AI 中文总结
研究二元输入加性高斯白噪声信道上RM码的CPA译码，通过分析其软输出密度函数建立密度进化模型，用硬判决译码近似简化，仿真结果显示该模型能捕捉软信息变化，渐近分析表明CPA译码对低码率RM码可实现零错误概率。
AI 中文摘要
Reed-Muller（RM）码已被证明在一系列信道上能达到容量，最近提出的投影聚合（PA）译码经实验显示能实现接近最大似然译码性能。这些成果推动了对PA译码的理论研究。本文分析了二元输入加性高斯白噪声（BIAWGN）信道上RM码的折叠投影聚合（CPA）译码软输出的密度函数，证明软判决CPA译码返回精确边缘概率且对称，据此建立了CPA译码的密度进化模型。为简化密度进化，用硬判决译码近似投影和快速哈达玛变换译码。BIAWGN信道的仿真结果表明，所提密度进化模型捕捉到了CPA译码软信息均值和方差的快速减小，定性解释了译码机制和快速收敛速度。基于所提密度进化进行渐近分析，表明CPA译码对码率趋于零的RM码能实现错误概率趋于零。
英文摘要
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.
