Physical Layer Network Coding Over Finite And Infinite Fields
  Direct application of network coding at the physical layer - physical layer network coding (PNC) - is a promising technique for two-way relay wireless networks. In a two-way relay network, relay nodes are used to relay two-way information flows between pairs of end nodes. This paper proposes a precise definition for PNC. Specifically, in PNC, a relay node does not decode the source information from the two ends separately, but rather directly maps the combined signals received simultaneously to a signal to be relayed. Based on this definition, PNC can be further sub-classed into two categories - PNCF (PNC over finite field) and PNCI (PNC over infinite field) - according to whether the network-code field (or groups, rings) adopted is finite or infinite. For each of PNCF and PNCI, we consider two specific estimation techniques for dealing with noise in the mapping process. The performance of the four schemes is investigated by means of analysis and simulation, assuming symbol-level synchronization only. 
Zhang Shengli, Soung chang Liew and Lu Lu