Density Evolution of Soft-Decision Collapsed Projection-Aggregation Decoding for Reed-Muller Codes over the BIAWGN Channel

2026-07-10

AI Summary

This work addresses the lack of effective analytical tools for characterizing the performance of soft-decision Collapsed Projection-Aggregation (CPA) decoding over the binary-input additive white Gaussian noise (BIAWGN) channel. It establishes, for the first time, that the soft-decision CPA output corresponds exactly to marginal posterior probabilities and exhibits symmetry. Leveraging these properties, the authors develop a density evolution framework tailored to Reed–Muller codes. By introducing a hard-decision approximation to simplify both the projection step and the fast Hadamard transform, the analysis complexity is substantially reduced. The proposed model accurately captures the rapid decay of the mean and variance of soft information, and asymptotic analysis reveals that as the code rate approaches zero, CPA decoding achieves vanishing error probability, thereby uncovering its fast convergence mechanism.

Abstract

Reed-Muller (RM) codes have been shown to achieve capacity over a range of channels, and recently proposed projection-aggregation (PA) decoding has been experimentally shown to achieve near-maximum-likelihood decoding performance. These recent achievements motivate theoretical research on PA decoding. In this work, we analyze the density function of the soft output from collapsed projection-aggregation (CPA) decoding for RM codes over the binary-input additive white Gaussian noise (BIAWGN) channel. We prove that soft-decision CPA decoding returns an exact marginal probability and is symmetric. Based on the analysis, we build a density evolution model for CPA decoding. To simplify the density evolution, we approximate the projection and the fast Hadamard transform decoding using hard-decision decoding. Simulation results over the BIAWGN channel show that our proposed density evolution model captures the fast reduction in the mean and the variance of the soft information returned from the CPA decoding, which qualitatively explains the decoding mechanism and the fast convergence speed of the CPA decoding. We perform an asymptotic analysis based on the proposed density evolution, and we show that CPA decoding can achieve a vanishing error probability for RM codes with a vanishing code rate.

Problem

• Reed-Muller codes
• density evolution
• soft-decision decoding
• BIAWGN channel
• collapsed projection-aggregation

Innovation

• density evolution
• collapsed projection-aggregation decoding
• Reed-Muller codes
• soft-decision decoding
• BIAWGN channel