DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Deterministic fixed-length binary block code and block error

Definition

Fix the finite-alphabet IID source in Finite-alphabet IID source with alphabet $A$ and one-letter probability vector $p$ from Finite distributions, entropy, divergence, and stochastic matrices; its finite block law is $p^{n}$ in Self-information and surprisal variance for a finite law. For $n \geq 1$ and an integer $k_{n} \geq 0$ , a deterministic fixed-length binary block code is a pair of maps $f_{n} : A^{n} \to {0, 1}^{k_{n}}, g_{n} : {0, 1}^{k_{n}} \to A^{n} .$ Every transmitted binary string has exactly $k_{n}$ bits. Its block error probability and rate in nats per source symbol are defined using Information units and logarithm-base convention: $e_{n} : = P (g_{n} (f_{n} (X^{n})) \neq X^{n}), R_{n} : = \frac{k_{n} \ln 2}{n} .$ A sequence of such codes achieves rate $R$ if $e_{n} \to 0$ and ${lim sup}_{n \to \infty} R_{n} \leq R$ .

Depends on

Used by

Fixed-length lossless source coding at the entropy rate Theorem

Dependency tree · two levels

9 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 3, §3.2 (standard reference, not scraped)
MIT OpenCourseWare 6.441 Information Theory, Spring 2010, Lecture 5: Codes, Kraft Inequality, and Optimal Codes (standard reference, not scraped)