Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph $G$ together with an inner code $C_0$. Expander codes are Tanner codes whose defining bipartite graph $G$ has good expansion property. The landmark work of Sipser and Spielman showed that every bipartite expander $G$ with expansion ratio $\delta>3/4$ together with a parity-check code defines an expander code which corrects $\Omega(n)$ errors in $O(n)$ time, where $n$ is the code length. Viderman showed that $\delta>2/3-\Omega(1)$ is already sufficient. Our paper is motivated by the following natural and fundamental problem in decoding expander codes: \textbf{Question:} What are the sufficient and necessary conditions that $\delta$ and $d_0$ should satisfy so that {\it every} bipartite expander $G$ with expansion ratio $\delta$ and {\it every} inner code $C_0$ with minimum distance $d_0$ together define an expander code which corrects $\Omega(n)$ errors in $O(n)$ time? We give a near-optimal solution to the question above, showing that $\delta d_0>3$ is sufficient and $\delta d_0>1$ is necessary. Our result significantly improves the previously known result of Dowling and Gao, who showed that $d_0=\Omega(c\delta^{-2})$ is sufficient, where $c$ is the left-degree of $G$. We suspect that $\delta d_0>1$ is also sufficient to solve the question above.

翻译：Tanner码是基于图的线性码，其校验矩阵可由二分图$G$和内码$C_0$共同刻画。扩展码是Tanner码的一种，其定义二分图$G$具有良好的扩展性质。Sipser与Spielman的开创性工作表明：任意扩展比$\delta>3/4$的二分扩展图$G$与奇偶校验码结合，可定义能纠正$\Omega(n)$个错误（$n$为码长）且解码时间复杂度为$O(n)$的扩展码。Viderman进一步指出$\delta>2/3-\Omega(1)$已足够。本文受扩展码解码中以下自然且基础的问题驱动：\textbf{问题：}为使{\it 任意}扩展比为$\delta$的二分扩展图$G$与{\it 任意}最小距离为$d_0$的内码$C_0$共同定义能纠正$\Omega(n)$个错误且时间复杂度为$O(n)$的扩展码，$\delta$与$d_0$需满足的充要条件是什么？我们对该问题给出近优解，证明$\delta d_0>3$为充分条件，而$\delta d_0>1$为必要条件。该结果显著改进了Dowling与Gao此前得到的充分条件$d_0=\Omega(c\delta^{-2})$（其中$c$为$G$的左度）。我们推测$\delta d_0>1$亦是上述问题的充分条件。