On the sum rate of a 2 × 2 interference network

In an M × N interference network, there are M transmitters and N receivers with each transmitter having independent messages for each of the 2^N - 1 possible non-empty subsets of the receivers. We consider the 2 × 2 interference network with 6 possible messages, of which the 2 × 2 interference channel and X channel are special cases obtained by using only 2 and 4 messages respectively. Starting from an achievable rate region similar to the Han-Kobayashi region, we obtain an achievable sum rate. For the Gaussian interference network, we determine which of the 6 messages are sufficient for maximizing the sum rate within this rate region for the low, mixed, and strong interference conditions. It is observed that 2 messages are sufficient in several cases. Finally, we show that sum capacity is achieved using only 2 messages for a subset of the mixed interference conditions.