When Can an Expander Code Correct Ω(n) Errors in O(n) Time?

Tanner codes are error-correcting codes built from a bipartite graph G and a short inner code $C_{0}$ . Expander codes are a special type of Tanner code, where the graph is highly interconnected, ensuring stronger error correction capabilities. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that $\delta \in [{0,1}]$ and $d_{0}\in \mathbb {N}$ must satisfy, so that every bipartite expander G with vertex expansion ratio $\delta $ and every linear inner code $C_{0}$ with minimum distance $d_{0}$ together define an expander code that corrects $\Omega (n)$ 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 $\delta \gt 3/4$ is sufficient; later, Viderman (ACM-TOCT’13) improved this to $\delta \gt 2/3-\Omega (1)$ and he also showed that $\delta \gt 1/2$ is necessary. For general linear code $C_{0}$ , the previously best-known result of Dowling and Gao (IEEE-TIT’18) showed that $d_{0}=\Omega (c\delta ^{-2})$ is sufficient, where c is the left-degree of G. We present a near-optimal solution to the above problem for general $C_{0}$ by showing that $\delta d_{0}\gt 3$ is sufficient and $\delta d_{0}\gt 1$ is necessary, thereby significantly improving Dowling-Gao’s result. To prove the sufficient condition, we present two novel algorithms for decoding arbitrary expander codes with $\delta d_{0}\gt 3$ , where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius. To prove the necessary condition, we generalize the aforementioned necessary result of Viderman, and construct for every pair of $\delta,d_{0}$ with $\delta d_{0}=1$ , an expander code with constant distance, that only corrects a constant number of errors.