In this study, the performance of generalized low-density parity-check (GLDPC) codes under the a posteriori probability (APP) decoder is analyzed. We explore the concentration, symmetry, and monotonicity properties of GLDPC codes under the APP decoder, extending the applicability of density evolution to GLDPC codes. We demonstrate that with an appropriate proportion of generalized constraint (GC) nodes, GLDPC codes can reduce the original gap to capacity compared to their original LDPC counterparts over the BEC and BI-AWGN channels. Additionally, on the BI-AWGN channel, we adopt Gaussian mixture distributions to approximate the message distributions from variable nodes and Gaussian distributions for those from constraint nodes. This approximation technique significantly enhances the precision of the channel parameter threshold compared to traditional Gaussian approximations while maintaining a low computational complexity similar to that of Gaussian approximations. Our simulation experiments provide empirical evidence that GLDPC codes, when decoded with the APP decoder and equipped with the right fraction of GC nodes, can achieve substantial performance improvements compared to low-density parity-check (LDPC) codes.

翻译：本研究分析了广义低密度奇偶校验（GLDPC）码在后验概率（APP）译码器下的性能。我们探讨了GLDPC码在APP译码器下的集中性、对称性和单调性性质，将密度演化的适用范围扩展至GLDPC码。我们证明，在二元删除信道（BEC）和二进制输入加性高斯白噪声（BI-AWGN）信道下，当采用适当比例的广义约束（GC）节点时，GLDPC码相较于原始LDPC码可缩小与信道容量的差距。此外，在BI-AWGN信道上，我们采用高斯混合分布近似变量节点的消息分布，并用高斯分布近似约束节点的消息分布。该近似技术相较于传统高斯近似显著提升了信道参数阈值的精度，同时保持了与高斯近似相当的低计算复杂度。仿真实验提供了实证证据表明，采用APP译码器并配备适当比例GC节点的GLDPC码，相较于低密度奇偶校验（LDPC）码能够实现显著的性能提升。