Constructing small-bias sets from algebraic-geometric codes

Avraham Ben-Aroya, Amnon Ta-Shma

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We give an explicit construction of an ∈-biased set over k bits of size O (k/∈2log(1/∈))5/4. This improves upon previous explicit constructions when ∈ is roughly (ignoring logarithmic factors) in the range [k-1.5, k-0.5]. The construction builds on an algebraic-geometric code. However, unlike previous constructions we use low-degree divisors whose degree is significantly smaller than the genus. Studying the limits of our technique, we arrive at a hypothesis that if true implies the existence of ∈-biased sets with parameters nearly matching the lower bound, and in particular giving binary error correcting codes beating the Gilbert-Varshamov bound.

Original languageEnglish
Title of host publicationProceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009
Pages191-197
Number of pages7
DOIs
StatePublished - 2009
Event50th Annual Symposium on Foundations of Computer Science, FOCS 2009 - Atlanta, GA, United States
Duration: 25 Oct 200927 Oct 2009

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference50th Annual Symposium on Foundations of Computer Science, FOCS 2009
Country/TerritoryUnited States
CityAtlanta, GA
Period25/10/0927/10/09

Keywords

  • Algebraic-geometric codes
  • Small-bias sets

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