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 C₀. 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 δ ∈ [0, 1] and d₀ ∈ N must satisfy, so that every bipartite expander G with vertex expansion ratio δ and every linear inner code C₀ with minimum distance d₀ together define an expander code that corrects Ω(n) errors in O(n) time? For C₀ 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 C₀, the previously best-known result of Dowling and Gao (IEEE-TIT’18) showed that d₀ = Ω(cδ^-2) is sufficient, where c is the left-degree of G. We present a near-optimal solution to the above problem for general C₀ by showing that δd₀ > 3 is sufficient and δd₀ > 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 δd₀ > 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 δ,d₀ with δd₀=1, an expander code with constant distance, that only corrects a constant number of errors.