Optimal Independent Encoding Schemes for Several Classes of Discrete Degraded Broadcast Channels

Let X → Y → Z be a discrete memoryless degraded broadcast channel (DBC) with marginal transition probability matrices TY X and TZX . Denote q as the distribution of the channel input X . For any given q, and H(Y |X) ≤ s ≤ H(Y ), where H(Y |X) is the conditional entropy of Y given X and H(Y ) is the entropy of Y , define the function F ∗ TY X ,TZX (q, s) as the infimum of H(Z|U), the conditional entropy of Z given U with respect to all discrete random variables U such that a)H(Y |U) = s, and b) U and Y, Z are conditionally independent given X . This paper studies the function F ∗, its properties and its calculation. This paper then applies these results to several classes of DBCs including the broadcast Z channel, the input-symmetric DBC, which includes the group-additive degraded broadcast channel, and the discrete multiplicative degraded broadcast channel. This paper provides independent-encoding schemes and demonstrates that each achieve the boundary of the capacity region for the corresponding class of DBCs. This paper first represents the capacity region of the DBC X → Y → Z with the function F ∗ TY X ,TZX . Secondly, this paper shows that the OR encoding approach, an independent-encoding scheme, achieves the boundary of the capacity region for the multi-user broadcast Z channel. This paper then defines and studies the input-symmetric DBC, introduces the permutation encoding approach, an independent-encoding scheme, for the input-symmetric DBC and proves its optimality. As a consequence, the group-addition encoding approach achieves the boundary of the capacity region for the group-additive DBC. Finally, this paper studies the discrete multiplicative DBC and shows that the multiplication encoding approach achieves the boundary of the capacity region for the discrete multiplicative DBC.