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 and 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 the dimension and covering radius of
$n/2 - c\sqrt{n\log n}$, for any constant c. This improves upon the well known construction of dual BCH codes that only guarantee covering radius of $\n/2 - c\sqrt{n}$.
Our work is motivated by questions of constructing interesting geometric objects, known to exist via probabilistic arguments.
We will discuss some of this motivation.
This is joint work with Amir Shpilka.
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