On Convexity of Error Rates in Digital Communications
Bibliographic record
Abstract
Convexity properties of error rates of a class of decoders, including the maximum-likelihood/min-distance one as a special case, are studied for arbitrary constellations, bit mapping, and coding. Earlier results obtained for the additive white Gaussian noise channel are extended to a wide class of noise densities, including unimodal and spherically invariant noise. Under these broad conditions, symbol and bit error rates are shown to be convex functions of the signal-to-noise ratio (SNR) in the high-SNR regime with an explicitly determined threshold, which depends only on the constellation dimensionality and minimum distance, thus enabling an application of the powerful tools of convex optimization to such digital communication systems in a rigorous way. It is the decreasing nature of the noise power density around the decision region boundaries that ensures the convexity of symbol error rates in the general case. The known high/low-SNR bounds of the convexity/concavity regions are tightened and no further improvement is shown to be possible in general. The high-SNR bound fits closely into the channel coding theorem: all codes, including capacity-achieving ones, whose decision regions include the hardened noise spheres (from the noise sphere hardening argument in the channel coding theorem), satisfy this high-SNR requirement and thus has convex error rates in both SNR and noise power. We conjecture that all capacity-achieving codes have convex error rates. Convexity properties in signal amplitude and noise power are also investigated. Some applications of the results are discussed. In particular, it is shown that fading is convexity-preserving and is never good in low dimensions under spherically invariant noise, which may also include any linear diversity combining.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.030 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".