Explicit construction of a small ∈-net for linear threshold functions

We give explicit constructions of ∈-nets for linear threshold functions on the binary cube and on the unit sphere. The size of the constructed nets is polynomial in the dimension n and in 1/∈. To the best of our knowledge no such constructions were previously known. Our results match, up to the exponent of the polynomial, the bounds that are achieved by probabilistic arguments. As a corollary we also construct subsets of the binary cube that have size polynomial in n and a covering radius of n/2 - c√n log n for any constant c. This improves upon the well-known construction of dual BCH codes that guarantee only a covering radius of n/2 - c√n.