Linear-time Erasure List-decoding of Expander Codes

摘要

We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let r 0 be any integer. Given an inner code C0 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, C0) of length nd from approximately δδrnd erasures in time n · poly(d2 r/δ), where δ and δr are the relative distance and the r’th generalized relative distance of C0, 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 2nd. 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.