  The analysis of modulation schemes for the physical layer network-coded two
way relaying scenario is presented which employs two phases: Multiple access
(MA) phase and Broadcast (BC) phase. It was shown 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. Depending on the signal set used
at the end nodes, deep fades occur for a finite number of channel fade states
referred as the singular fade states. The singular fade states fall into the
following two classes: The ones which are caused due to channel outage and
whose harmful effect cannot be mitigated by adaptive network coding are
referred as the \textit{non-removable singular fade states}. The ones which
occur due to the choice of the signal set and whose harmful effects can be
removed by a proper choice of the adaptive network coding map are referred as
the \textit{removable} singular fade states. In this paper, we derive an upper
bound on the average end-to-end Symbol Error Rate (SER), with and without
adaptive network coding at the relay, for a Rician fading scenario. It is shown
that without adaptive network coding, at high Signal to Noise Ratio (SNR), the
contribution to the end-to-end SER comes from the following error events which
fall as $\text{SNR}^{-1}$: the error events associated with the removable
singular fade states, the error events associated with the non-removable
singular fade states and the error event during the BC phase. In contrast, for
the adaptive network coding scheme, the error events associated with the
removable singular fade states contributing to the average end-to-end SER fall
as $\text{SNR}^{-2}$ and as a result the adaptive network coding scheme
provides a coding gain over the case when adaptive network coding is not used.
