On Maximum Linearly Achievable Degrees of Freedom Region of 2 $,\times,$ 2 $,\times,$ 2 Interference

We obtain the exact maximum Degrees of Freedom (DoF) region achieved by linear transceivers without symbol extensions of the 2 × 2 × 2 multiple-input multiple-output (MIMO) interference network, which is comprised of two sources, two relays and two destinations, each with arbitrary number of antennas. We prove that the maximum achievable DoF region for the networks under some antenna configurations coincide with the cut-set bound of the networks, while there is only one DoF gap from the bound for the networks under other configurations, which can not be bridged by linear transceivers. The basic idea of the proof is to construct a two-tier transceiver, where the first tier decomposes the general networks to several basic networks with special configurations, and the second tier achieves the maximal DoF tuples for these basic networks. A mechanism for constructing the second tier transceiver is proposed, which is referred to as channel scaling assisted interference elimination. Combined with interference avoidance at sources and interference cancelation at destinations depending on the antenna configurations, interference-free transmission can be ensured.