LP decoding of expander codes: a simpler proof

TL;DR

This paper simplifies the proof of LP decoding performance for expander codes, extending the error correction guarantee to all expansion parameters greater than two-thirds.

Contribution

It provides a simpler proof of existing LP decoding error correction bounds for expander codes, applicable to all expansion parameters above two-thirds.

Findings

Simplified proof of LP decoding correctness for expander codes.

Extended the error correction guarantee to all  > 2/3.

Confirmed the robustness of LP decoding with broader parameters.

Abstract

A code $C \subseteq \F_2^n$ is a $(c,\epsilon,\delta)$-expander code if it has a Tanner graph, where every variable node has degree $c$, and every subset of variable nodes $L_0$ such that $|L_0|\leq \delta n$ has at least $\epsilon c |L_0|$ neighbors. Feldman et al. (IEEE IT, 2007) proved that LP decoding corrects $\frac{3\epsilon-2}{2\epsilon-1} \cdot (\delta n-1)$ errors of $(c,\epsilon,\delta)$-expander code, where $\epsilon > 2/3+\frac{1}{3c}$. In this paper, we provide a simpler proof of their result and show that this result holds for every expansion parameter $\epsilon > 2/3$.