  How low can the joint entropy of $n$ $d$-wise independent (for $d\ge2$)
discrete random variables be, subject to given constraints on the individual
distributions (say, no value may be taken by a variable with probability
greater than $p$, for $p<1$)? This question has been posed and partially
answered in a recent work of Babai.
  In this paper we improve some of his bounds, prove new bounds in a wider
range of parameters and show matching upper bounds in some special cases. In
particular, we prove tight lower bounds for the min-entropy (as well as the
entropy) of pairwise and three-wise independent balanced binary variables for
infinitely many values of $n$.
