Density optimisation of generator matrices of quasi-cyclic low-density parity-check codes and their rank analysis

The efficient encoding of quasi-cyclic (QC) low-density parity-check (LDPC) codes is based on generator matrices in systematic-circulant (SC) form. The cost of the encoders of QC-LDPC codes mainly depends on the number of non-zero entries in the SC generator matrices. This study introduces a novel construction of SC generator matrices based on matrix transformations via Galois Fourier transform. By revealing the structure of SC generator matrices in the transform domain, an algorithm is proposed to reduce the density of the generator matrices of QC-LDPC codes. Furthermore, a tight upper bound on ranks of QC matrices is derived. Based on the bound, rank distributions of parity-check matrices and generator matrices in the transform domain illustrate the efficiency of the proposed algorithm. Simulation results show that the density of their SC generator matrices can be significantly decreased with moderate computational complexity.