From Local to Robust Testing via Agreement Testing

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

Research output: Contribution to journalArticlepeer-review

Abstract

A local tester for an error-correcting code is a probabilistic procedure that queries a small (sublinear) 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 said to be robust if for non-codewords it satisfies the stronger condition 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. We show that for certain codes, any (natural) local tester can be converted to a robust 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.

Original languageEnglish
Article number12
JournalTheory of Computing
Volume18
DOIs
StatePublished - 2022

Bibliographical note

Publisher Copyright:
© 2022 Irit Dinur, Prahladh Harsha, Tali Kaufman, and Noga Ron-Zewi b Licensed under a Creative Commons Attribution License (CC-BY).

Funding

Supported by ERC-CoG grant number 772839. Supported by the Department of Atomic Energy, Government of India, under project no. RTI4001 and in part by the UGC-ISF grant and the Swarnajayanti Fellowship. Supported by ERC and BSF grants. Supported by ISF grant 735/20. A conference version of this paper appeared in the Proceedings of the 10th Innovations in Theoretical Computer Science Conference, 2019 [15]. *Supported by ERC-CoG grant number 772839. †Supported by the Department of Atomic Energy, Government of India, under project no. RTI4001 and in part by the UGC-ISF grant and the Swarnajayanti Fellowship. ‡Supported by ERC and BSF grants. §Supported by ISF grant 735/20.

FundersFunder number
ERC-CoG772839
UGC-ISF
European Research Council
Department of Atomic Energy, Government of IndiaRTI4001
United States-Israel Binational Science Foundation
Israel Science Foundation735/20

    Keywords

    • affine-invariant codes
    • agreement testing
    • lifted codes
    • local testing
    • robust soundness

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