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Protograph low-density-parity-check (LDPC) are considered to designnear-capacity low-rate codes over the binary erasure channel (BEC) and binaryadditive white Gaussian noise (BIAWGN) channel. For protographs with degree-onevariable nodes and doubly-generalized LDPC (DGLDPC) codes, conditions arederived to ensure equality of bit-error threshold and block-error threshold.Using this condition, low-rate codes with block-error threshold close tocapacity are designed and shown to have better error rate performance thanother existing codes.

Protograph LDPC Codes with Block Thresholds: Extension to Degree-1 and Generalized Nodes