An Analysis of RPA Decoding of Reed-Muller Codes Over the BSC
V. Arvind Rameshwar, V. Lalitha
TL;DR
This work provides a formal theoretical underpinning for the Recursive Projection-Aggregation (RPA) decoder applied to binary Reed-Muller codes over the Binary Symmetric Channel. By deriving explicit upper bounds on the decoding error that hold when a single RPA iteration suffices per recursion, and by bounding the error contributions from both the first-order (ML/FHT) decoding and the aggregation steps, the paper demonstrates vanishing error probability in the large-blocklength limit for RM codes with order growing roughly like $O(\log m)$ (with $N=2^m$). The analysis begins with second-order RM codes, establishing high-probability success after one (and two) RPA iterations, and then extends via projection trees to general RM$(m,r)$, including extensions to higher-dimensional subspaces with careful concentration arguments. The results provide theoretical justification for the practicality of RPA at large scales and offer insight into the decoding-radius and rate trade-offs for RM codes under this decoding paradigm.
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 $m$.