Reed-Muller codes were introduced in 1954, with a simple explicit construction based on polynomial evaluations, and have long been conjectured to achieve Shannon capacity on symmetric channels. Major progress was made towards a proof over the last decades; using combinatorial weight enumerator bounds, a breakthrough on the erasure channel from sharp thresholds, hypercontractivity arguments, and polarization theory. Another major progress recently established that the bit error probability vanishes slowly below capacity. However, when channels allow for errors, the results of Bourgain-Kalai do not apply for converting a vanishing bit to a vanishing block error probability, neither do the known weight enumerator bounds. The conjecture that RM codes achieve Shannon capacity on symmetric channels, with high probability of recovering the codewords, has thus remained open. This paper closes the conjecture's proof. It uses a new recursive boosting framework, which aggregates the decoding of codeword restrictions on `subspace-sunflowers', handling their dependencies via an $L_p$ Boolean Fourier analysis, and using a list-decoding argument with a weight enumerator bound from Sberlo-Shpilka. The proof does not require a vanishing bit error probability for the base case, but only a non-trivial probability, obtained here for general symmetric codes. This gives in particular a shortened and tightened argument for the vanishing bit error probability result of Reeves-Pfister, and with prior works, it implies the strong wire-tap secrecy of RM codes on pure-state classical-quantum channels.

翻译：Reed-Muller码于1954年提出，其构造简单显式，基于多项式求值。长期以来，人们猜测该码能在对称信道上达到香农容量。过去几十年中，在证明方面取得了重大进展：利用组合权重枚举界、基于尖锐阈值的擦除信道突破、超收缩性论证以及极化理论。近期另一项重大进展确立了误比特率在低于容量时缓慢趋于零。然而，当信道允许错误时，Bourgain-Kalai的结果不适用于将渐近零的误比特率转化为渐近零的误块率，已知的权重枚举界亦不适用。因此，关于RM码以高概率恢复码字并在对称信道上达到香农容量的猜想依然悬而未决。本文完成了该猜想的证明。它采用一种新的递归提升框架，该框架聚合了“子空间-向日葵”结构上码字限制的解码，通过$L_p$布尔傅里叶分析处理其依赖关系，并利用来自Sberlo-Shpilka的权重枚举界进行列表解码论证。该证明不要求基础情形具有渐近零的误比特率，仅需非平凡概率（本文为一般对称码获得）。这特别地缩短并强化了Reeves-Pfister关于误比特率渐近为零的结果，并且结合先前工作，该结果还蕴含RM码在纯态经典-量子信道上具有强保密性。