CCapacity Atlas
physically-degraded-broadcast-channel

Physically degraded two-receiver broadcast channel

One transmitter sends private messages to receivers whose outputs form a degradation chain.

Broadcast Finite alphabet Discrete memoryless Degraded Capacity region Exact Single-letter characterization

Channel and question

Input
\(X\in\mathcal X\)
Output
Receiver outputs \(Y_1\) and \(Y_2\).
Law
A memoryless channel satisfying \(X\to Y_1\to Y_2\).
Quantity
Capacity region \(\mathcal C\), measured in rate pairs in bits per channel use.

Criterion. Vanishing average error at both receivers.

  • Receiver 1 is stronger.
  • Each receiver requests a private message.

Current status

\[\mathcal C=\bigcup_{p(u,x)}\{(R_1,R_2):R_1\le I(X;Y_1|U),\ R_2\le I(U;Y_2)\}\]

Conditions. \(U\to X\to Y_1\to Y_2\).

Known results and bounds for Physically degraded two-receiver broadcast channel
ResultRelationMethodYear
Inner Region\(R_1\le I(X;Y_1|U),\quad R_2\le I(U;Y_2)\)Superposition coding and successive decoding.1973
Outer Region\(R_1\le I(X;Y_1|U),\quad R_2\le I(U;Y_2)\)Single-letter converse exploiting degradation.1973

Formal verification

Lean coverageFormally stated

Concrete operational definitions and admitted research statements are present. Existing proofs are preserved. New statements require mathematical review and proof completion.

Claims

  • The private-message operational region of a degraded finite broadcast channel.
    operational-capacity · exact capacity · solved · Formally stated · v1
Lean declarations (1)

References

  1. Peter P. Bergmans (1973). Random Coding Theorem for Broadcast Channels with Degraded Components. IEEE Transactions on Information Theory. DOI 10.1109/TIT.1973.1054980.
  2. Abbas El Gamal and Young-Han Kim (2011). Network Information Theory. Cambridge University Press. DOI 10.1017/CBO9781139030687.

Discussion

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Related problems

Blackwell deterministic broadcast channel

The three-input Blackwell channel is a concrete nondegraded deterministic broadcast channel with an exact entropy capacity region.

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A less-noisy ordering compares every finite stochastic prefix and yields the exact superposition-coding capacity region.

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A more-capable ordering compares the receivers for every input distribution and yields an exact superposition-coding capacity region.

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