Set-codes with small intersections and small discrepancies

We address the new problem of designing large families of subsets of a common labeled ground set that simultaneously have small pairwise intersections and the property that the maximum discrepancy of the label values within each of the subsets is less than or equal to one. Our results include an upper bound on the size of such families, and constructions based on transversal designs, packings, and new forms of Latin rectangles. The constructions jointly optimize the size of the family of sets and the labeling scheme and 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 intersecting sets discrepancy problem is motivated by emerging applications in coding for molecular data storage.