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A coded modulation belief propagation (CMBP) decoder is proposed for decoding LDPC codes with multilevel modulations. The decoder takes into account statistical dependencies among bits originating in the same symbol, providing better performance than the marginal BP (MBP) decoder. Asymptotically it converges to MAP decoding. The CMBP decoder is based on a single-level coding (SLC) scheme and does not suffer from practical disadvantages of multi-level coding (MLC) schemes. Furthermore, the CMBP decoder can close the capacity gap of the bit interleaved coded modulation (BICM) SLC scheme. The BICM capacity gap increases when the modulation size increases and in scenarios where gray mapping is not possible.