Distance properties and performance of physical layer network coding with binary linear codes for Gaussian two-way relay channels

We investigate joint channel and physical layer network coding (CPNC) for Gaussian two-way relay channels. The two users´ messages are encoded using the same binary linear code and are transmitted simultaneously with equal power. At the relay node, the network-coded message is recovered directly from the received signal sequence, and is then broadcast to the users. We propose a new methodology to explicitly find the distance spectrum of the coding scheme. Based on that, we analyze the error probability at the relay and derive an asymptotically tight performance bound (for high SNRs). We show that, with a general binary linear code, the CPNC scheme is subject to an asymptotic SNR loss of approximately ln 2 relative to the single-user case, regardless of the coding rate. Numerical results show that our analysis matches very well with the performance of the CPNC scheme.