Channel capacity in the non-asymptotic regime: Taylor-type expansion and computable benchmarks
Bibliographic record
Abstract
In this paper, the non-asymptotic counterpart of Shannon capacity is investigated for any discrete input memory-less channel with discrete or continuous output (DIMC). Given any block length n and word error probability ϵ, let R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> (ϵ) be the best channel coding rate achievable with the block length n subject to the error probability e. Based on the non-asymptotic equipartition properties (NEP) established recently by Yang and Meng, a quantity δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t, n</sub> (ϵ) is first defined to measure the relative magnitude of error probability ϵ and block length n with respect to a given DIMC and an input distribution t. Then, by combining the non-asymptotic achievability and converse established recently by Yang and Meng via jar decoding, it is shown that, given n and ϵ <; 1/2, R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> (ϵ) has a "Taylor-type expansion" with respect to δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t, n</sub> (ϵ), with the first two terms of the expansion being max <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> [I (t; P)-δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t, n</sub> (ϵ)] = I(t*;P)-δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t*, n</sub> (ϵ) for some optimal distribution t*, and the third order term being O(δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t*, n</sub> ) + O(ln n/n). Finally, based on the Taylor-type expansion and the non-asymptotic converse, two easy to compute approximation formulas for R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> (ϵ) (dubbed “SO” and “NEP”) are provided. Numerical results show that both the SO and NEP approximation formulas provide reliable and accurate estimation, in contrast with the normal approximation which sometimes falls below achievable bounds and sometimes rises above converses. An important implication arising from the Taylor-type expansion of R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> (ϵ) is that in the practical non-asymptotic regime, the optimal marginal codeword symbol distribution is not necessarily a Shannon capacity achieving distribution.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".