  The design of modulation schemes for the physical layer network-coded two way
relaying scenario was studied in [1], [3], [4] and [5]. In [7] it was shown
that every network coding map that satisfies the exclusive law is representable
by a Latin Square and conversely, and this relationship can be used to get the
network coding maps satisfying the exclusive law. But, only the scenario in
which the end nodes use $M$-PSK signal sets is addressed in [7] and [8]. In
this paper, we address the case in which the end nodes use $M$-QAM signal sets.
In a fading scenario, for certain channel conditions $\gamma e^{j \theta}$,
termed singular fade states, the MA phase performance is greatly reduced. By
formulating a procedure for finding the exact number of singular fade states
for QAM, we show that square QAM signal sets give lesser number of singular
fade states compared to PSK signal sets. This results in superior performance
of $M$-QAM over $M$-PSK. It is shown that the criterion for partitioning the
complex plane, for the purpose of using a particular network code for a
particular fade state, is different from that used for $M$-PSK. Using a
modified criterion, we describe a procedure to analytically partition the
complex plane representing the channel condition. We show that when $M$-QAM ($M
>4$) signal set is used, the conventional XOR network mapping fails to remove
the ill effects of $\gamma e^{j \theta}=1$, which is a singular fade state for
all signal sets of arbitrary size. We show that a doubly block circulant Latin
Square removes this singular fade state for $M$-QAM.
