Statistical mechanics of low-density parity check error-correcting codes over Galois fields
K Nakamura, Y Kabashima, D Saad · Europhysics Letters (EPL) · 2001
A variation of low-density parity check (LDPC) error-correcting codes defined over Galois fields ( GF ( q )) is investigated using statistical physics. A code of this type is characterised by a sparse random parity check matrix composed of C non-zero elements per column. We examine the dependence of the code performance on the value of q , for finite and infinite C values, both in terms of the thermodynamical transition point and the practical decoding phase characterised by the existence of a unique (ferromagnetic) solution. We find different q -dependence in the cases of C = 2 and C ⩾ 3; the analytical solutions are in agreement with simulation results, providing a quantitative measure to the improvement in performance obtained using non-binary alphabets.