  We address in this paper decoding aspects of the Compute-and-Forward (CF)
physical-layer network coding strategy. It is known that the original decoder
for the CF is asymptotically optimal. However, its performance gap to optimal
decoders in practical settings are still not known. In this work, we develop
and assess the performance of novel decoding algorithms for the CF operating in
the multiple access channel. For the fading channel, we analyze the ML decoder
and develop a novel diophantine approximation-based decoding algorithm showed
numerically to outperform the original CF decoder. For the Gaussian channel, we
investigate the maximum a posteriori (MAP) decoder. We derive a novel MAP
decoding metric and develop practical decoding algorithms proved numerically to
outperform the original one.
