Wireless Network-Coded Multi-Way Relaying Using Latin Hyper-Cubes

Physical layer network-coding for the $n$-way wireless relaying scenario is dealt with, where each of the $n$ user nodes $X_1,$ $X_2,...,X_n$ wishes to communicate its messages to all the other $(n-1)$ nodes with the help of the relay node R. The given scheme, based on the denoise-and-forward scheme proposed for two-way relaying by Popovski et al. in \cite{PoY1}, employs two phases: Multiple Access (MA) phase and Broadcast (BC) phase with each phase utilizing one channel use and hence totally two channel uses. Physical layer network-coding using the denoise-and-forward scheme was done for the two-way relaying scenario in\cite{KPT}, for three-way relaying scenario in \cite{SVR}, and for four-way relaying scenario in \cite{ShR}. This paper employs denoise-and-forward scheme for physical layer network coding of the $n$-way relaying scenario illustrating with the help of the case $n = 5$ not dealt with so far. It is observed that adaptively changing the network coding map used at the relay according to the channel conditions reduces the impact of multiple access interference which occurs at the relay during the MA phase. These network coding maps are chosen so that they satisfy a requirement called \textit{exclusive law}. We show that when the $n$ users transmit points from the same $M$-PSK $(M=2^{\lambda})$ constellation, every such network coding map that satisfies the exclusive law can be represented by a $n$-fold Latin Hyper-Cube of side $M$. The singular fade subspaces resulting from the scheme are described and enumerated for general values of $n$ and $M$ and are classified based on their removability in the given scenario. A network code map to be used by the relay for the BC phase aiming at reducing the effect of interference at the MA stage is obtained.