  In this paper, we study a compute-and-forward (CAF) relaying scheme with
low-density parity-check (LDPC) codes, a special case of physical layer network
coding, under the quadrature phase shift keying (QPSK) modulation. The novelty
of this paper is the theoretical analysis of decoding performance of the CAF
scheme and traditional separation decoding (SD) scheme with joint decoding or
with successive interference cancellation (SIC) decoding when the reception
powers from both senders are not equal but close to each other. First, we study
the decoding performance of linear random coding (LRC) in the CAF scheme whose
decoder is based on the degraded channel. When rotation angles of
constellations of two senders are controlled, we show that they can achieve
rates well beyond the multiple access channel (MAC) with LRC with optimal
rotation angles. Second, we analyze the practical feasibility of CAF using LDPC
codes in terms of computational costs and decoding performance of belief
propagation (BP) decoders. The calculation complexity of the BP decoder for the
CAF scheme is almost equal to that of the SIC BP decoder, and smaller than the
joint BP decoder in the SD scheme. Decoding performance is evaluated by the
asymptotic decodable region using density evolution. The results show that,
with code rate fixed, the CAF scheme has better performance than the SD scheme
with joint BP decoding and SIC BP decoding in the high rate region.
