On the Existence of Asymptotically Good Linear Codes in Minor-Closed Classes
Bibliographic record
Abstract
Let C = (C1, C2, ...) be a sequence of codes such that each Ciis a linear [ni, ki, di]-code over some fixed finite field F, where niis the length of the code words, kiis the dimension, and diis the minimum distance. We say that C is asymptotically good if, for some ε > 0 and for all i ∈ ℤ>0, we have ni≥ i and min(ki/ni, di/ni) ≥ ε. Sequences of asymptotically good codes exist. We prove that if C is a class of GF(pn)-linear codes (where p is prime and n ≥ 1), closed under puncturing and shortening, and if C contains an asymptotically good sequence, then C must contain all GF(p)-linear codes. Our proof relies on a powerful new result from matroid structure theory.
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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 | 0.003 | 0.046 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.008 | 0.002 |
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".