## 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 |
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Article number | 9449885 |

Pages (from-to) | 5827-5839 |

Number of pages | 13 |

Journal | IEEE Transactions on Information Theory |

Volume | 67 |

Issue number | 9 |

DOIs | |

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