The capacity region of the class of Gaussian MIMO three-receiver multilevel broadcast channels with degraded message sets

Nair and Gamal established the capacity region for the class of three-receiver multilevel broadcast channels (MBC) with degraded message sets. In this channel, the output at receiver 2 is a degraded version of the output at receiver 1. The transmitter sends a common message to all three receivers and a private message to receiver 1. By considering a specific discrete-memoryless example, they showed that a direct extension of the Körner-Marton region (for the two-receiver broadcast channels with degraded message sets) is strictly suboptimal. They also considered a three-receiver Gaussian product MBC and showed that, restricted to Gaussian inputs, a direct extension of the Körner-Marton region is again strictly suboptimal. However, the question as to whether Gaussian inputs are optimal remained unresolved. In this paper, we show that Gaussian inputs, along with time-sharing between rate points obtained with Gaussian inputs, achieves the capacity region of the class of three-receiver Gaussian MIMO MBC with degraded message sets.