We construct the first (locally computable, approximately) locally list decodable codes with rate, efficiency, and error tolerance approaching the information theoretic limit, a core regime of interest for the complexity theoretic task of hardness amplification. Our algorithms run in polylogarithmic time and sub-logarithmic depth, which together with classic constructions in the unique decoding (low-noise) regime leads to the resolution of several long-standing problems in coding and complexity theory: 1. Near-optimally input-preserving hardness amplification (and corresponding fast PRGs) 2. Constant rate codes with $\log(N)$-depth list decoding (RNC$^1$) 3. Complexity-preserving distance amplification Our codes are built on the powerful theory of (local-spectral) high dimensional expanders (HDX). At a technical level, we make two key contributions. First, we introduce a new framework for ($\mathrm{polylog(N)}$-round) belief propagation on HDX that leverages a mix of local correction and global expansion to control error build-up while maintaining high rate. Second, we introduce the notion of strongly explicit local routing on HDX, local algorithms that given any two target vertices, output a random path between them in only polylogarithmic time (and, preferably, sub-logarithmic depth). Constructing such schemes on certain coset HDX allows us to instantiate our otherwise combinatorial framework in polylogarithmic time and low depth, completing the result.

翻译：我们首次构造了（局部可计算、近似）局部列表可译码码，其码率、效率和容错能力逼近信息论极限——这是复杂性理论中硬度放大任务的核心关注区域。我们的算法在多重对数时间和对数深度下运行，结合经典唯一译码（低噪声）区域的构造，解决了编码与复杂性理论中若干长期悬而未决的问题：1. 近乎最优的输入保持型硬度放大（及相应的快速伪随机生成器）2. 具有$\log(N)$深度列表译码（RNC$^1$）的恒定码率编码 3. 复杂度保持型距离放大。我们的编码建立在（局部谱）高维扩展器（HDX）的强大理论基础上。在技术层面，我们做出两项关键贡献：首先，我们提出了一种基于HDX的（$\mathrm{polylog(N)}$轮）置信传播新框架，通过融合局部校正与全局扩展来控制误差累积，同时保持高码率。其次，我们引入了HDX上的强显式局部路由概念——这种局部算法能在仅多重对数时间（且优选亚对数深度）内，针对任意两个目标顶点输出其间的随机路径。在特定陪集HDX上构建此类方案，使我们能够以多重对数时间和低深度实例化原本的组合框架，从而完成该成果。