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Optimal DoF Region of the MIMO Butterfly Network

A butterfly network is a classical model to demonstrate the throughput advantage of the network coding technique over wireline networks. We are curious about what happens if the wireless network is considered instead, especially for the multiple-antenna scenarios. In this letter, we study this problem and aim to fully characterize the degrees of freedom (DoF) region for the multiple-input-multiple-output (MIMO) butterfly network in the general setting. It turns out that significant DoF gains can be achieved if more advanced interference management schemes than the conventional time-sharing one are employed for the MIMO butterfly network, which is quite similar to the case of its wireline counterpart. Moreover, an existence proof based on the algebraic topology is provided in this letter, which may be of independent interest.