CCapacity Atlas
binary-skew-symmetric-broadcast-channel

Binary skew-symmetric broadcast channel

This minimal binary-input broadcast channel isolates a persistent gap between Marton-type inner bounds and UV-type outer bounds.

Broadcast Finite alphabet Discrete memoryless Binary Asymmetric Capacity region Bounds only

Channel and question

Input
A binary symbol \(X\).
Output
Binary outputs \(Y_1,Y_2\).
Law
For \(X=0\), \(Y_1=0\) and \(Y_2\) is fair; for \(X=1\), \(Y_1\) is fair and \(Y_2=1\).
Quantity
Private-message capacity region \(\mathcal C_{\mathrm{BSSC}}\), measured in ordered pairs of bits per channel use.

Criterion. Closure of achievable private-message rate pairs.

  • Independent private messages are sent to the receivers.
  • Average probability that either receiver errs vanishes.

Current status

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

Equality of the named inner and outer descriptions is the canonical formal target.

ResultRelationMethodYear
Inner Region\(\mathcal R_{\mathrm{Marton}}\subseteq\mathcal C_{\mathrm{BSSC}}\)Marton coding specialized to binary input and symmetric outputs.1979
Outer Region\(\mathcal C_{\mathrm{BSSC}}\subseteq\mathcal R_{\mathrm{UV}}\)Auxiliary-variable broadcast outer bound.2007

Research frontier

Close the gap between the best Marton-type inner region and UV-type outer region for the fixed BSSC law.

Why it remains open. Known auxiliary-variable optimizations do not coincide on all supporting hyperplanes.

What would count as progress

  • Certify a strict separating rate pair or prove the two optimized descriptions equal.

Lean formalization

Canonical statementStatement

Version 1 · Lean. The transition law is concrete; the proposition interface fixes the inner-versus-outer equality target without supplying a proof.

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. Yanlin Geng, Chandra Nair, Shlomo Shamai, and Zizhou Vincent Wang (2010). On Broadcast Channels with Binary Inputs and Symmetric Outputs. arXiv preprint.

Discussion

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