On the Error Probability of RPA Decoding of Reed-Muller Codes over BMS Channels

TL;DR

This paper proves that the RPA decoder for Reed-Muller codes achieves vanishing error probabilities over all BMS channels when the code parameters scale appropriately, extending previous results from the BSC case.

Contribution

It generalizes the analysis of RPA decoding performance from the BSC to all BMS channels without restrictive assumptions, using a novel equivalence with polar code channel combining.

Findings

Vanishing error probability for RPA decoding over BMS channels

Error probability bounds apply to all BMS channels

RM codes with order scaling as log log n achieve reliable decoding

Abstract

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.