License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2020.1
URN: urn:nbn:de:0030-drops-125531
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12553/
Ben-Aroya, Avraham ;
Doron, Dean ;
Ta-Shma, Amnon
Near-Optimal Erasure List-Decodable Codes
Abstract
A code ? ⊆ {0,1}^n̅ is (s,L) erasure list-decodable if for every word w, after erasing any s symbols of w, the remaining n̅-s symbols have at most L possible completions into a codeword of ?. Non-explicitly, there exist binary ((1-τ)n̅,L) erasure list-decodable codes with rate approaching τ and tiny list-size L = O(log 1/(τ)). Achieving either of these parameters explicitly is a natural open problem (see, e.g., [Guruswami and Indyk, 2002; Guruswami, 2003; Guruswami, 2004]). 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) [Guruswami, 2003]. We construct an explicit binary ((1-τ)n̅,L) erasure list-decodable code having rate τ^(1+γ) (for any constant γ > 0 and small τ) and list-size poly(log 1/τ), 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 2log(1/ε) or entropy-loss smaller than 2log(1/ε), 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(1/ε) and entropy-loss O(log log1/ε).
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 [Raz, 2005] 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 [Li, 2015; Chattopadhyay and Zuckerman, 2019; Cohen, 2016]) from achieving the above results.
BibTeX - Entry
@InProceedings{benaroya_et_al:LIPIcs:2020:12553,
author = {Avraham Ben-Aroya and Dean Doron and Amnon Ta-Shma},
title = {{Near-Optimal Erasure List-Decodable Codes}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {1:1--1:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Shubhangi Saraf},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12553},
URN = {urn:nbn:de:0030-drops-125531},
doi = {10.4230/LIPIcs.CCC.2020.1},
annote = {Keywords: Dispersers, Erasure codes, List decoding, Ramsey graphs, Two-source extractors}
}
|
Keywords: |
|
Dispersers, Erasure codes, List decoding, Ramsey graphs, Two-source extractors |
|
Collection: |
|
35th Computational Complexity Conference (CCC 2020) |
|
Issue Date: |
|
2020 |
|
Date of publication: |
|
17.07.2020 |
