On the Gap to Channel Capacity of Generalized Low-Density Parity-Check Codes

In this study, the gap to channel capacity of generalized low-density parity-check (GLDPC) codes under the a posteriori probability (APP) decoder on the binary input additive white Gaussian noise (BI-AWGN) channel is analyzed. Building on the density evolution for LDPC codes, we extend this to GLDPC codes by generalizing the properties of concentration, symmetry, and monotonicity to accommodate the characteristics of GLDPC codes. Specifically, we propose a methodology to simplify the computation of density evolution for GLDPC codes under APP decoding over BI-AWGN channels. Firstly, we identify a class of subcodes that can greatly simplify the performance analysis and practical decoding of GLDPC codes, which we refer to as message-invariant subcodes. Secondly, based on the characteristics of GLDPC codes, we develop a Gaussian mixture approximation algorithm to approximate the message distributions in density evolution. Compared to Gaussian approximation, the proposed Gaussian mixture approximation approach can greatly enhance accuracy while maintaining low computational complexity. Based on the above techniques, we demonstrate that with an appropriate proportion of generalized constraint (GC) nodes, despite the rate loss when single parity-check (SPC) nodes are replaced by GC nodes, GLDPC codes can reduce the original gap to capacity compared to their original LDPC counterparts. Our simulation experiments validate the performance analysis.