When can an expander code correct $Ω(n)$ errors in $O(n)$ time?
Kuan Cheng, Minghui Ouyang, Chong Shangguan, Yuanting Shen
TL;DR
This work analyzes when expander-based Tanner codes $T(G,C_0)$ can correct a linear fraction of errors in linear time, focusing on the interplay between the graph expansion $\delta$ and the inner code's minimum distance $d_0$. It proves near-optimal bounds: $\delta d_0>3$ suffices for deterministic linear-time decoding and $\delta d_0>1$ is necessary, improving prior results that tied decoding guarantees to stricter conditions. The authors introduce two decoding paradigms—deterministic MainDecode and linear-time randomized decoding—built on a novel flip strategy (EasyFlip/DeepFlip) and a HardSearch refinement that collectively ensure linear-time recovery from $\Theta(n)$ errors. The results close a substantial portion of the gap between known sufficient conditions and necessary conditions, with explicit decoding radii and concrete time bounds, while outlining future directions to tighten constants and extend to broader inner-code families. The work has potential impact on practical linear-time decodable codes with strong distance properties and informs the limits of graph-based decoding strategies for expander codes.
Abstract
Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph $G$ together with a linear inner code $C_0$. Expander codes are Tanner codes whose defining bipartite graph $G$ has good expansion property. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that $δ$ and $d_0$ must satisfy, so that \textit{every} bipartite expander $G$ with vertex expansion ratio $δ$ and \textit{every} linear inner code $C_0$ with minimum distance $d_0$ together define an expander code that corrects $Ω(n)$ errors in $O(n)$ time? For $C_0$ being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that $δ>3/4$ is sufficient; later Viderman (ACM-TOCT'13) improved this to $δ>2/3-Ω(1)$ and he also showed that $δ>1/2$ is necessary. For general linear code $C_0$, the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that $d_0=Ω(cδ^{-2})$ is sufficient, where $c$ is the left-degree of $G$. In this paper, we give a near-optimal solution to the above question for general $C_0$ by showing that $δd_0>3$ is sufficient and $δd_0>1$ is necessary, thereby also significantly improving Dowling-Gao's result. We present two novel algorithms for decoding expander codes, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius.