Linear-Time Erasure List-Decoding of Expander Codes

Noga Ron-Zewi; Mary Wootters; Gilles Zemor

doi:10.1109/TIT.2021.3086805

Linear-Time Erasure List-Decoding of Expander Codes

Noga Ron-Zewi, Mary Wootters, Gilles Zemor

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let r > 0 be any integer. Given an inner code {mathcal {C}}{0} of length d , and a d -regular bipartite expander graph G with n vertices on each side, we give an algorithm to list-decode the code {mathcal {C}}= {mathcal {C}}(G, {mathcal {C}}{0}) of length nd from approximately delta delta {r} nd erasures in time n cdot mathrm {poly} (d2{r} / delta) , where delta and delta {r} are the relative distance and the r 'th generalized relative distance of {mathcal {C}}{0} , respectively. To the best of our knowledge, this is the first linear-time algorithm that can list-decode expander codes from erasures beyond their (designed) distance of approximately delta {2}~nd. To obtain our results, we show that an approach similar to that of (Hemenway and Wootters, Information and Computation, 2018) can be used to obtain such an erasure-list-decoding algorithm with an exponentially worse dependence of the running time on r and delta ; then we show how to improve the dependence of the running time on these parameters.

Original language	English
Article number	9449885
Pages (from-to)	5827-5839
Number of pages	13
Journal	IEEE Transactions on Information Theory
Volume	67
Issue number	9
DOIs	https://doi.org/10.1109/TIT.2021.3086805
State	Published - Sep 2021

Bibliographical note

Funding Information:
Manuscript received April 22, 2020; revised May 2, 2021; accepted May 11, 2021. Date of publication June 9, 2021; date of current version August 25, 2021. The work of Noga Ron-Zewi was supported in part by the US-Israel Binational Science Foundation (BSF) under Grant 2017732 and in part by the Israel Science Foundation (ISF) under Grant 735/20. The work of Mary Wootters was supported in part by the NSF CAREER Award under Grant CCF-1844628, in part by the NSF-BSF Award under Grant CCF-1814629, and in part by the Sloan Research Fellowship. The work of Gilles Zémor was supported in part by the ANR Project CBCRYPT under Project ANR-17-CE39-0007. This article was presented at the 2020 IEEE International Symposium on Information Theory. (Corresponding author: Noga Ron-Zewi.) Noga Ron-Zewi is with the Department of Computer Science, University of Haifa, Haifa 3498838, Israel (e-mail: nogazewi@gmail.com).

Publisher Copyright:
© 1963-2012 IEEE.

Keywords

Error-correcting codes
erasure decoding
expander codes
list decoding

ASJC Scopus subject areas

Information Systems
Computer Science Applications
Library and Information Sciences

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10.1109/TIT.2021.3086805

Cite this

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title = "Linear-Time Erasure List-Decoding of Expander Codes",

abstract = "We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let r > 0 be any integer. Given an inner code {mathcal {C}}{0} of length d , and a d -regular bipartite expander graph G with n vertices on each side, we give an algorithm to list-decode the code {mathcal {C}}= {mathcal {C}}(G, {mathcal {C}}{0}) of length nd from approximately delta delta {r} nd erasures in time n cdot mathrm {poly} (d2{r} / delta) , where delta and delta {r} are the relative distance and the r 'th generalized relative distance of {mathcal {C}}{0} , respectively. To the best of our knowledge, this is the first linear-time algorithm that can list-decode expander codes from erasures beyond their (designed) distance of approximately delta {2}~nd. To obtain our results, we show that an approach similar to that of (Hemenway and Wootters, Information and Computation, 2018) can be used to obtain such an erasure-list-decoding algorithm with an exponentially worse dependence of the running time on r and delta ; then we show how to improve the dependence of the running time on these parameters. ",

keywords = "Error-correcting codes, erasure decoding, expander codes, list decoding",

author = "Noga Ron-Zewi and Mary Wootters and Gilles Zemor",

note = "Funding Information: Manuscript received April 22, 2020; revised May 2, 2021; accepted May 11, 2021. Date of publication June 9, 2021; date of current version August 25, 2021. The work of Noga Ron-Zewi was supported in part by the US-Israel Binational Science Foundation (BSF) under Grant 2017732 and in part by the Israel Science Foundation (ISF) under Grant 735/20. The work of Mary Wootters was supported in part by the NSF CAREER Award under Grant CCF-1844628, in part by the NSF-BSF Award under Grant CCF-1814629, and in part by the Sloan Research Fellowship. The work of Gilles Z{\'e}mor was supported in part by the ANR Project CBCRYPT under Project ANR-17-CE39-0007. This article was presented at the 2020 IEEE International Symposium on Information Theory. (Corresponding author: Noga Ron-Zewi.) Noga Ron-Zewi is with the Department of Computer Science, University of Haifa, Haifa 3498838, Israel (e-mail: nogazewi@gmail.com). Publisher Copyright: {\textcopyright} 1963-2012 IEEE.",

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N2 - We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let r > 0 be any integer. Given an inner code {mathcal {C}}{0} of length d , and a d -regular bipartite expander graph G with n vertices on each side, we give an algorithm to list-decode the code {mathcal {C}}= {mathcal {C}}(G, {mathcal {C}}{0}) of length nd from approximately delta delta {r} nd erasures in time n cdot mathrm {poly} (d2{r} / delta) , where delta and delta {r} are the relative distance and the r 'th generalized relative distance of {mathcal {C}}{0} , respectively. To the best of our knowledge, this is the first linear-time algorithm that can list-decode expander codes from erasures beyond their (designed) distance of approximately delta {2}~nd. To obtain our results, we show that an approach similar to that of (Hemenway and Wootters, Information and Computation, 2018) can be used to obtain such an erasure-list-decoding algorithm with an exponentially worse dependence of the running time on r and delta ; then we show how to improve the dependence of the running time on these parameters.

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Linear-Time Erasure List-Decoding of Expander Codes

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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