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 {\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.