New results on GDDs, covering, packing and directable designs with block size 5
Bibliographic record
Abstract
Abstract This article looks at (5,λ) GDDs and ( v ,5,λ) pair packing and pair covering designs. For packing designs, we solve the (4 t ,5,3) class with two possible exceptions, solve 16 open cases with λ odd, and improve the maximum number of blocks in some ( v , 5, λ) packings when v small (here, the Schönheim bound is not always attainable). When λ=1, we construct v =432 and improve the spectrum for v =14, 18 (mod 20). We also extend one of Hanani's conditions under which the Schönheim bound cannot be achieved (this extension affects (20 t +9,5,1), (20 t +17,5,1) and (20 t +13,5,3)) packings. For covering designs we find the covering numbers C (280,5,1), C (44,5,17) and C (44,5,λ) with λ=13 (mod 20). We also know that the covering number, C ( v , 5, 2), exceeds the Schönheim bound by 1 for v =9, 13 and 15. For GDDs of type g n , we have one new design of type 30 9 when λ=1, and three new designs for λ=2, namely, types g 15 with g ∈{13, 17, 19}. If λ is even and a (5, λ) GDD of type g u is known, then we also have a directable (5,λ) GDD of type g u . © 2010 Wiley Periodicals, Inc. J Combin Designs 18:337–368, 2010
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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 | 0.001 | 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 | 0.001 |
| 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".