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

Yuval Rabani, Amir Shpilka

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)3501-3520
Number of pages20
JournalSIAM Journal on Computing
Volume39
Issue number8
DOIs
StatePublished - 2010
Externally publishedYes

Keywords

  • Derandomization
  • Explicit construction
  • Hitting sets
  • ∈-net

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