# P3106: Exact capacity region of the two-user Gaussian interference channel

- ID: `P3106`
- Reference: `gaussian-interference-channel-capacity`
- Page: https://theoremdb.org/statements/P3106
- Record maturity: Reviewed problem with recorded work

## Problem

Determine the exact capacity region of the memoryless real two-user Gaussian interference channel \(Y_1=X_1+aX_2+Z_1\) and \(Y_2=bX_1+X_2+Z_2\) for arbitrary fixed \(a,b\in\mathbb R\), independent standard Gaussian noises, and average power constraints \(\mathbb E[X_i^2]\le P_i\).

### Context

Known frontier: The Han-Kobayashi scheme with modern analysis achieves rates within one bit of capacity universally; several parameter regimes are exact.

Open boundary: A matching exact capacity region for arbitrary gains and powers remains unknown.

### Problem setup

- **Definition (capacity region).** All rate pairs achievable by block codes with error tending to zero.
- **Definition (interference channel).** Each receiver observes its intended signal plus the other sender's signal and noise.
- **Remark.** The capacity region is the closure of simultaneously achievable rate pairs with vanishing error. Exact formulas are known in strong and several other regimes; the general weak and mixed regimes are known within a constant gap.

### What counts as a solution

- Give matching single-letter or computable inner and outer bounds for all parameters a,b,P₁,P₂.
- Or prove that no proposed finite-letter characterization can hold under a precisely stated formulation and replace it with an exact operational characterization.

## Status

OPEN as checked on 2026-08-01. Strongest checked neighboring result: The Han-Kobayashi scheme with modern analysis achieves rates within one bit of capacity universally; several parameter regimes are exact. Exact unresolved remainder: A matching exact capacity region for arbitrary gains and powers remains unknown. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (Current status and exact unresolved remainder).** OPEN as checked on 2026-08-01. Strongest checked neighboring result: The Han-Kobayashi scheme with modern analysis achieves rates within one bit of capacity universally; several parameter regimes are exact. Exact unresolved remainder: A matching exact capacity region for arbitrary gains and powers remains unknown.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: The Han-Kobayashi scheme with modern analysis achieves rates within one bit of capacity universally; several parameter regimes are exact.

The exact unresolved remainder is: A matching exact capacity region for arbitrary gains and powers remains unknown.

A complete resolution must meet the following acceptance conditions:
- Give matching single-letter or computable inner and outer bounds for all parameters a,b,P₁,P₂.
- Or prove that no proposed finite-letter characterization can hold under a precisely stated formulation and replace it with an exact operational characterization.

### Background and intake notes

- Original intake status: OPEN as checked on 2026-08-01. Strongest checked neighboring result: The Han-Kobayashi scheme with modern analysis achieves rates within one bit of capacity universally; several parameter regimes are exact. Exact unresolved remainder: A matching exact capacity region for arbitrary gains and powers remains unknown.
- The release review checked 2 structured sources on 2026-08-01.
- Equivalent-formulation queries: general two user Gaussian interference channel exact capacity remains open 2026; Gaussian interference channel capacity within one bit exact region
- Strongest checked neighboring result: The Han-Kobayashi scheme with modern analysis achieves rates within one bit of capacity universally; several parameter regimes are exact.
- Exact unresolved remainder: A matching exact capacity region for arbitrary gains and powers remains unknown.

### Other known results

- **Claim 2** (supported): The Han-Kobayashi scheme with modern analysis achieves rates within one bit of capacity universally; several parameter regimes are exact. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The Han-Kobayashi scheme with modern analysis achieves rates within one bit of capacity universally; several parameter regimes are exact. Unresolved remainder: A matching exact capacity region for arbitrary gains and powers remains unknown. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): A matching exact capacity region for arbitrary gains and powers remains unknown.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `gaussian-interference-channel-capacity`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>R. Etkin, D. Tse, and H. Wang, Gaussian interference channel capacity to within one bit, IEEE Transactions on Information Theory 54 (2008). main one-bit theorem https://arxiv.org/abs/cs/0702045
   - Also cited at R. Etkin, D. Tse, and H. Wang, Gaussian interference channel capacity to within one bit, IEEE Transactions on Information Theory 54 (2008). main one-bit theorem
   - preprint; primary source; arXiv:cs/0702045, checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Approximates the full capacity region to within one bit for all parameters while stating exact capacity remains open.
   - Source used to assess the problem's recorded status.
   - For Exact capacity region of the two-user Gaussian interference channel: This is the dated publication status for the canonical target Exact capacity region of the two-user Gaussian interference channel.
   - Source named by the research packet.
2. <a id="reference-2"></a>Te Han and K. Kobayashi, “A new achievable rate region for the interference channel”. IEEE Transactions on Information Theory 27(1) (1981), 49-60. DOI 10.1109/TIT.1981.1056307. Han-Kobayashi region https://doi.org/10.1109/TIT.1981.1056307
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Provides the central achievable region against which converses are compared.
   - Source used to assess the problem's recorded status.
   - For Exact capacity region of the two-user Gaussian interference channel: Provides the central achievable region against which converses are compared.