Affine extractors over prime fields

Abstract

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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    J. Bourgain: On the construction of affine extractors, Geometric and Functional Analysis 17(1) (2007), 33–57.

    MathSciNet  MATH  Article  Google Scholar 

  2. [2]

    J. Bourgain: Multilinear exponential sum bounds with optimal entropy assignments, Geometric and Functional Analysis 18(5) (2009), 1477–1502.

    MathSciNet  Article  Google Scholar 

  3. [3]

    J. Bourgain: On exponential sums in finite fields, manuscript, 2009.

  4. [4]

    J. Bourgain, A. A. Glibichuk and S. V. Konyagin: Estimates for the number of sums and products and for exponential sums in fields of prime order, Journal of the London Mathematical Society 73(2) (2006), 380–398.

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    A. Gabizon and R. Raz: Deterministic extractors for affine sources over large fields, in: Proceedings of the 46th FOCS, pages 407–418, 2005.

  6. [6]

    O. Goldreich: Three XOR-lemmas — an exposition, manuscript, 1997.

  7. [7]

    A. Rao: Extractors for low-weight affine sources, in: Proceedings of CCC, pages 95–101, 2009.

  8. [8]

    R. Raz and A. Yehudayoff: Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors; in: proceedings of FOCS, pages 273–282, 2008.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Amir Yehudayoff.

Additional information

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

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Yehudayoff, A. Affine extractors over prime fields. Combinatorica 31, 245 (2011). https://doi.org/10.1007/s00493-011-2604-9

Download citation

Mathematics Subject Classification (2000)

  • 11T23