Orthogonal designs via computational algebra
Why this work is in the frame
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Bibliographic record
Abstract
Abstract We detail the Williamson array construction based on quaternions, following the description by Baumert and Hall. By analogy, we extend the construction to larger arrays using matrix representations of the algebras of octonions and sedenions. In the case of octonions, we obtain the full orthogonal design OD (8;1,1,1,1,1,1,1,1) or order 8 with 8 variables. In the case of sedenions we obtain the full orthogonal design OD (16;1,1,7,7) of order 16 with 4 variables and the full orthogonal design OD (16;1,1,2,2,2,2,2,2,2) of order 16 with 9 variables. We use OD (16;1,1,2,2,2,2,2,2,2) to search for inequivalent Hadamard matrices of orders 112, 144, 176 and we establish constructively three new lower bounds for the numbers of inequivalent Hadamard matrices of these three orders. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 351–362, 2006
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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