LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-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.

Expurgation converts small average error to small maximal error

Statement

Every length- $n$ code with average error $ε_{a v}$ has a subcode with at least $M / 2$ messages and maximal error at most $2 ε_{a v}$ .

Facts & Assumptions

Given: A length- $n$ code with $M \geq 1$ messages and per-message errors $e_{1}, \dots, e_{M}$ .

[F1]

Average and maximal error of a fixed-length code: $ε_{a v} = M^{- 1} \sum_{m} e_{m}$ and $ε_{\max} = \max_{m} e_{m}$ .

[F2]

Fixed-length codes for a finite DMC: restricting the encoder to a nonempty subset of messages and mapping decoder outputs to the retained message labels gives another fixed-length code.

Proof

1.1F1givenalgebra

Let $S : = {m : e_{m} \leq 2 ε_{a v}}$ . If fewer than $M / 2$ messages belonged to $S$ , then more than $M / 2$ messages would each have error greater than $2 ε_{a v}$ , forcing $\sum_{m} e_{m} > M ε_{a v}$ , a contradiction. Hence $∣ S ∣ \geq M / 2$ .

2.1F1F2step 1.1constructalgebra∎

Restrict the encoder to $S$ and relabel its messages. Define the new decoder to keep any old decoded message in $S$ and to output a fixed retained message otherwise. For each transmitted $m \in S$ , every output previously decoded correctly as $m$ is still decoded correctly, so its new error is at most $e_{m} \leq 2 ε_{a v}$ . Thus the subcode has maximal error at most $2 ε_{a v}$ ; its rate loss is at most $n^{- 1} \ln 2$ because $∣ S ∣ \geq M / 2$ .

Depends on

Used by

Gallager random-coding exponent lower bound Theorem
Operational capacity formula for a finite DMC Theorem

Dependency tree · two levels

4 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

Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Chapter 19 §19.2 (standard reference, not scraped)
Robert Gallager, Principles of Digital Communication I, Chapter 8 §8.7.4, MIT OpenCourseWare (standard reference, not scraped)