Group Divisible Covering Designs with Block Size 4: A Type of Covering Array with Row Limit
Bibliographic record
Abstract
Abstract A k‐GDCD, group divisible covering design, of type is a triple , where V is a set of gu elements, is a partition of V into u sets of size g, called groups, and is a collection of k‐subsets of V, called blocks, such that every pair of elements in V is either contained in a unique group or there is at least one block containing it, but not both. This family of combinatorial objects is equivalent to a special case of the graph covering problem and a generalization of covering arrays, which we call CARLs. In this paper, we show that there exists an integer such that for any positive integers g and , there exists a 4‐GDCD of type which in the worst case exceeds the Schönheim lower bound by δ blocks, except maybe when (1) and , or (2) , , and or . To show this, we develop constructions of 4‐GDCDs, which depend on two types of ingredients: essential, which are used multiple times, and auxiliary, which are used only once in the construction. If the essential ingredients meet the lower bound, the products of the construction differ from the lower bound by as many blocks as the optimal size of the auxiliary ingredient differs from the lower bound.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".