepting at least (32/33) • 2^t inputs is approximation resistant. The results extend from the Boolean domain to larger finite domains.