Pair covering and other designs with block size 6
Why this work is in the frame
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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 C λ ( v , k , t ) is the minimum value of b in any t ‐( v , k , λ) covering design. In this article, we determine C 1 ( v , 6, 2) for v ≡ 2 (mod 5), showing C 1 ( v , 6, 2) attains the Schönheim bound. We also show C 1 ( v , 6, 2) attains the Schönheim bound for v ≡ 1 (mod 5) whenever v ≥ 23986. If U λ ( v , k , t ) denotes the Schönheim bound for t ‐( v , k ,λ) packing designs, then we show that D 1 ( v , 6, 2) = U 1 ( v , 6, 2) − 1 and D 1 ( v − 1, 6, 2) = U 1 ( v − 1, 6, 2) if v ≡ 11 (mod 15) and v ≥ 23441, and D 1 ( v , 6, 2) = U 1 ( v , 6, 2) and D 1 ( v − 1, 6, 2) = U 1 ( 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 = H 1(5) = {k:k ≡ 1 (mod 5)} and when K = {6} ∪ ( H 1(5) ∩ {prime powers}). © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 511–533, 2007
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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 | 0.001 | 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 it