  We study multi-user multi-way relay networks where $N$ user nodes exchange
their information through a single relay node. We use network coding in the
relay to increase the throughput. Due to the limitation of complexity, we only
consider the binary multi-user network coding (BMNC) in the relay. We study
BMNC matrix (in GF(2)) and propose several design criteria on the BMNC matrix
to improve the symbol error probability (SEP) performance. Closed-form
expressions of the SEP of the system are provided. Moreover, an upper bound of
the SEP is also proposed to provide further insights on system performance.
Then BMNC matrices are designed to minimize the error probabilities.
