High-performance binary and non-binary Low-density parity-check codes based on affine permutation matrices

Low-density parity-check (LDPC) codes from affine permutation matrices, called APM-LDPC codes, are a class of LDPC codes whose parity-check matrices consist of zero matrices or APMs of the same orders. APM-LDPC codes are not quasi-cyclic (QC), in general. In this study, necessary and sufficient conditions are provided for an APM-LDPC code to have cycles of length 2l, l ≥ 2, and a deterministic algorithm is given to generate APM-LDPC codes with a given girth. Unlike Type-I conventional QC-LDPC codes, the constructed (J, L) APM-LDPC codes with the J × L all-one base matrix can achieve minimum distance greater than (J + 1)! and girth larger than 12. Moreover, the lengths of the constructed APM-LDPC codes, in some cases, are smaller than the best known lengths reported for QC-LDPC codes with the same base matrices. Another significant advantage of the constructed APM-LDPC codes is that they have remarkably fewer cycle multiplicities compared with QC-LDPC codes with the same base matrices and the same lengths. Simulation results show that the binary and non-binary constructed APM-LDPC codes with lower girth outperform QC-LDPC codes with larger girth.