Thresholds of precoded (3,6) LDPC codes under iterative and maximum-likelihood decodings

The technique of precoding is recently applied to low-density parity-check (LDPC) codes by Divsalar et al, resulting in precoded LDPC codes. It is shown that precoding can improve not only the iterative decoding thresholds of regular LDPC codes but also their maximum-likelihood (ML) decoding thresholds. In this paper, we consider thresholds of precoded (3,6) LDPC codes under both iterative and ML decodings. For iterative decoding, the tool of EXIT charts is employed. With the EXIT chart, thresholds of precoded (3,6) LDPC codes with different precoding ratios are obtained. It is shown that the iterative decoding threshold is reduced with the increase of the precoding ratio until the precoding ratio reaches at 0.25. When the precoding ratio is greater than 0.25, the increase of precoding ratio may impair the iterative decoding threshold. Thus, the optimal precoding ratio under iterative decoding is revealed to be 0.25, i.e., half of the information bits are to be precoded. For ML decoding analysis, we first drive the asymptotic ensemble distance spectrum of precoded (3,6) LDPC codes. Using the derived asymptotic ensemble distance spectrum along with Divsalar´s simple bound, we calculate the thresholds of precoded (3,6) LDPC codes with different precoding ratios under ML decoding. It is shown that the increase of precoding ratio always leads to an improved ML decoding threshold and thus the optimal ML decoding threshold is achieved at a precoding ratio of 0.5, i.e., all information bits are to be precoded.