Iterative symbol-by-symbol decoding of LDPC codes and constellation mapping for multilevel modulation

Iterative probabilistic decoding of binary low-density parity-check (LDPC) codes have been studied extensively. Non-binary LDPC codes have recently attracted an increasing attention. Most of the existing non-binary codes are built over GF(2
^q
), and decoding methods developed for binary LDPC codes cannot be used directly with multilevel modulations. In this paper, we first extend the binary parity-check codes to the case with multiple symbols over modular arithmetic. Then, we develop a sum-product algorithm to decode this new type of codes at the symbol level. Finally, we propose an effective constellation mapping method for multilevel modulations. Error performances of this type of codes with 4-PAM, 4-PSK, and 16-PSK modulations over AWGN channels are provided. Compared with uncoded systems, the coding gain of a medium-size regular LDPC code of rate 8/9 with 4-PAM and 4-PSK modulations is about 5 dB. With 16-PSK and an appropriate constellation mapping at a code rate 3/4, the proposed code's performance is comparable to that of trellis codes.