Strong locally testable codes with relaxed local decoders

Oded Goldreich, Tom Gur, Ilan Komargodski

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

12 Scopus citations

Abstract

Locally testable codes (LTCs) are error-correcting codes that admit very efficient codeword tests. An LTC is said to be strong if it has a proximity-oblivious tester; that is, a tester that makes only a constant number of queries and reject non-codewords with probability that depends solely on their distance from the code. Locally decodable codes (LDCs) are complimentary to LTCs. While the latter allow for highly efficient rejection of strings that are far from being codewords, LDCs allow for highly efficient recovery of individual bits of the information that is encoded in strings that are close to being codewords. Constructions of strong-LTCs with nearly-linear length are known, but the existence of a constant-query LDC with polynomial length is a major open problem. In an attempt to bypass this barrier, Ben-Sasson et al. (SICOMP 2006) introduced a natural relaxation of local decodability, called relaxed-LDCs. This notion requires local recovery of nearly all individual informationbits, yet allows for recovery-failure (but not error) on the rest. Ben-Sasson et al. constructed a constant-query relaxed-LDC with nearly-linear length (i.e., length k1+α for an arbitrarily small constant α > 0, where k is the dimension of the code). This work focuses on obtaining strong testability and relaxed decodability simultaneously. We construct a family of binary linear codes of nearly-linear length that are both strong-LTCs (with one-sided error) and constant-query relaxed-LDCs. This improves upon the previously known constructions, which either obtain weak LTCs or require polynomial length. Our construction heavily relies on tensor codes and PCPs. In particular, we provide strong canonical PCPs of proximity for membership in any linear code with constant rate and relative distance. Loosely speaking, these are PCPs of proximity wherein the verifier is proximity oblivious (similarly to strong-LTCs) and every valid statement has a unique canonical proof. Furthermore, the verifier is required to reject non-canonical proofs (even for valid statements). As an application, we improve the best known separation result between the complexity of decision and verification in the setting of property testing.

Original languageAmerican English
Title of host publication30th Conference on Computational Complexity, CCC 2015
EditorsDavid Zuckerman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages1-41
Number of pages41
ISBN (Electronic)9783939897811
DOIs
StatePublished - 1 Jun 2015
Externally publishedYes
Event30th Conference on Computational Complexity, CCC 2015 - Portland, United States
Duration: 17 Jun 201519 Jun 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume33
ISSN (Print)1868-8969

Conference

Conference30th Conference on Computational Complexity, CCC 2015
Country/TerritoryUnited States
CityPortland
Period17/06/1519/06/15

Bibliographical note

Publisher Copyright:
© Oded Goldreich, Tom Gur, and Ilan Komargodski; licensed under Creative Commons License CC-BY.

Keywords

  • Locally decodable codes
  • Locally testable codes
  • PCPs of proximity

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