Design of Outage-Limit-Approaching Protograph LDPC Codes via Generalized Rootchecks

This paper presents a new protograph-based LDPC code design framework that simultaneously achieves full diversity over block-fading channels (BFCs) and near-capacity performance over additive white Gaussian noise channels. By leveraging a Boolean approximation-based analysis-Diversity Evolution-we derive structural constraints with generalized rootchecks that guarantee full diversity. Building on these constraints, we propose a diversity-aligned protograph template tailored for the two-block BFC (M=2) that ensures full diversity under iterative belief propagation decoding. Furthermore, a genetic algorithm guided by density evolution is employed to optimize the protograph edges within this family for improved coding gain. The resulting codes, termed DA-GRP-LDPC codes, simultaneously achieve full diversity and enhanced coding gain, reaching a 0.8 dB gap to the outage limit for the two-block BFC at a block length of 16,896. This demonstrates that the proposed framework effectively bridges the gap between diversity optimality in non-ergodic channels and high coding gain in ergodic channels.