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Revision #1 to TR20-013 | 24th May 2021 21:23

#### Linear-time Erasure List-decoding of Expander Codes

Revision #1
Authors: Noga Ron-Zewi, Mary Wootters, Gilles Z\'{e}mor
Accepted on: 24th May 2021 21:23
Keywords:

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 $\cC_0$ 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 $\cC = \cC(G, \cC_0)$ of length $nd$ from approximately $\delta \delta_r nd$ erasures in time $n \cdot \poly(d2^r / \delta)$, where $\delta$ and $\delta_r$ are the relative distance and the $r$'th generalized relative distance of $\cC_0$, 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, \emph {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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Revision per final journal version

### Paper:

TR20-013 | 17th February 2020 20:35

#### Linear-time Erasure List-decoding of Expander Codes

TR20-013
Authors: Noga Ron-Zewi, Mary Wootters, Gilles Z\'{e}mor
Publication: 18th February 2020 17:44
We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let $r > 0$ be any integer. Given an inner code $\cC_0$ 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 expander code $\cC = \cC(G, \cC_0)$ of length $nd$ from approximately $\delta \delta_r nd$ erasures in time $n \cdot \poly(d2^r / \delta)$, where $\delta$ and $\delta_r$ are the relative distance and the $r$'th generalized relative distance of $\cC_0$, 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, \emph {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.