Optimal precoding for multi-way relay channels with QAM constellations

Complex field network coding has been shown to achieve a throughput of 1/2 symbol per user per channel use in a multi-way relay channel (MWRC). To achieve this throughput, a superimposed combination of user symbols must be distinguishable at the relay. In [1], precoding vectors were designed for an MWRC with all users employing the same rectangular quadrature amplitude modulation (QAM) constellation. These precoding vectors guarantee that every superimposed combination of user symbols at the relay is distinguishable and a QAM constellation is received. There are many precoding vectors with such properties. In this paper, we consider a more general problem in which each user can employ an arbitrary rectangular QAM constellation and we design a precoding vector for the users such that the power efficiency of the received rectangular QAM constellation at the relay is maximized.