We show that Gallager's ensemble of Low-Density Parity Check (LDPC) codes achieves list-decoding capacity with high probability. These are the first graph-based codes shown to have this property. This result opens up a potential avenue towards truly linear-time list-decodable codes that achieve list-decoding capacity. Our result on list decoding follows from a much more general result: any $\textit{local}$ property satisfied with high probability by a random linear code is also satisfied with high probability by a random LDPC code from Gallager's distribution. Local properties are properties characterized by the exclusion of small sets of codewords, and include list-decodability, list-recoverability and average-radius list-decodability. In order to prove our results on LDPC codes, we establish sharp thresholds for when local properties are satisfied by a random linear code. More precisely, we show that for any local property $\mathcal{P}$, there is some $R^*$ so that random linear codes of rate slightly less than $R^*$ satisfy $\mathcal{P}$ with high probability, while random linear codes of rate slightly more than $R^*$, with high probability, do not. We also give a characterization of the threshold rate $R^*$.

翻译：我们显示, Gallager 的低密度对数值检查( LDPC) 代码合起来的低密度对数值检查( LDPC) 代码可以实现列表解码能力, 概率很高。 这些是第一个基于图形的代码显示具有此属性的特性。 这个结果为建立真正线性时间列表解码能力打开了一条可能的路径。 我们的列表解码结果来自一个更一般性的结果: 任意线性代码满足了高概率的任何$\ text{ local} 地产也满足了来自 Gallager 分布的随机 LDPC 代码的高概率。 本地属性的特性特征是排除了小组代码的特性, 包括列表可变性、 列表可覆盖性和平均线性列表可辨码。 为了证明我们在 LDPC 代码上的结果, 我们为本地属性通过随机线性代码满足时, 更准确地说, 任何本地属性 $\ mathcall {P} 都有一定的 $( $R%) 的随机线性代码, 略低于 $ R$ 美元, 也满足了高概率。