Summary: We give explicit constructions of \(\epsilon\)-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 \(\frac{1}{\epsilon}\). 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 \(\frac{n}{2}-c\sqrt{n\log n}\) for any constant \(c\). This improves upon the well-known construction of dual BCH codes that guarantee only a covering radius of \(\frac{n}{2}-c\sqrt{n}\).