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Lower bounds for 2-query LCCs over large alphabet

Bhattacharyya, A and Gopi, S and Tal, A (2017) Lower bounds for 2-query LCCs over large alphabet. In: 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017, 16 - 18 August 2017, Berkeley.

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A locally correctable code (LCC) is an error correcting code that allows correction of any arbitrary coordinate of a corrupted codeword by querying only a few coordinates. We show that any 2- query locally correctable code C : 0, 1k ! -n that can correct a constant fraction of corrupted symbols must have n > exp(k/ log) under the assumption that the LCC is zero-error. We say that an LCC is zero-error if there exists a non-adaptive corrector algorithm that succeeds with probability 1 when the input is an uncorrupted codeword. All known constructions of LCCs are zero-error. Our result is tight upto constant factors in the exponent. The only previous lower bound on the length of 2-query LCCs over large alphabet was((k/ log |-|)2) due to Katz and Trevisan (STOC 2000). Our bound implies that zero-error LCCs cannot yield 2-server private information retrieval (PIR) schemes with sub-polynomial communication. Since there exists a 2-server PIR scheme with sub-polynomial communication (STOC 2015) based on a zero-error 2-query locally decodable code (LDC), we also obtain a separation between LDCs and LCCs over large alphabet. © Arnab Bhattacharyya, Sivakanth Gopi, and Avishay Tal.

Item Type: Conference Paper
Publication: Leibniz International Proceedings in Informatics, LIPIcs
Publisher: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information: The copyright for this article belongs to the Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing.
Keywords: Approximation algorithms; Codes (symbols); Combinatorial optimization; Errors; Information retrieval; Optimization; Random processes, Constant factors; Error correcting code; Large alphabets; Locally correctable codes; Locally-decodable codes; Lower bounds; Private information retrieval; Regularity lemma, C (programming language)
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Date Deposited: 17 Jul 2022 05:54
Last Modified: 17 Jul 2022 05:54
URI: https://eprints.iisc.ac.in/id/eprint/74499

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