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In this paper, a performance analysis of network-coded bidirectional amplify-and-forward (BD-AF) multi-relay networks is presented. In such networks, communication is performed over two phases: the broadcasting phase, and the cooperation phase. In the broadcasting phase, both source nodes broadcast their signals simultaneously to the intermediate JV relay nodes. The cooperation phase is based on one of two schemes: (1) multiple-access (MA) and (2) single relay selection (RS) transmission. In the MA-BD-AF scheme, all N relay nodes simultaneously transmit linearly coded signals to both source nodes. A simple suboptimal relay selection scheme (i.e. SRS-BD-AF) is proposed such that the relay that maximizes the sum-of-rates is selected to transmit its network coded signal to both source nodes. Finally, a suboptimal sum-of-rates maximizing relay power allocation under the MA-BD-AF scheme is formulated and shown to reduce to the SRS-BD-AF scheme. Also, a symbol error rate performance analysis is provided, where it is shown that the SRS-BD-AF scheme achieves full diversity. Simulation results are provided to complement the theoretical analysis.