An Analysis of RPA Decoding of Reed-Muller Codes Over the BSC

TL;DR

This paper provides an explicit upper bound on the probability of decoding errors for the RPA decoder applied to Reed-Muller codes over the BSC, demonstrating its effectiveness in large blocklength regimes.

Contribution

It introduces a new upper bound on RPA decoding error probability over BSC, focusing on single-iteration convergence and error estimates for first-order RM codes.

Findings

RPA decoder achieves vanishing error probabilities for large blocklengths

Explicit bounds on decoding error probabilities are derived

Decoding success is shown for RM codes with orders growing logarithmically in blocklength

Abstract

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 \emph{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 $m$.