Affine extractors over prime fields


An affine extractor is a map that is balanced on every affine subspace of large enough dimension. We construct an explicit affine extractor AE from \(\mathbb{F}^n \) to \(\mathbb{F}\), \(\mathbb{F}\) a prime field, so that AE(x) is exponentially close to uniform when x is chosen uniformly at random from an arbitrary affine subspace of \(\mathbb{F}^n \) of dimension at least δn, 0<δ≤1 a constant. Previously, Bourgain constructed such affine extractors when the size of \(\mathbb{F}\) is two. Our construction is in the spirit of but different than Bourgain’s construction. This allows for simpler analysis and better quantitative results.

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Correspondence to Amir Yehudayoff.

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Research partially supported by NSF grants CCF-0832797 and DMS-0835373.

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Yehudayoff, A. Affine extractors over prime fields. Combinatorica 31, 245 (2011).

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Mathematics Subject Classification (2000)

  • 11T23