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On the power of relaxed local decoding algorithms

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Gur, Tom and Lachish, Oded (2021) On the power of relaxed local decoding algorithms. SIAM Journal on Computing, 50 (2). pp. 788-813. doi:10.1137/19M1307834

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Official URL: https://doi.org/10.1137/19M1307834

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Abstract

A locally decodable code (LDC) $C \colon \{0,1\}^k \to \{0,1\}^n$ is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and in practice, ranging from probabilistically checkable proofs to distributed storage. However, despite nearly two decades of extensive study, the best known constructions of $O(1)$-query LDCs have superpolynomial blocklength. The notion of relaxed LDCs is a natural relaxation of LDCs, which aims to bypass the foregoing barrier by requiring local decoding of nearly all individual message bits, yet allowing decoding failure (but not error) on the rest. State of the art constructions of $O(1)$-query relaxed LDCs achieve blocklength $n = O\left(k^{1+ \gamma}\right)$ for an arbitrarily small constant $\gamma$. We prove a lower bound which shows that $O(1)$-query relaxed LDCs cannot achieve blocklength $n = k^{1+ o(1)}$. This resolves an open problem raised by Goldreich in 2004.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software
Divisions: Faculty of Science, Engineering and Medicine > Science > Computer Science
Library of Congress Subject Headings (LCSH): Data encryption (Computer science), Computer networks, Computer algorithms, Algorithms
Journal or Publication Title: SIAM Journal on Computing
Publisher: Society for Industrial and Applied Mathematics
ISSN: 0097-5397
Official Date: 18 April 2021
Dates:
DateEvent
18 April 2021Published
5 January 2021Accepted
Volume: 50
Number: 2
Page Range: pp. 788-813
DOI: 10.1137/19M1307834
Status: Peer Reviewed
Publication Status: Published
Reuse Statement (publisher, data, author rights): First Published in SIAM Journal on Computing in 50(2), published by the Society for Industrial and Applied Mathematics (SIAM). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
Access rights to Published version: Restricted or Subscription Access
Copyright Holders: © 2021, Society for Industrial and Applied Mathematics
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
MR/S031545/1UK Research and Innovationhttp://dx.doi.org/10.13039/100014013
Version or Related Resource: https://dl.acm.org/doi/10.5555/3381089.3381172
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