  In this paper, we propose a novel linear physicallayer network coding (LPNC)
for Rayleigh fading 2-way relay channels (2-WRC). Rather than the simple
modulo-2 (bit-XOR) operation, the relay directly maps the superimposed signal
of the two users into the linear network coded combination in GF(2^2) by
multiplying the user data by properly selected generator matrix. We derive the
constellation constrained capacities for LPNC and 5QAM denoise-and forward
(5QAM-DNF) [2] and further explicitly characterize the capacity difference
between LPNC and 5QAM-DNF. Based on our analysis and simulation, we highlight
that without employing the irregular 5QAM mapping and sacrificing the spectral
efficiency, our LPNC in GF(2^2) is superior to 5QAM-DNF scheme in low SNR
regime while they achieve equal performance in the the moderate-to-high SNR
regime.
