In this paper, we revisit the Recursive Projection-Aggregation (RPA) decoder, of Ye and Abbe (2020), for Reed-Muller (RM) codes. Our main contribution is an explicit upper bound on the probability of incorrect decoding, using the RPA decoder, over a binary symmetric channel (BSC). Importantly, we focus on the events where a single iteration of the RPA decoder, in each recursive call, is sufficient for convergence. Key components of our analysis are explicit estimates of the probability of incorrect decoding of first-order RM codes using a maximum likelihood (ML) decoder, and estimates of the error probabilities during the aggregation phase of the RPA decoder. Our results allow us to show that for RM codes with blocklength $N = 2^m$, the RPA decoder can achieve vanishing error probabilities, in the large blocklength limit, for RM orders that grow roughly logarithmically in $m$.

翻译：本文重新审视了Ye与Abbe（2020）提出的用于Reed-Muller（RM）码的递归投影聚合（RPA）译码器。我们的主要贡献是推导了在二进制对称信道（BSC）下使用RPA译码器进行译码的错误概率的显式上界。值得注意的是，我们重点分析了在每次递归调用中仅需单次RPA译码迭代即可收敛的情形。我们分析的关键组成部分包括：使用最大似然（ML）译码器对一阶RM码进行译码的错误概率的显式估计，以及对RPA译码器聚合阶段错误概率的估计。我们的结果表明，对于码块长度$N = 2^m$的RM码，在码长趋于无穷大的极限下，当RM码的阶数以大致对数速率随$m$增长时，RPA译码器能够实现趋近于零的错误概率。