  This paper deals with physical layer network-coding for the four-way wireless
relaying scenario where four nodes A, B, C and D wish to communicate their
messages to all the other nodes with the help of the relay node R. The scheme
given in the paper is based on the denoise-and-forward scheme proposed first by
Popovski et al. Intending to minimize the number of channel uses, the protocol
employs two phases: Multiple Access (MA) phase and Broadcast (BC) phase with
each phase utilizing one channel use. This paper does the equivalent for the
four-way relaying scenario as was done for the two-way relaying scenario by
Koike-Akino et al., and for three-way relaying scenario in [3]. It is observed
that adaptively changing the network coding map used at the relay according to
the channel conditions greatly reduces the impact of multiple access
interference which occurs at the relay during the MA phase. These network
coding maps are so chosen so that they satisfy a requirement called exclusive
law. We show that when the four users transmit points from the same M-PSK
constellation, every such network coding map that satisfies the exclusive law
can be represented by a 4-fold Latin Hyper-Cube of side M. The network code map
used by the relay for the BC phase is explicitly obtained and is aimed at
reducing the effect of interference at the MA stage.
