[Show abstract][Hide abstract] ABSTRACT: In this paper, we study the design of network coding for the multiple two-way relaying channels where multiple pairs of sources wish to exchange information with their partners. All nodes are equipped with multiple antennas, and we focus on a particular scenario where the sources have less antennas than the relay. For such a case, the application of existing protocols developed for the scenario with a single pair has to rely on the time sharing approach, which will result in some loss of reliability and throughput. In this paper, we develop a new network coding transmission protocol which can be viewed as a combination of traditional beamforming and the recently developed approaches of signal alignment. By using such a protocol, all M pairs of source nodes can accomplish information exchanging within two time slots. We have developed analytical results, such as outage probability and diversity-multiplexing tradeoff, which demonstrate that the proposed transmission protocol can achieve a larger multiplexing gain than the time sharing approach. Furthermore, the proposed network coding scheme is extended to the special case where source nodes only have a single antenna. Simulation results have been provided to demonstrate the performance of the proposed network coding schemes.