Efficient list-decoding with constant alphabet and list sizes

Zeyu Guo, Noga Ron-Zewi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations

Abstract

We present an explicit and efficient algebraic construction of capacity-achieving list decodable codes with both constant alphabet and constant list sizes. More specifically, for any R e (0,1) and ?>0, we give an algebraic construction of an infinite family of error-correcting codes of rate R, over an alphabet of size (1/?)O(1/?2), that can be list decoded from a (1-R-?)-fraction of errors with list size at most exp(poly(1/?)). Moreover, the codes can be encoded in time poly(1/?, n), the output list is contained in a linear subspace of dimension at most poly(1/?), and a basis for this subspace can be found in time poly(1/?, n). Thus, both encoding and list decoding can be performed in fully polynomial-time poly(1/?, n), except for pruning the subspace and outputting the final list which takes time exp(poly(1/?)) · poly(n). In contrast, prior explicit and efficient constructions of capacity-achieving list decodable codes either required a much higher complexity in terms of 1/? (and were additionally much less structured), or had super-constant alphabet or list sizes. Our codes are quite natural and structured. Specifically, we use algebraic-geometric (AG) codes with evaluation points restricted to a subfield, and with the message space restricted to a (carefully chosen) linear subspace. Our main observation is that the output list of AG codes with subfield evaluation points is contained in an affine shift of the image of a block-triangular-Toeplitz (BTT) matrix, and that the list size can potentially be reduced to a constant by restricting the message space to a BTT evasive subspace, which is a large subspace that intersects the image of any BTT matrix in a constant number of points. We further show how to explicitly construct such BTT evasive subspaces, based on the explicit subspace designs of Guruswami and Kopparty (Combinatorica, 2016), and composition.

Original languageEnglish
Title of host publicationSTOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
EditorsSamir Khuller, Virginia Vassilevska Williams
PublisherAssociation for Computing Machinery
Pages1502-1515
Number of pages14
ISBN (Electronic)9781450380539
DOIs
StatePublished - 15 Jun 2021
Externally publishedYes
Event53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy
Duration: 21 Jun 202125 Jun 2021

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
Country/TerritoryItaly
CityVirtual, Online
Period21/06/2125/06/21

Keywords

  • algebraic codes
  • algebraic-geometric codes
  • error-correcting codes
  • explicit constructions
  • list decoding
  • pseudorandomness

ASJC Scopus subject areas

  • Software

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