Linear-time Erasure List-decoding of Expander Codes

We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let $r > 0$ be any integer. Given an inner code $C_0$ 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 expander code $C = C(G, C_0)$ of length $nd$ from approximately $δδ_r nd$ erasures in time $n \cdot \mathrm{poly}(d2^r / δ)$, where $δ$ and $δ_r$ are the relative distance and the $r$'th generalized relative distance of $C_0$, 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 $δ^2 nd$. 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 $δ$; then we show how to improve the dependence of the running time on these parameters.

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