Randomness-efficient low degree tests and short PCPs via epsilon-biased sets

Eli Ben-Sasson*, Salil Vadhan, Madhu Sudan, Avi Wigderson

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

106 Scopus citations

Abstract

We present the first explicit construction of Probabilistically Checkable Proofs (PCPs) and Locally Testable Codes (LTCs) of fixed constant query complexity which have almost-linear (= n · 2Õ(√log n) size. Such objects were recently shown to exist (nonconstructively) by Goldreich and Sudan. Previous explicit constructions required size n1+ω(ε) with 1/ε queries. The key to these constructions is a nearly optimal randomness-efficient version of the low degree test. In a similar way we give a randomness-efficient version of the BLR linearity test (which is used, for instance, in locally testing the Hadamard code). The derandomizations are obtained through ε-biased sets for vector spaces over finite fields. The analysis of the derandomized tests rely on alternative views of ε-biased sets - as generating sets of Cayley expander graphs for the low degree test, and as defining linear error-correcting codes for the linearity test.

Original languageEnglish
Pages (from-to)612-621
Number of pages10
JournalConference Proceedings of the Annual ACM Symposium on Theory of Computing
DOIs
StatePublished - 2003
Event35th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States
Duration: 9 Jun 200311 Jun 2003

Keywords

  • Linearity Testing
  • Locally Testable Codes
  • Low Degree Testing
  • Probabilistically Checkable Proofs
  • Property Testing

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