Quasilinear time list-decodable codes for space bounded channels

Jad Silbak, Swastik Kopparty, Ronen Shaltiel

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

Abstract

We consider codes for space bounded channels. This is a model for communication under noise that was studied by Guruswami and Smith (J. ACM 2016) and lies between the Shannon (random) and Hamming (adversarial) models. In this model, a channel is a space bounded procedure that reads the codeword in one pass, and modifies at most a p fraction of the bits of the codeword. Guruswami and Smith, and later work by Shaltiel and Silbak (RANDOM 2016), gave constructions of list-decodable codes with rate approaching 1-H(p) against channels with space s=c log n, with encoding/decoding time poly(2s)= poly(nc). In this paper we show that for every constant 0 ≤ p <, and every sufficiently small constant ϵ>0, there are codes with rate R ≥ 1-H(p)-ϵ, list size poly(1/ϵ), and furthermore: • Our codes can handle channels with space s=nΩ(1), which is much larger than O(log n) achieved by previous work. • We give encoding and decoding algorithms that run in time n polylog(n). Previous work achieved large and unspecified poly(n) time (even for space s=1 ⋅ log n channels). • We can handle space bounded channels that read the codeword in any order, whereas previous work considered channels that read the codeword in the standard order. Our construction builds on the machinery of Guruswami and Smith (with some key modifications) replacing some nonconstructive codes and pseudorandom objects (that are found in exponential time by brute force) with efficient explicit constructions. For this purpose we exploit recent results of Haramaty, Lee and Viola (SICOMP 2018) on pseudorandom properties of 't-wise independence + low weight noise'' which we quantitatively improve using techniques by Forbes and Kelly (FOCS 2018). To make use of such distributions, we give new explicit constructions of binary linear codes that have dual distance of nΩ(1), and are also polynomial time list-decodable from relative distance -ϵ, with list size poly(1/ϵ). To the best of our knowledge, no such construction was previously known. Somewhat surprisingly, we show that Reed-Solomon codes with dimension k<√n, have this property if interpreted as binary codes (in some specific interpretation) which we term: 'Raw Reed-Solomon Codes''. A key idea is viewing Reed-Solomon codes as 'bundles'' of certain dual-BCH codewords.

Original languageEnglish
Title of host publicationProceedings - 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019
PublisherIEEE Computer Society
Pages302-333
Number of pages32
ISBN (Electronic)9781728149523
DOIs
StatePublished - Nov 2019
Event60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019 - Baltimore, United States
Duration: 9 Nov 201912 Nov 2019

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2019-November
ISSN (Print)0272-5428

Conference

Conference60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019
Country/TerritoryUnited States
CityBaltimore
Period9/11/1912/11/19

Bibliographical note

Publisher Copyright:
© 2019 IEEE.

Keywords

  • Error-correcting codes
  • Online-space channels
  • Pseudorandomness

ASJC Scopus subject areas

  • Computer Science (all)

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