Improved List-Decodability and List-Recoverability of Reed-Solomon Codes via Tree Packings: [Extended Abstract]

Zeyu Guo, Ray Li, Chong Shangguan, Itzhak Tamo, Mary Wootters

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

Abstract

This paper shows that there exist Reed-Solomon (RS) codes, over large finite fields, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving list-decoding capacity. In particular, we show that for any ϵ E (0,1] there exist RS codes with rate Ω(1\not(1/in)+1}) that are list-decodable from radius of 1-ϵ. We generalize this result to list-recovery, showing that there exist (1-,ell, O(ℓ)-list-recoverable RS codes with rate Ω≤ft(√\ell}(log(1/)+1)right). Along the way we use our techniques to give a new proof of a result of Blackburn on optimal linear perfect hash matrices, and strengthen it to obtain a construction of strongly perfect hash matrices. To derive the results in this paper we show a surprising connection of the above problems to graph theory, and in particular to the tree packing theorem of Nash-Williams and Tutte. We also state a new conjecture that generalizes the tree-packing theorem to hypergraphs, and show that if this conjecture holds, then there would exist RS codes that are optimally (non-asymptotically) list-decodable.11A full version of this paper is available online at https://arxiv.org/abs/2011.04453.

Original languageEnglish
Title of host publicationProceedings - 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science, FOCS 2021
PublisherIEEE Computer Society
Pages708-719
Number of pages12
ISBN (Electronic)9781665420556
DOIs
StatePublished - 2022
Event62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021 - Virtual, Online, United States
Duration: 7 Feb 202210 Feb 2022

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2022-February
ISSN (Print)0272-5428

Conference

Conference62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021
Country/TerritoryUnited States
CityVirtual, Online
Period7/02/2210/02/22

Keywords

  • Johnson radius
  • Nash-Williams-Tutte Theo-rem
  • Reed-Solomon codes
  • list decoding
  • list recovery
  • perfect hash matrix

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