How low can the joint entropy of $n$ $d$-wise independent (for $dge2$) 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 $n$.
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