We analyze the performance of the Recursive Projection-Aggregation (RPA) decoder of Ye and Abbe (2020), for Reed-Muller (RM) codes, over general binary memoryless symmetric (BMS) channels. Our work is a significant generalization of a recent result of Rameshwar and Lalitha (2025) that showed that the RPA decoder provably achieves vanishing error probabilities for "low-rate" RM codes, over the binary symmetric channel (BSC). While a straightforward generalization of the proof strategy in that paper will require additional, restrictive assumptions on the BMS channel, our technique, which employs an equivalence between the RPA projection operation and a part of the "channel combining" phase in polar codes, requires no such assumptions. Interestingly, such an equivalence allows for the use of a generic union bound on the error probability of the first-order RM code (the "base case" of the RPA decoder), under maximum-likelihood decoding, which holds for any BMS channel. We then exploit these observations in the proof strategy outlined in the work of Rameshwar and Lalitha (2025), and argue that, much like in the case of the BSC, one can obtain vanishing error probabilities, in the large $n$ limit (where $n$ is the blocklength), for RM orders that scale roughly as $\log \log n$, for all BMS channels.

翻译：本文分析了Ye和Abbe（2020）提出的递归投影聚合（RPA）译码器在一般二进制无记忆对称（BMS）信道上对Reed-Muller（RM）码的译码性能。我们的工作是Rameshwar和Lalitha（2025）近期研究结果的重要推广，该结果表明RPA译码器在二进制对称信道（BSC）上能够为“低码率”RM码实现可证明的渐近消失误码率。虽然直接推广该论文的证明策略需要对BMS信道施加额外限制性假设，但我们的技术利用了RPA投影操作与极化码“信道合并”阶段中某一部分的等价性，从而无需此类假设。有趣的是，这种等价性允许对一阶RM码（RPA译码器的“基础情形”）在最大似然译码下的误码率使用通用联合界，该界适用于任意BMS信道。随后，我们基于Rameshwar和Lalitha（2025）工作中概述的证明策略，利用这些观察结果论证：与BSC情形类似，对于所有BMS信道，当码长$n$趋于无穷大时，对于阶数大致按$\log \log n$增长的RM码，均可获得渐近消失的误码率。