  Density evolution for protograph Low-Density Parity-Check (LDPC) codes is
considered, and it is shown that the message-error rate falls
double-exponentially with iterations whenever the degree-2 subgraph of the
protograph is cycle-free and noise level is below threshold. Conditions for
stability of protograph density evolution are established and related to the
structure of the protograph. Using large-girth graphs, sequences of protograph
LDPC codes with block-error threshold equal to bit-error threshold and
block-error rate falling near-exponentially with blocklength are constructed
deterministically. Small-sized protographs are optimized to obtain thresholds
near capacity for binary erasure and binary-input Gaussian channels.
