RELAXED LOCALLY CORRECTABLE CODES WITH NEARLY-LINEAR BLOCK LENGTH AND CONSTANT QUERY COMPLEXITY.
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| Title: | RELAXED LOCALLY CORRECTABLE CODES WITH NEARLY-LINEAR BLOCK LENGTH AND CONSTANT QUERY COMPLEXITY. |
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| Authors: | CHIESA, ALESSANDRO1 alexch@berkeley.edu, GUR, TOM2 tom.gur@warwick.ac.uk, SHINKAR, IGOR3 ishinkar@sfu.ca |
| Source: | SIAM Journal on Computing. 2022, Vol. 51 Issue 6, p1839-1865. 27p. |
| Subjects: | Block codes, Computer science conferences, Coding theory |
| Geographic Terms: | Thailand |
| Abstract: | Locally correctable codes (LCCs) are error correcting codes C:Σk→Σn which admit local algorithms that correct any individual symbol of a corrupted codeword via a minuscule number of queries. For systematic codes, this notion is stronger than that of locally decodable codes (LDCs), where the goal is to only recover individual symbols of the message. One of the central problems in algorithmic coding theory is to construct O(1) -query LCCs and LDCs with minimal block length. Alas, state-of-the-art of such codes requires super-polynomial block length to admit O(1) -query algorithms for local correction and decoding, despite much attention during the last two decades. The study of relaxed LCCs and LDCs, which allow the correction algorithm to abort (but not err) on a small fraction of the locations, provides a way to circumvent this barrier. This relaxation turned out to allow constant-query correcting and decoding algorithms for codes with polynomial block length. Focusing on local correction, Gur, Ramnarayan, and Rothblum [Proceedings of the 9th Innovations in Theoretical Computer Science Conference, ITCS'18, 2018, pp. 1-27] showed that there exist O(1) -query relaxed LCCs that achieve nearly-quartic block length n=k4+α, for an arbitrarily small constant α>0. We construct an O(1) -query relaxed LCC with nearly-linear block length n=k1+α, for an arbitrarily small constant α>0. This significantly narrows the gap between the lower bound which states that there are no O(1) -query relaxed LCCs with block length n=k1+o(1). In particular, our construction matches the parameters achieved by Ben-Sasson et al. [SIAM J. Comput., 36 (2006), pp. 889-974], who constructed relaxed LDCs with the same parameters. This resolves an open problem raised by Gur, Ramnarayan, and Rothblum [Proceedings of the 9th Innovations in Theoretical Computer Science Conference, ITCS'18, 2018, pp. 1-27]. [ABSTRACT FROM AUTHOR] |
| Copyright of SIAM Journal on Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 161682062 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: RELAXED LOCALLY CORRECTABLE CODES WITH NEARLY-LINEAR BLOCK LENGTH AND CONSTANT QUERY COMPLEXITY. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22CHIESA%2C+ALESSANDRO%22">CHIESA, ALESSANDRO</searchLink><relatesTo>1</relatesTo><i> alexch@berkeley.edu</i><br /><searchLink fieldCode="AR" term="%22GUR%2C+TOM%22">GUR, TOM</searchLink><relatesTo>2</relatesTo><i> tom.gur@warwick.ac.uk</i><br /><searchLink fieldCode="AR" term="%22SHINKAR%2C+IGOR%22">SHINKAR, IGOR</searchLink><relatesTo>3</relatesTo><i> ishinkar@sfu.ca</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Computing%22">SIAM Journal on Computing</searchLink>. 2022, Vol. 51 Issue 6, p1839-1865. 27p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Block+codes%22">Block codes</searchLink><br /><searchLink fieldCode="DE" term="%22Computer+science+conferences%22">Computer science conferences</searchLink><br /><searchLink fieldCode="DE" term="%22Coding+theory%22">Coding theory</searchLink> – Name: SubjectGeographic Label: Geographic Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Thailand%22">Thailand</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Locally correctable codes (LCCs) are error correcting codes C:Σk→Σn which admit local algorithms that correct any individual symbol of a corrupted codeword via a minuscule number of queries. For systematic codes, this notion is stronger than that of locally decodable codes (LDCs), where the goal is to only recover individual symbols of the message. One of the central problems in algorithmic coding theory is to construct O(1) -query LCCs and LDCs with minimal block length. Alas, state-of-the-art of such codes requires super-polynomial block length to admit O(1) -query algorithms for local correction and decoding, despite much attention during the last two decades. The study of relaxed LCCs and LDCs, which allow the correction algorithm to abort (but not err) on a small fraction of the locations, provides a way to circumvent this barrier. This relaxation turned out to allow constant-query correcting and decoding algorithms for codes with polynomial block length. Focusing on local correction, Gur, Ramnarayan, and Rothblum [Proceedings of the 9th Innovations in Theoretical Computer Science Conference, ITCS'18, 2018, pp. 1-27] showed that there exist O(1) -query relaxed LCCs that achieve nearly-quartic block length n=k4+α, for an arbitrarily small constant α>0. We construct an O(1) -query relaxed LCC with nearly-linear block length n=k1+α, for an arbitrarily small constant α>0. This significantly narrows the gap between the lower bound which states that there are no O(1) -query relaxed LCCs with block length n=k1+o(1). In particular, our construction matches the parameters achieved by Ben-Sasson et al. [SIAM J. Comput., 36 (2006), pp. 889-974], who constructed relaxed LDCs with the same parameters. This resolves an open problem raised by Gur, Ramnarayan, and Rothblum [Proceedings of the 9th Innovations in Theoretical Computer Science Conference, ITCS'18, 2018, pp. 1-27]. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of SIAM Journal on Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1137/20M135515X Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 27 StartPage: 1839 Subjects: – SubjectFull: Block codes Type: general – SubjectFull: Computer science conferences Type: general – SubjectFull: Coding theory Type: general – SubjectFull: Thailand Type: general Titles: – TitleFull: RELAXED LOCALLY CORRECTABLE CODES WITH NEARLY-LINEAR BLOCK LENGTH AND CONSTANT QUERY COMPLEXITY. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: CHIESA, ALESSANDRO – PersonEntity: Name: NameFull: GUR, TOM – PersonEntity: Name: NameFull: SHINKAR, IGOR IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 11 Text: 2022 Type: published Y: 2022 Identifiers: – Type: issn-print Value: 00975397 Numbering: – Type: volume Value: 51 – Type: issue Value: 6 Titles: – TitleFull: SIAM Journal on Computing Type: main |
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