Locally Testable Codes Require Redundant Testers

Eli Ben-Sasson, Venkatesan Guruswami, Tali Kaufman, Madhu Sudan, Michael Viderman

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

10 Scopus citations

Abstract

Locally testable codes (LTCs) are errorcorrecting codes for which membership, in the code, of a given word can be tested by examining it in very few locations. Most known constructions of locally testable codes are linear codes, and give error-correcting codes whose duals have (superlinearly) many small weight codewords. Examining this feature appears to be one of the promising approaches to proving limitation results for (i.e., upper bounds on the rate of) LTCs. Unfortunately till now it was not even known if LTCs need to be non-trivially redundant, i.e., need to have one linear dependency among the low-weight codewords in its dual. in this paper we give the first lower bound of this form, by showing that every positive rate constant query strong LTC must have linearly many redundant low-weight codewords in its dual. We actually prove the stronger claim that the actual test itself must use a linear number of redundant dual codewords (beyond the minimum number of basis elements required to characterize the code); in other words, non-redundant (in fact, low redundancy) local testing is impossible.

Original languageEnglish
Title of host publicationProceedings of the 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009
Pages52-61
Number of pages10
DOIs
StatePublished - 2009
Externally publishedYes
Event2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009 - Paris, France
Duration: 15 Jul 200918 Jul 2009

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159

Conference

Conference2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009
Country/TerritoryFrance
CityParis
Period15/07/0918/07/09

Keywords

  • Dual codes
  • Ldpc codes
  • Linear codes
  • Lower bounds
  • Property testing

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