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 inputsymmetric DBC, which includes the degraded broadcast group-addition channel, and the discrete degraded multiplication 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 approach, an independent encoding scheme, achieves the optimal boundary of the capacity region for the multi-user broadcast Z channel. This paper then studies the inputsymmetric DBC, introduces the permutation approach, an independent encoding scheme, for the inputsymmetric DBC and proves its optimality. As a consequence, the group-addition approach achieves the optimal boundary of the capacity region for the degraded broadcast group-addition channel. Finally, this paper studies the discrete degraded broadcast multiplication channel and shows that the multiplication approach achieves the boundary of the capacity region for the discrete degraded broadcast multiplication channel.