This paper shows that a class of codes such as Reed-Muller (RM) codes have vanishing bit-error probability below capacity on symmetric channels. The proof relies on the notion of `camellia codes': a class of symmetric codes decomposable into `camellias', i.e., set systems that differ from sunflowers by allowing for scattered petal overlaps. The proof then follows from a boosting argument on the camellia petals with second moment Fourier analysis. For erasure channels, this gives a self-contained proof of the bit-error result in Kudekar et al.'17, without relying on sharp thresholds for monotone properties Friedgut-Kalai'96. For error channels, this gives a shortened proof of Reeves-Pfister'23 with an exponentially tighter bound, and a proof variant of the bit-error result in Abbe-Sandon'23. The control of the full (block) error probability still requires Abbe-Sandon'23 for RM codes.

翻译：本文证明，在全球对称信道上，一类码（如Reed-Muller（RM）码）在容量以下具有消失的比特错误概率。该证明依赖于“山茶花码”的概念：一类可分解为“山茶花”的对称码，即允许花瓣交错的集合系统，不同于向日葵结构。随后，证明通过对山茶花花瓣进行二阶矩傅里叶分析的提升论证完成。对于擦除信道，这给出了Kudekar等人（2017年）中比特错误结果的一个自包含证明，无需依赖单调性质的尖锐阈值（Friedgut-Kalai，1996年）。对于错误信道，这给出了Reeves-Pfister（2023年）的一个缩短证明，且界指数级更紧；同时给出了Abbe-Sandon（2023年）中比特错误结果的一个变体证明。对于RM码的全（块）错误概率控制仍需依赖Abbe-Sandon（2023年）的工作。