In 1948, Shannon used a probabilistic argument to show the existence of codes achieving a maximal rate defined by the channel capacity. In 1954, Muller and Reed introduced a simple deterministic code construction, based on polynomial evaluations, conjectured shortly after to achieve capacity. The conjecture led to decades of activity involving various areas of mathematics and the recent settlement by [AS23] using flower set boosting. In this paper, we provide an alternative proof of the weak form of the capacity result, i.e., that RM codes have a vanishing local error at any rate below capacity. Our proof relies on the recent Polynomial Freiman-Ruzsa conjecture's proof [GGMT23] and an entropy extraction approach similar to [AY19]. Further, a new additive combinatorics conjecture is put forward which would imply the stronger result with vanishing global error. We expect the latter conjecture to be more directly relevant to coding applications.

翻译：1948年，香农通过概率论证证明了存在能达到信道容量定义的最大速率的编码。1954年，Muller与Reed基于多项式求值提出了一种简单的确定性编码构造，随后不久被推测能达到容量。该猜想引发了涉及多个数学领域的数十年研究，并最终由[AS23]通过花集提升技术近期解决。本文针对容量结果的弱形式提供了另一种证明，即Reed-Muller码在任意低于容量的速率下具有趋近于零的局部误差。我们的证明依赖于近期多项式Freiman-Ruzsa猜想的证明[GGMT23]以及类似于[AY19]的熵提取方法。此外，本文提出了一个新的加法组合学猜想，若该猜想成立则可推出具有趋近于零全局误差的更强结果。我们预期后者猜想将更直接地适用于编码应用领域。