Molecular biology increasingly relies on large screens where enormous numbers of specimens are systematically assayed in the search for a particular, rare outcome. These screens include the systematic testing of small molecules for potential drugs and testing the association between genetic variation and a phenotype of interest. While these screens are ``hypothesis-free,'' they can be wasteful; pooling the specimens and then testing the pools is more efficient. We articulate in precise mathematical ways the type of structures useful in combinatorial pooling designs so as to eliminate waste, to provide light weight, flexible, and modular designs. We show that Reed-Solomon codes, and more generally linear codes, satisfy all of these mathematical properties. We further demonstrate the power of this technique with Reed-Solomon-based biological experiments. We provide general purpose tools for experimentalists to construct and carry out practical pooling designs with rigorous guarantees for large screens.