LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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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.

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A low-error random code can be pruned to satisfy a strict average cost constraint

Statement

Fix a blocklength $n$ , a message count $M \geq 2$ , a codebook ensemble with exchangeable message codewords, and a decoder for each sampled codebook. Let $B_{n}$ be the random fraction of codewords whose normalized cost exceeds $Γ$ , and let ${\overset{ˉ}{ε}}_{n}$ be the ensemble codebook's average error. If $q_{n} : = E B_{n}$ and $a_{n} : = E {\overset{ˉ}{ε}}_{n}$ satisfy $q_{n} + a_{n} < 1$ , then some deterministic subcode has every retained codeword of normalized cost at most $Γ$ , average error at most $\sqrt{q_{n} + a_{n}} / (1 - \sqrt{q_{n} + a_{n}})$ , and at least $M (1 - \sqrt{q_{n} + a_{n}})$ messages.

If this construction is available for each sufficiently large $n$ , with $q_{n} + a_{n} \to 0$ , then the retained code's rate differs from $n^{- 1} \ln M_{n}$ by a quantity tending to zero.

Facts & Assumptions

Given: A random length- $n$ codebook with $M \geq 2$ exchangeable message codewords, nonnegative normalized codeword cost, and a decoder whose per-message errors define ${\overset{ˉ}{ε}}_{n}$ .

[F1]

Chebyshev-Markov inequality for the integral: for a nonnegative measurable random variable $V$ and $a > 0$ , $\Pr [V \geq a] \leq E [V] / a$ .

[F2]

The finite union bound: the probability of a union of two events is at most the sum of their probabilities.

[F3]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: the natural logarithm is continuous and increasing on $(0, \infty)$ , with $\ln 1 = 0$ and $\ln (a b) = \ln a + \ln b$ for $a, b > 0$ .

[F4]

A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere: a nonnegative measurable random variable with expectation zero vanishes almost surely.

Proof

1.1F1F2F4givencasesalgebra

Put $s_{n} = \sqrt{q_{n} + a_{n}} < 1$ , including $s_{n} = 0$ when $q_{n} + a_{n} = 0$ . If $q_{n} + a_{n} = 0$ , both nonnegative variables $B_{n}$ and ${\overset{ˉ}{ε}}_{n}$ have expectation zero, so [F4] makes each zero almost surely. Their zero events intersect by [F2], giving a realization with both variables zero. If $q_{n} + a_{n} > 0$ , then $s_{n} > 0$ . By [F1], $\Pr [B_{n} > s_{n}] \leq q_{n} / s_{n}$ and $\Pr [{\overset{ˉ}{ε}}_{n} > s_{n}] \leq a_{n} / s_{n}$ . The union bound [F2] makes the probability of either failure at most $s_{n} < 1$ , so a realization exists with both quantities at most $s_{n}$ .

2.1step 1.1algebracases

Retain exactly the codewords in that realization whose normalized cost is at most $Γ$ , and let $M^{'}$ be their number. Since $B_{n} \leq s_{n}$ , $M^{'} \geq M (1 - s_{n}) > 0$ . Change the decoder only by mapping decisions for removed messages to one fixed retained message. An originally correct decision for a retained message is unchanged, so its error probability cannot increase; hence the retained average error is at most $M {\overset{ˉ}{ε}}_{n} / M^{'} \leq s_{n} / (1 - s_{n})$ . Each retained word satisfies the cost constraint by construction.

3.1F3step 1.1step 2.1givenalgebra∎

If $q_{n} + a_{n} \to 0$ along blocklengths, then $s_{n} \to 0$ and $1 - s_{n} > 0$ . Since $M (1 - s_{n}) \leq M^{'} \leq M$ , the order and product laws in [F3] give $0 \leq (\ln M - \ln M^{'}) / n \leq - \ln (1 - s_{n}) / n$ . Continuity of $\ln$ at $1$ and $\ln 1 = 0$ in [F3] yield $\ln (1 - s_{n}) \to 0$ ; hence the rate loss tends to zero. The error bound in step 2.1 also tends to zero. The conclusion is per blocklength, so no sequence of witnesses is selected in this lemma.

Depends on

Used by

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Sources

Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Chapter 20 §20.1 (standard reference, not scraped)
Robert Gallager, Principles of Digital Communication I, Chapter 8 §8.7.4, MIT OpenCourseWare (standard reference, not scraped)