Channel and question
- Input
- Elements of a fixed nonempty finite abelian group.
- Output
- An element of the same group at each time.
- Law
- Y_t=X_t+S_t. The noise state then transitions according to a fixed Markov kernel K.
- Quantity
- Finite-group additive channel with Markov noise capacity \(C\), measured in bits per channel use.
Criterion. Vanishing average block error, with the constraints specified in the model.
- The initial noise law p is stationary for K.
- K is primitive: all entries of every sufficiently large power are strictly positive.
- The noise process is independent of the message. Neither terminal observes the state.
- There is no feedback and no input cost constraint.
Current status
\[C=\log_2|G|-\sum_s p(s)H(K(\cdot\mid s))\]
| Result | Relation | Method | Year |
|---|---|---|---|
| Exact | \(C=\log_2|G|-\sum_s p(s)H(K(\cdot\mid s))\) | Additive-noise information-rate coding and an entropy-rate converse. | 1995 |
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
- Stationary primitive Markov additive noise subtracts its entropy rate from log alphabet size.
operational-capacity· exact capacity · solved · Formally stated · v2
Lean declarations (1)
CapacityAtlas.Claims.markovNoiseclaim · operational-capacity
lean/CapacityAtlas/Claims/MarkovNoise.lean — Stationary primitive Markov additive noise subtracts its entropy rate from log alphabet size.
References
- Fady Alajaji (1995). Feedback Does Not Increase the Capacity of Discrete Channels with Additive Noise. IEEE Transactions on Information Theory. DOI 10.1109/18.370168.