CCapacity Atlas
general-two-receiver-broadcast-channel

General two-receiver discrete memoryless broadcast channel

The capacity region for two arbitrary broadcast receivers remains unknown outside important ordered subclasses.

Broadcast Finite alphabet Discrete memoryless Capacity region Bounds only

Channel and question

Input
One transmitter chooses \(X\in\mathcal X\).
Output
Receivers observe \(Y_1\) and \(Y_2\).
Law
An arbitrary memoryless law \(p(y_1,y_2|x)\).
Quantity
Capacity region \(\mathcal C_{\mathrm{BC}}\), measured in rate pairs in bits per channel use.

Criterion. Vanishing average error at both receivers.

  • Each receiver requests a private message.
  • No degradedness or receiver ordering is assumed.

Current status

\[\mathcal R_{\mathrm{Marton}}\subseteq\mathcal C_{\mathrm{BC}}\subseteq\mathcal R_{\mathrm{UV}}\]

The two regions coincide for many special classes but not in general.

ResultRelationMethodYear
Inner Region\(\mathcal R_{\mathrm{Marton}}\subseteq\mathcal C\)Correlated auxiliaries, random binning, and superposition.1979
Outer Region\(\mathcal C\subseteq\mathcal R_{\mathrm{UV}}\)A single-letter outer region using two auxiliary variables.2007

Research frontier

Determine the private-message capacity region of the general two-receiver DMC broadcast channel.

Why it remains open. The transmitter must coordinate incompatible receiver-specific descriptions, and known converses do not capture the full binning structure of the best inner bounds.

What would count as progress

  • Find a channel separating or matching Marton's inner region and current outer bounds.
  • Discover a tighter computable outer region.
  • Formalize a sharply stated benchmark subclass.

Lean formalization

Canonical statementNone

Version 1 · Lean. The canonical statement should follow a shared broadcast-code and rate-region API.

Substantial proofs0 linked

No external Lean proof is registered. Proofs longer than roughly 50 lines or requiring problem-specific infrastructure should live in a dedicated repository and link back to this statement version.

References

  1. Katalin Marton (1979). A Coding Theorem for the Discrete Memoryless Broadcast Channel. IEEE Transactions on Information Theory. DOI 10.1109/TIT.1979.1056046.
  2. Chandra Nair and Abbas El Gamal (2007). An Outer Bound to the Capacity Region of the Broadcast Channel. IEEE Transactions on Information Theory. DOI 10.1109/TIT.2006.887492.
  3. Abbas El Gamal and Young-Han Kim (2011). Network Information Theory. Cambridge University Press. DOI 10.1017/CBO9781139030687.

Discussion

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