  A k-wise independent distribution on n bits is a joint distribution of the
bits such that each k of them are independent. In this paper we consider k-wise
independent distributions with identical marginals, each bit has probability p
to be 1. We address the following question: how high can the probability that
all the bits are 1 be, for such a distribution? For a wide range of the
parameters n,k and p we find an explicit lower bound for this probability which
matches an upper bound given by Benjamini et al., up to multiplicative factors
of lower order. The question we investigate can be seen as a relaxation of a
major open problem in error-correcting codes theory, namely, how large can a
linear error correcting code with given parameters be?
  The question is a type of discrete moment problem, and our approach is based
on showing that bounds obtained from the theory of the classical moment problem
provide good approximations for it. The main tool we use is a bound controlling
the change in the expectation of a polynomial after small perturbation of its
zeros.
