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High-Rate Locally Correctable and Locally Testable Codes with Sub-Polynomial Query Complexity

Authors:

Swastik Kopparty,

Shubhangi SarafAuthors Info & Claims

Journal of the ACM (JACM), Volume 64, Issue 2

Article No.: 11, Pages 1 - 42

https://doi.org/10.1145/3051093

Published: 25 May 2017 Publication History

Abstract

Locally correctable codes (LCCs) and locally testable codes (LTCs) are error-correcting codes that admit local algorithms for correction and detection of errors. Those algorithms are local in the sense that they only query a small number of entries of the corrupted codeword. The fundamental question about LCCs and LTCs is to determine the optimal tradeoff among their rate, distance, and query complexity.

In this work, we construct the first LCCs and LTCs with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist LCCs and LTCs with block length n, constant rate (which can even be taken arbitrarily close to 1), and constant relative distance, whose query complexity is exp(Õ(√log n)) (for LCCs) and (log n)^{O(log log n)} (for LTCs).

In addition to having small query complexity, our codes also achieve better tradeoffs between the rate and the relative distance than were previously known to be achievable by LCCs or LTCs. Specifically, over large (but constant size) alphabet, our codes approach the Singleton bound, that is, they have almost the best-possible relationship between their rate and distance. Over the binary alphabet, our codes meet the Zyablov bound. Such tradeoffs between the rate and the relative distance were previously not known for any o(n) query complexity. Our results on LCCs also immediately give locally decodable codes with the same parameters.

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Index Terms

High-Rate Locally Correctable and Locally Testable Codes with Sub-Polynomial Query Complexity
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Error-correcting codes

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 64, Issue 2

April 2017

277 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3080497

Editor:
Éva Tardos
Cornell University

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Copyright © 2017 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 25 May 2017

Accepted: 01 February 2017

Revised: 01 January 2017

Received: 01 May 2016

Published in JACM Volume 64, Issue 2

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Kumar VMon GMohar BShinkar IO'Donnell R(2024)Relaxed Local Correctability from Local TestingProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649611(1585-1593)Online publication date: 10-Jun-2024
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https://doi.org/10.1109/FOCS61266.2024.00112
Cohen GCohen IMaor G(2024)Tight Bounds for the Zig-Zag Product2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00094(1470-1499)Online publication date: 27-Oct-2024
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Dikstein YHopkins M(2024)Chernoff Bounds and Reverse Hypercontractivity on HDX2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00060(870-919)Online publication date: 27-Oct-2024
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Block ABlocki JCheng KGrigorescu ELi XZheng YZhu M(2023)On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion ErrorsProceedings of the conference on Proceedings of the 38th Computational Complexity Conference10.4230/LIPIcs.CCC.2023.14(1-25)Online publication date: 17-Jul-2023
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Kopparty SRon-Zewi NSaraf SWootters M(2023)Improved List Decoding of Folded Reed-Solomon and Multiplicity CodesSIAM Journal on Computing10.1137/20M137021552:3(794-840)Online publication date: 15-Jun-2023
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Dinur IEvra SLivne RLubotzky AMozes SLeonardi SGupta A(2022)Locally testable codes with constant rate, distance, and localityProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520024(357-374)Online publication date: 9-Jun-2022
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