Channel finite-blocklength normal approximation

Second-order approximation to maximal code size at blocklength n and error epsilon.

Specification version 1Published 2010-05-011 evidence records

About this problem

The question, scope, and sources behind this Registry record.

Formal question

Characterize the first two asymptotic orders of maximal code size at fixed nonzero error probability.

Scope & constraints

channel: regular memoryless channel
capacity: C
dispersion: V
blocklength: n
errorProbability: 0<epsilon<1

Primary sources

Channel coding rate in the finite blocklength regime — Channel finite-blocklength normal approximation Normal approximation and dispersion theorems

What we know

Current frontiers derived from accepted Claims.

Exact established valuelog2 M*(n,epsilon) = nC - sqrt(nV) Q^-1(epsilon) + O(log n)

The accepted equality closes this optimization result.

Claims

Assertions tied to evidence, attribution, and review.

2026

CLM-INFO-003213= log2 M*(n,epsilon) = nC - sqrt(nV) Q^-1(epsilon) + O(log n)EXACT VALUEThe cited theorem establishes log2 M*(n,epsilon) = nC - sqrt(nV) Q^-1(epsilon) + O(log n) under the current specification; this does not assert universal empirical attainability.Source: Channel coding rate in the finite blocklength regime — Channel finite-blocklength normal approximation ↗

PROVEN

Timeline

The frontier as it changed over time.

How this frontier is derived

Only accepted Claims matching the current specification contribute to the displayed bounds. Strict inequalities remain open; contradictory Claims require editorial review.

—No breakthrough events recorded yetEvents appear here when an accepted Claim tightens a bound or closes the frontier.

Specification

What question is being asked?

Constraints: channel=regular memoryless channel · capacity=C · dispersion=V · blocklength=n · errorProbability=0<epsilon<1

Read full specification ↗

Corrections & disputes

Record history

No unresolved corrections or disputes are recorded for this published version.

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Editorial status

Evidence-backed record

1 accepted Claim, with 1 linked evidence records.

Human-reviewed · No AI-only claims

Verified prizes & bounties

Active verified incentives

No active verified bounties are linked to this Limit.

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Reproducibility

Machine-checked history

No accepted machine-checked reproductions are recorded for this Limit.

Cite this LimitLimits Registry. LR-003213. Channel finite-blocklength normal approximation. 2026.