Near-optimal erasure list-decodable codes

Avraham Ben-Aroya, Dean Doron, Amnon Ta-Shma

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

1 Scopus citations

Abstract

A code C ⊆ {0, 1}n¯ is (s, L) erasure list-decodable if for every word w, after erasing any s symbols of w, the remaining n¯ − s symbols have at most L possible completions into a codeword of C. Non-explicitly, there exist binary ((1 − τ)¯ n, L) erasure list-decodable codes with rate approaching τ and tiny list-size L = O(log τ1 ). Achieving either of these parameters explicitly is a natural open problem (see, e.g., [26, 24, 25]). While partial progress on the problem has been achieved, no prior nontrivial explicit construction achieved rate better than Ω(τ2) or list-size smaller than Ω(1/τ). Furthermore, Guruswami showed no linear code can have list-size smaller than Ω(1/τ) [24]. We construct an explicit binary ((1 − τ)¯ n, L) erasure list-decodable code having rate τ1+γ (for any constant γ > 0 and small τ) and list-size poly(log τ1 ), answering simultaneously both questions, and exhibiting an explicit non-linear code that provably beats the best possible linear code. The binary erasure list-decoding problem is equivalent to the construction of explicit, low-error, strong dispersers outputting one bit with minimal entropy-loss and seed-length. For error ε, no prior explicit construction achieved seed-length better than 2 log(1ε ) or entropy-loss smaller than 2 log(1ε ), which are the best possible parameters for extractors. We explicitly construct an ε-error one-bit strong disperser with near-optimal seed-length (1 + γ) log(1ε ) and entropy-loss O(log log 1ε ). The main ingredient in our construction is a new (and almost-optimal) unbalanced two-source extractor. The extractor extracts one bit with constant error from two independent sources, where one source has length n and tiny min-entropy O(log log n) and the other source has length O(log n) and arbitrarily small constant min-entropy rate. When instantiated as a balanced two-source extractor, it improves upon Raz's extractor [39] in the constant error regime. The construction incorporates recent components and ideas from extractor theory with a delicate and novel analysis needed in order to solve dependency and error issues that prevented previous papers (such as [27, 9, 13]) from achieving the above results.

Original languageEnglish
Title of host publication35th Computational Complexity Conference, CCC 2020
EditorsShubhangi Saraf
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771566
DOIs
StatePublished - 1 Jul 2020
Externally publishedYes
Event35th Computational Complexity Conference, CCC 2020 - Virtual, Online, Germany
Duration: 28 Jul 202031 Jul 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume169
ISSN (Print)1868-8969

Conference

Conference35th Computational Complexity Conference, CCC 2020
Country/TerritoryGermany
CityVirtual, Online
Period28/07/2031/07/20

Keywords

  • Dispersers
  • Erasure codes
  • List decoding
  • Ramsey graphs
  • Two-source extractors

ASJC Scopus subject areas

  • Software

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