  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. On the binary memoryless symmetric channels, using the BEC and
BI-AWGN channels as two examples, 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.
Additionally, on the BI-AWGN channel, we apply and improve the Gaussian
approximation algorithm in the density evolution of GLDPC codes. By adopting
Gaussian mixture distributions to approximate the message distributions from
variable nodes and Gaussian distributions for those from constraint nodes, the
precision of the channel parameter threshold can be significantly enhanced
while maintaining a low computational complexity similar to that of Gaussian
approximations. Furthermore, 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. Using the aforementioned techniques,
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.
