Generalised array low-density parity-check codes

In this study, using Group Permutation Low-Density Parity-Check (GP-LDPC) codes, the authors generalise the concept of array Low-Density Parity-Check (LDPC) codes from fields of prime order to those of prime power order. In fact, they consider the additive group of the finite field GF(q), q a prime power, as the underlying group for the GP-LDPC code construction and since when q is a prime, the author´s code construction method coincides with that of quasi-cyclic array LDPC codes, they call their codes, generalised array LDPC (GA-LDPC) codes. First, they prove that, like array LDPC codes, GA-LDPC codes are quasi-cyclic codes. Then, they analyse the girth of GA-LDPC codes in a way similar to that for array LDPC codes and introduce some shortened GA-LDPC codes with girths 8, 10 and 12. For many values of g, J and L, the lengths of (J, L)-regular shortened GA-LDPC codes of girth g and rate at least 1 - J/L, constructed in this study, are smaller than the lengths of (J, L)-regular LDPC codes of girth g and rate at least 1 - J/L, constructed in the literature. Also, simulation results show that GA-LDPC codes perform well with the iterative message-passing decoding.