Construction of Multi-Rate QC-LDPC Codes Based on Permutation Method

This paper proposes a systematic permutation-based construction method for multi-rate quasi-cyclic low-density parity-check (QC-LDPC) codes. We first present a graph-theoretic framework in which any regular QC-LDPC code can be normalized to a canonical base matrix that is uniquely determined by a permutation π. This normalization reduces the complex code design to a single combinatorial optimization problem over the symmetric group. Based on this normalization, we analyze the cycle structure of the lifted Tanner graph and derive necessary and sufficient conditions for 4-cycle elimination in terms of the permutation difference function. We develop two complementary algorithms: a simulated annealing algorithm that searches for permutations that minimize a weighted sum of 4-cycles and 6-cycles, and a progressive column-ordering algorithm that ensures every prefix subgraph maintains high girth. This approach yields a nested family of rate-compatible codes. Simulation results show that the constructed codes outperform the 5G-LDPC codes. The nested base matrix structure facilitates seamless rate switching, which makes the proposed code family well suited for adaptive transmission systems in future wireless networks.