Simultaneous Multi-Rate Threshold Optimization of Protograph-Based Raptor-Like LDPC Codes via Genetic Algorithm

Protograph-based Raptor-like (PBRL) LDPC codes, adopted in the 5G NR eMBB data channel, support a wide range of code rates by generating incremental redundancy through XOR operations. As the protograph structure directly influences performance, its efficient design is crucial. This paper introduces a novel framework for optimizing the IRC submatrix by minimizing the Euclidean norm of capacity gaps, a new evaluation metric that captures overall threshold behavior across all supported rates, in contrast to the conventional rate-by-rate greedy optimization. The approach formulates this as a global optimization problem solved via a genetic algorithm (GA) with simulated annealing (SA) applied for improved convergence. Numerical results show notable threshold reductions, including a 58.4% improvement in the capacity gap norm for the 5G Base Graph 1. The method is also compatible with other rate-compatible structures, such as rate-compatible spatially coupled LDPC codes, which are promising candidates for coding schemes in next-generation mobile systems.