  We are concerned with the problem of designing large families of subsets over
a common labeled ground set that have small pairwise intersections and the
property that the maximum discrepancy of the label values within each of the
sets is less than or equal to one. Our results, based on transversal designs,
factorizations of packings and Latin rectangles, show that by jointly
constructing the sets and labeling scheme, one can achieve optimal family sizes
for many parameter choices. Probabilistic arguments akin to those used for
pseudorandom generators lead to significantly suboptimal results when compared
to the proposed combinatorial methods. The design problem considered is
motivated by applications in molecular data storage and theoretical computer
science.
