  In this paper, we analyze the tradeoff between coding rate and asymptotic
performance of a class of generalized low-density parity-check (GLDPC) codes
constructed by including a certain fraction of generalized constraint (GC)
nodes in the graph. The rate of the GLDPC ensemble is bounded using classical
results on linear block codes, namely Hamming bound and Varshamov bound. We
also study the impact of the decoding method used at GC nodes. To incorporate
both bounded-distance (BD) and Maximum Likelihood (ML) decoding at GC nodes
into our analysis without resorting on multi-edge type of degree distributions
(DDs), we propose the probabilistic peeling decoding (P-PD) algorithm, which
models the decoding step at every GC node as an instance of a Bernoulli random
variable with a successful decoding probability that depends on both the GC
block code as well as its decoding algorithm. The P-PD asymptotic performance
over the BEC can be efficiently predicted using standard techniques for LDPC
codes such as density evolution (DE) or the differential equation method.
Furthermore, for a class of GLDPC ensembles, we demonstrate that the simulated
P-PD performance accurately predicts the actual performance of the GLPDC code
under ML decoding at GC nodes. We illustrate our analysis for GLDPC code
ensembles with regular and irregular DDs. In all cases, we show that a large
fraction of GC nodes is required to reduce the original gap to capacity, but
the optimal fraction is strictly smaller than one. We then consider techniques
to further reduce the gap to capacity by means of random puncturing, and the
inclusion of a certain fraction of generalized variable nodes in the graph.
