  In this paper we develop a low-complexity coding scheme and system design
framework for the half duplex relay channel based on the
Quantize-Map-and-Forward (QMF) relay- ing scheme. The proposed framework allows
linear complexity operations at all network terminals. We propose the use of
binary LDPC codes for encoding at the source and LDGM codes for mapping at the
relay. We express joint decoding at the destination as a belief propagation
algorithm over a factor graph. This graph has the LDPC and LDGM codes as
subgraphs connected via probabilistic constraints that model the QMF relay
operations. We show that this coding framework extends naturally to the high
SNR regime using bit interleaved coded modulation (BICM). We develop density
evolution analysis tools for this factor graph and demonstrate the design of
practical codes for the half-duplex relay channel that perform within 1dB of
information theoretic QMF threshold.
