ML-performance of low-density parity check codes

In this paper, we derive the maximum likelihood (ML) performance for low-density parity check (LDPC) codes, considering BPSK and QPSK transmission over a Gaussian channel. We compare the theoretical ML performance with the performance of the iterative decoding algorithm. It turns out that the performance of the iterative decoding algorithm is close to the ML performance when the girth of the code is sufficiently high. When the girth of the code is equal to or smaller than 6, the decoding algorithm performs sub-optimal: the bit error rate (BER) obtained with the iterative decoding algorithm is much higher than the optimal BER that can be obtained when using ML decoding.