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In this paper, we study the degrees of freedom (DoF) of the MIMO relaynetworks. We start with a general Y channel, where each user has $M_i$ antennasand aims to exchange messages with the other two users via a relay equippedwith $N$ antennas. Then, we extend our work to a general 4-user MIMO relaynetwork. Unlike most previous work which focused on the total DoF of thenetwork, our aim here is to characterize the achievable DoF region as well. Wedevelop an outer bound on the DoF region based on the notion of one sidedgenie. Then, we define a new achievable region using the Signal Space Alignment(SSA) and the Detour Schemes. Our achievable scheme achieves the upper boundfor certain conditions relating $M_i$'s and $N$.

Achievable Degrees of Freedom Region of the MIMO Relay Networks using the Detour Schemes