Linear-time list recovery of high-rate expander codes

TL;DR

This paper demonstrates that high-rate expander codes can be efficiently list recovered in linear time, advancing the construction of codes suitable for list-decoding and related applications.

Contribution

It introduces a method to construct high-rate expander codes with linear-time list recovery algorithms, surpassing previous rate limitations.

Findings

Codes can have rate close to 1 with linear-time list recovery

Applicable to list-decoding, compressive sensing, and group testing

Approaching optimal trade-offs between rate and list size

Abstract

We show that expander codes, when properly instantiated, are high-rate list recoverable codes with linear-time list recovery algorithms. List recoverable codes have been useful recently in constructing efficiently list-decodable codes, as well as explicit constructions of matrices for compressive sensing and group testing. Previous list recoverable codes with linear-time decoding algorithms have all had rate at most 1/2; in contrast, our codes can have rate $1 - \epsilon$ for any $\epsilon > 0$. We can plug our high-rate codes into a construction of Meir (2014) to obtain linear-time list recoverable codes of arbitrary rates, which approach the optimal trade-off between the number of non-trivial lists provided and the rate of the code. While list-recovery is interesting on its own, our primary motivation is applications to list-decoding. A slight strengthening of our result would implies linear-time and optimally list-decodable codes for all rates, and our work is a step in the direction of solving this important problem.