Random linear codes are a workhorse in coding theory, and are used to show the existence of codes with the best known or even near-optimal trade-offs in many noise models. However, they have little structure besides linearity, and are not amenable to tractable error-correction algorithms. In this work, we prove a general derandomization result applicable to random linear codes. Namely, in settings where the coding-theoretic property of interest is "local" (in the sense of forbidding certain bad configurations involving few vectors -- code distance and list-decodability being notable examples), one can replace random linear codes (RLCs) with a significantly derandomized variant with essentially no loss in parameters. Specifically, instead of randomly sampling coordinates of the (long) Hadamard code (which is an equivalent way to describe RLCs), one can randomly sample coordinates of any code with low bias. Over large alphabets, the low bias requirement can be weakened to just large distance. Furthermore, large distance suffices even with a small alphabet in order to match the current best known bounds for RLC list-decodability. In particular, by virtue of our result, all current (and future) achievability bounds for list-decodability of random linear codes extend automatically to random puncturings of any low-bias (or large alphabet) "mother" code. We also show that our punctured codes emulate the behavior of RLCs on stochastic channels, thus giving a derandomization of RLCs in the context of achieving Shannon capacity as well. Thus, we have a randomness-efficient way to sample codes achieving capacity in both worst-case and stochastic settings that can further inherit algebraic or other algorithmically useful structural properties of the mother code.

翻译：随机线性码是编码理论中的核心工具，用于证明在多种噪声模型下具有最佳已知甚至接近最优折中的码的存在性。然而，除线性性质外，它们缺乏其他结构，且难以实现可处理的纠错算法。本文证明了一个适用于随机线性码的通用去随机化结果：在编码理论中关注的“局部”性质（即禁止涉及少量向量的特定不良配置——码距离和列表可译性是其典型例子）的场景下，可以用一种显著去随机化的变体替代随机线性码，且参数几乎没有损失。具体而言，无需随机采样（长）哈达玛码的坐标（这是描述随机线性码的等价方式），而是可以随机采样任意低偏置码的坐标。在大字母表上，低偏置要求可放宽至仅需大距离。此外，即使在小字母表上，仅需大距离即可匹配当前已知的最佳随机线性码列表可译性界限。特别地，基于我们的结果，当前（及未来）关于随机线性码列表可译性的所有可达性界限会自动推广到任意低偏置（或大字母表）“母码”的随机穿刺。我们还证明，我们的穿刺码在随机信道上的行为与随机线性码一致，从而在实现香农容量的背景下实现了随机线性码的去随机化。因此，我们获得了一种随机性高效的方式，能够采样在极端情形和随机设置下均达到容量的码，且可进一步继承母码的代数或其他算法上有用的结构性质。