  This paper investigates the fundamental building blocks of physical-layer
network coding (PNC). Most prior work on PNC focused on its application in a
simple two-way-relay channel (TWRC) consisting of three nodes only. Studies of
the application of PNC in general networks are relatively few. This paper is an
attempt to fill this gap. We put forth two ideas: 1) A general network can be
decomposed into small building blocks of PNC, referred to as the PNC atoms, for
scheduling of PNC transmissions. 2) We identify nine PNC atoms, with TWRC being
one of them. Three major results are as follows. First, using the decomposition
framework, the throughput performance of PNC is shown to be significantly
better than those of the traditional multi-hop scheme and the conventional
network coding scheme. For example, under heavy traffic volume, PNC can achieve
100% throughput gain relative to the traditional multi-hop scheme. Second, PNC
decomposition based on a variety of different PNC atoms can yield much better
performance than PNC decomposition based on the TWRC atom alone. Third, three
out of the nine atoms are most important to good performance. Specifically, the
decomposition based on these three atoms is good enough most of the time, and
it is not necessary to use the other six atoms.
