DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Fixed-length finite DMC code with perfect causal output feedback

Definition

For a finite DMC $W : X \to Y$ as in Finite discrete memoryless channels, for integers $n, M \geq 1$ , a length- $n$ feedback code has a uniform message $M \in [M]$ , deterministic causal encoders $f_{i} : [M] \times Y^{i - 1} \to X$ for $1 \leq i \leq n$ , and a decoder $g : Y^{n} \to [M]$ . The encoder at time $i$ sees only $Y^{i - 1}$ , sets $X_{i} = f_{i} (M, Y^{i - 1})$ , and the causal joint law is $\Pr (M = m, Y^{n} = y^{n}) = \frac{1}{M} \prod_{i = 1}^{n} W (y_{i} ∣ f_{i} (m, y^{i - 1})) .$ Equivalently, at each positive-probability message-and-past-output history, $Y_{i}$ has conditional law $W (\cdot ∣ f_{i} (m, y^{i - 1}))$ . Its average error is $\Pr [g (Y^{n}) \neq M]$ , rate is $n^{- 1} \ln M$ nats/use, and the feedback capacity is the closure of rates achievable with error tending to zero. There is no input-cost constraint.

Facts & Assumptions

Given: A finite DMC, positive integers $n, M$ , the message set $[M]$ and deterministic causal maps.

[F1]

Finite discrete memoryless channels: the single-use transition kernel is the stochastic row $W (\cdot ∣ x)$ on the finite output alphabet for each input $x$ .

Proof

1.1givenF1algebra

Recursively in $i$ , the message and past-output history determine $X_{i} = f_{i} (M, Y^{i - 1})$ , after which [F1] specifies a probability vector for $Y_{i}$ . Multiplying these causal row probabilities and the uniform message mass $1 / M$ gives the displayed joint law. Summing first over $y_{n}$ , then $y_{n - 1}$ down to $y_{1}$ , each last row sums to one; summing over the $M$ messages then gives total mass one. Thus the law is normalized, and its positive-probability history conditionals are the stated rows.

2.1givenstep 1.1algebra∎

Hence the decoder error, rate and capacity convention are well-defined for every such finite code.

Depends on

Finite discrete memoryless channels

Used by

Perfect output feedback does not increase finite DMC capacity Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd ed., Chapter 7, §7.12 (standard reference, not scraped)
Abbas El Gamal and Young-Han Kim, Network Information Theory, Lecture Notes 18, feedback DMC (standard reference, not scraped)