Optimal natural encoding scheme for discrete multiplicative degraded broadcast channels

Certain degraded broadcast channels (DBCs) have the property that the boundary of the capacity region can be achieved by an encoder that combines independent codebooks (one for each receiver) using the same single-letter function that adds distortion to the channel. We call this the natural encoder for the DBC. Natural encoders are known to achieve the capacity region boundary of the broadcast Gaussian channel, and the broadcast binary-symmetric channel. Recently, they have also been shown to achieve the capacity region of the broadcast Z channel. This paper shows that natural encoding achieves the capacity region boundary for discrete multiplicative DBCs. The optimality of the natural encoder also leads to a relatively simple expression for the capacity region for discrete multiplicative DBCs.