In a system employing physical-layer network coding, the superimposed constellation at the relay may have ambiguity where one received signal is associated with more than one possible combination of users' data. We find the necessary and sufficient condition on a user bit mapping which removes this ambiguity for phase shift keying modulation. Further, we introduce the concept of semi-Gray mapping which improves the system BER performance and achievable rate. The necessary and sufficient condition of having a semi-Gray mapping is also found. Interestingly, the widely used binary reflected Gray mapping does not satisfy the semi-Gray mapping condition resulting in a poorer performance compared to our proposed semi-Gray scheme.

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