Optimal Encoding for Discrete Degraded Broadcast Channels

Consider a memoryless degraded broadcast channel (DBC) in which the channel output is a single-letter function of the channel input and the channel noise. As examples, for the Gaussian broadcast channel (BC), this single-letter function is real scalar addition and for the binary-symmetric BC, this single-letter function is modulo-two addition. This paper identifies several classes of discrete memoryless DBCs for which a relatively simple encoding scheme, which we call natural encoding, achieves capacity. Natural encoding (NE) combines symbols from independent codebooks (one for each receiver) using the same single-letter function that adds distortion to the channel. The alphabet size of each NE codebook is bounded by that of the channel input. This paper also defines the input-symmetric DBC, introduces permutation encoding for the input-symmetric DBC, and proves its optimality. Because it is a special case of permutation encoding, NE is capacity achieving for the two-receiver group-operation DBC. Combining the broadcast Z channel and group-operation DBC results yields a proof that NE is also optimal for the discrete multiplication DBC. Along the way, the paper also provides explicit parametric expressions for the two-receiver binary-symmetric DBC and broadcast Z channel.