TY - JOUR

T1 - Linear-Time Erasure List-Decoding of Expander Codes

AU - Ron-Zewi, Noga

AU - Wootters, Mary

AU - Zemor, Gilles

N1 - 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.

PY - 2021/9/1

Y1 - 2021/9/1

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.

AB - 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.

KW - Error-correcting codes

KW - erasure decoding

KW - expander codes

KW - list decoding

UR - http://www.scopus.com/inward/record.url?scp=85111022587&partnerID=8YFLogxK

U2 - 10.1109/TIT.2021.3086805

DO - 10.1109/TIT.2021.3086805

M3 - Article

AN - SCOPUS:85111022587

VL - 67

SP - 5827

EP - 5839

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 9

M1 - 9449885

ER -