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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.

Construction of Near-Capacity Protograph LDPC Code Sequences With Block-Error Thresholds