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modulo-additive-two-way-channel

Modulo-additive two-way channel

Independent additive noises allow simultaneous communication in both directions without an adaptation gain.

Two-way Finite alphabet Discrete memoryless Additive noise Feedback Capacity region Exact

Channel and question

Input
Each terminal chooses an element of a common finite abelian group.
Output
Terminal one sees X1+X2+Z1 and terminal two sees X1+X2+Z2.
Law
Group addition with independent memoryless noises having arbitrary fixed group distributions.
Quantity
Modulo-additive two-way channel capacity \(\mathcal C\), measured in bits per channel use.

Criterion. Vanishing average block error, with the constraints specified in the model.

  • The two messages are independent and uniform.
  • Each encoder knows its own message and strictly past local outputs.
  • Each decoder knows its own message and its entire local output word.
  • R1 is the rate from terminal one to terminal two, and R2 is the reverse rate.

Current status

\[0\le R_1\le\log_2|G|-H(Z_2),\quad 0\le R_2\le\log_2|G|-H(Z_1)\]
Known results and bounds for Modulo-additive two-way channel
ResultRelationMethodYear
Exact\(0\le R_1\le\log_2|G|-H(Z_2),\quad 0\le R_2\le\log_2|G|-H(Z_1)\)Self-input cancellation and conditional-entropy converses.2016

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

  • Independent memoryless additive noises give a rectangular capacity region even with adaptation.
    operational-capacity · exact capacity · solved · Formally stated · v1
Lean declarations (1)

References

  1. Lin Song, Fady Alajaji, and Tamas Linder (2016). Adaptation is Useless for Two Discrete Additive-Noise Two-Way Channels. IEEE International Symposium on Information Theory.

Discussion

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