From local to robust testing via agreement testing

Irit Dinur, Prahladh Harsha, Tali Kaufman, Noga Ron-Zewi

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

Abstract

A local tester for an error-correcting code is a probabilistic procedure that queries a small subset of coordinates, accepts codewords with probability one, and rejects non-codewords with probability proportional to their distance from the code. The local tester is robust if for non-codewords it satisfies the stronger property that the average distance of local views from accepting views is proportional to the distance from the code. Robust testing is an important component in constructions of locally testable codes and probabilistically checkable proofs as it allows for composition of local tests. In this work we show that for certain codes, any (natural) local tester can be converted to a roubst tester with roughly the same number of queries. Our result holds for the class of affine-invariant lifted codes which is a broad class of codes that includes Reed-Muller codes, as well as recent constructions of high-rate locally testable codes (Guo, Kopparty, and Sudan, ITCS 2013). Instantiating this with known local testing results for lifted codes gives a more direct proof that improves some of the parameters of the main result of Guo, Haramaty, and Sudan (FOCS 2015), showing robustness of lifted codes. To obtain the above transformation we relate the notions of local testing and robust testing to the notion of agreement testing that attempts to find out whether valid partial assignments can be stitched together to a global codeword. We first show that agreement testing implies robust testing, and then show that local testing implies agreement testing. Our proof is combinatorial, and is based on expansion / sampling properties of the collection of local views of local testers. Thus, it immediately applies to local testers of lifted codes that query random affine subspaces in 𝔽mq , and moreover seems amenable to extension to other families of locally testable codes with expanding families of local views.

Original languageEnglish
Title of host publication10th Innovations in Theoretical Computer Science, ITCS 2019
EditorsAvrim Blum
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770958
DOIs
StatePublished - 1 Jan 2019
Event10th Innovations in Theoretical Computer Science, ITCS 2019 - San Diego, United States
Duration: 10 Jan 201912 Jan 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume124
ISSN (Print)1868-8969

Conference

Conference10th Innovations in Theoretical Computer Science, ITCS 2019
Country/TerritoryUnited States
CitySan Diego
Period10/01/1912/01/19

Bibliographical note

Funding Information:
Supported by ERC-CoG grant number 772839. 2 Research supported in part by the UGC-ISF grant and the Swarnajayanti Fellowship. Part of the work was done when the author was visiting the Weizmann Institute of Science.

Publisher Copyright:
© Irit Dinur, Prahladh Harsha, Tal Kaufman, and Noga Ron-Zewi.

Keywords

  • Affine-invariant codes
  • Agreement testing
  • Lifted codes
  • Local testing
  • Robust testing

ASJC Scopus subject areas

  • Software

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