Pair covering and other designs with block size 6
Bibliographic record
Abstract
Abstract A t‐(v, k, λ) covering design is a set of b blocks of size k such that each t‐set of points occurs in at least λ blocks, and the covering number Cλ(v, k, t) is the minimum value of b in any t‐(v, k, λ) covering design. In this article, we determine C1(v, 6, 2) for v ≡ 2 (mod 5), showing C1(v, 6, 2) attains the Schönheim bound. We also show C1(v, 6, 2) attains the Schönheim bound for v ≡ 1 (mod 5) whenever v ≥ 23986. If Uλ(v, k, t) denotes the Schönheim bound for t‐(v, k,λ) packing designs, then we show that D1(v, 6, 2) = U1(v, 6, 2) − 1 and D1(v − 1, 6, 2) = U1(v − 1, 6, 2) if v ≡ 11 (mod 15) and v ≥ 23441, and D1(v, 6, 2) = U1(v, 6, 2) and D1(v − 1, 6, 2) = U1(v − 1, 6, 2) if v ≡ 1, 6, (mod 15) and v ≥ 811. In addition, we improve the existence results for (v, 6, 1( BIBDs and (v, K, 1) PBDs when K = H1(5) = {k:k ≡ 1 (mod 5)} and when K = {6} ∪ (H1(5) ∩ {prime powers}). © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 511–533, 2007
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.004 | 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".