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

2026-01-14

AI Summary

This work investigates the error probability performance of the recursive projection-aggregation (RPA) decoder for Reed–Muller (RM) codes over arbitrary binary memoryless symmetric (BMS) channels. By establishing an equivalence between the RPA projection operation and the channel combining phase in polar codes, and leveraging a union bound analysis of maximum-likelihood decoding together with a recursive proof strategy, the authors derive a general union bound applicable to any BMS channel. Notably, they prove—without imposing additional constraints on the channel—that the decoding error probability of RPA vanishes as the blocklength \(n \to \infty\) when the RM code order is approximately \(\log\log n\). This result substantially generalizes prior findings, which were limited to specific channel models or required extra assumptions.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Relational Probabilistic Models
• Multiagent Systems: Multiagent Systems under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• RPA decoding
• Reed-Muller codes
• error probability
• BMS channels
• vanishing error probability

Innovation

• RPA decoding
• Reed-Muller codes
• BMS channels
• channel polarization
• error probability