  In this work, we derive the random coding error exponent for the uplink phase
of a two-way relay system where physical layer network coding (PNC) is
employed. The error exponent is derived for the practical (yet sub-optimum) XOR
channel decoding setting. We show that the random coding error exponent under
optimum (i.e., maximum likelihood) PNC channel decoding can be achieved even
under the sub-optimal XOR channel decoding. The derived achievability bounds
provide us with valuable insight and can be used as a benchmark for the
performance of practical channel-coded PNC systems employing low complexity
decoders when finite-length codewords are used.
