  The design of modulation schemes for the physical layer network-coded two way
relaying scenario has been extensively studied recently with the protocol which
employs two phases: Multiple access (MA) Phase and Broadcast (BC) Phase. It was
observed by Koike-Akino et al. 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 and all these network coding maps should satisfy a requirement called the
exclusive law. Only the scenario in which the end nodes use M-PSK signal sets
is extensively studied in \cite{NVR} using Latin Sqaures. In this paper, we
address the case in which the end nodes use M-QAM signal sets (where M is of
the form $2^{2\lambda}$, $\lambda$ being any positive integer). In a fading
scenario, for certain channel conditions $\gamma e^{j \theta}$, termed singular
fade states, the MA phase performance is greatly reduced. We show that the
square QAM signal sets give lesser number of singular fade states compared to
PSK signal sets. Because of this, the complexity at the relay is enormously
reduced. Moreover, lesser number of overhead bits are required in the BC phase.
The fade state $\gamma e^{j \theta}=1$ is singular for all constellations of
arbitrary size including PSK and QAM. For arbitrary PSK constellation it is
well known that the Latin Square obtained by bit-wise XOR mapping removes this
singularity. We show that XOR mapping fails to remove this singularity for QAM
of size more greater than 4 and show that a doubly block circulant Latin Square
removes this singularity. Simulation results are presented to show the
superiority of QAM over PSK.
