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, which was conjectured and eventually proven to achieve capacity. Meanwhile, polarization theory emerged as an analytic framework to prove capacity results for a variation of RM codes - the polar codes. Polarization theory further gave a powerful framework for various other code constructions, but it remained unfulfilled for RM codes. In this paper, we settle the establishment of a polarization theory for RM codes, which implies in particular that RM codes have a vanishing local error below capacity. Our proof puts forward a striking connection with the recent proof of the Polynomial Freiman-Ruzsa conjecture [40] and an entropy extraction approach related to [2]. It further puts forward a small orbit localization lemma of potential broader applicability in combinatorial number theory. Finally, a new additive combinatorics conjecture is put forward, with potentially broader applications to coding theory.

翻译：1948年，香农通过概率论证证明了存在能达到信道容量所定义最大速率的编码。1954年，Muller和Reed提出了一种基于多项式求值的简单确定性编码构造，该构造被推测并最终被证明能达到信道容量。与此同时，极化理论作为分析框架出现，用于证明RM码的一种变体——极化码的容量可达性。极化理论进一步为其他多种编码构造提供了强大框架，但该理论在RM码中始终未能建立。本文解决了RM码极化理论的建立问题，特别证明了RM码在低于容量的情况下具有趋于零的局部错误概率。我们的证明揭示了与近期多项式Freiman-Ruzsa猜想证明[40]的显著联系，以及与文献[2]相关的熵提取方法。研究还提出了一个在组合数论中可能具有更广泛适用性的小轨道局部化引理。最后，本文提出了一个新的加法组合学猜想，该猜想可能对编码理论产生更广泛的应用价值。