  Compute-and-forward (CAF) relaying is effective to increase bandwidth
efficiency of wireless two-way relay channels. In a CAF scheme, a relay is
designed to decode a linear combination composed of transmitted messages from
other terminals or relays. Design for error-correcting codes and its decoding
algorithms suitable for CAF relaying schemes remain as an important issue to be
studied. As described in this paper, we will present an asymptotic performance
analysis of LDPC codes over two-way relay channels based on density evolution
(DE). Because of the asymmetric characteristics of the channel, we use the
population dynamics DE combined with DE formulas for asymmetric channels to
obtain BP thresholds. Additionally, we also evaluate the asymptotic performance
of spatially coupled LDPC codes for two-way relay channels. The results
indicate that the spatial coupling codes yield improvements in the BP threshold
compared with corresponding uncoupled codes for two-way relay channels.
Finally, we will compare the mutual information rate and rate achievability
between the CAF scheme and the MAC separation decoding scheme. We demonstrate
the possibility that the CAF scheme has higher reliability in the high-rate
region.
