Construction of girth-eight QC-LDPC codes from arithmetic progression sequence with large column weight

For any row weight L, a novel class of (8, L) quasi-cyclic low-density parity-check (QC-LDPC) codes is deterministically constructed with girth eight. Compared with a similar construction, the proposed (7, L) and (8, L) QC-LDPC codes exist for any circulant permutation matrix of size P ≥ (L - 1)(L² + 3L/4 + 3/4) + 1 and P ≥ (L - 1)(L² + 3L/4 + 7/4) +1, which remarkably improve the existing bounds of (L - 1)(L² + L) +1 and (L - 1)(L² + L + 1) +1, respectively. Moreover, for any column weight J, any row weight L and any common difference, a construction for (J, L) QC-LDPC codes with girth eight is also proposed. This construction has high flexibility with respect to the design of code rate and code length. Simulation results show that the novel codes with moderate lengths perform well in the additive white Gaussian noise channel.