CCapacity Atlas
general-two-user-interference-channel

General two-user discrete memoryless interference channel

Two transmitter-receiver pairs interfere, and the exact capacity region is unknown in general.

Interference Finite alphabet Discrete memoryless Capacity region Bounds only

Channel and question

Input
Independent inputs \(X_1,X_2\).
Output
Receivers observe \(Y_1,Y_2\).
Law
An arbitrary memoryless law \(p(y_1,y_2|x_1,x_2)\).
Quantity
Capacity region \(\mathcal C_{\mathrm{IC}}\), measured in rate pairs in bits per channel use.

Criterion. Vanishing average error at both receivers.

  • Receiver i requests only message i.
  • No strong- or weak-interference ordering is assumed.

Current status

\[\mathcal R_{\mathrm{HK}}\subseteq\mathcal C_{\mathrm{IC}}\subseteq\mathcal R_{\mathrm{outer}}\]

No universally tight single-letter outer bound is known.

ResultRelationMethodYear
Inner Region\(\mathcal R_{\mathrm{HK}}\subseteq\mathcal C\)Split each message into common and private parts.1981
Outer Region\(\mathcal C\subseteq\mathcal R_{\mathrm{outer}}\)Cut-set, genie-aided, and receiver-cooperation converses.1981

Research frontier

Determine the capacity region of the general two-user interference channel.

Why it remains open. The optimal extent of partial interference decoding depends delicately on the channel, while existing outer bounds lose the distributed decoding structure.

What would count as progress

  • Close the region for a new nontrivial subclass.
  • Separate Han-Kobayashi from a known outer bound on an explicit channel.
  • Formalize one exact-capacity regime before the general problem.

Lean formalization

Canonical statementNone

Version 1 · Lean. This needs a shared interference-network model and independent-message code definitions.

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. Te Sun Han and Kingo Kobayashi (1981). A New Achievable Rate Region for the Interference Channel. IEEE Transactions on Information Theory. DOI 10.1109/TIT.1981.1056307.
  2. Abbas El Gamal and Young-Han Kim (2011). Network Information Theory. Cambridge University Press. DOI 10.1017/CBO9781139030687.

Discussion

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