Hadamard matrices from Goethals — Seidel difference families with a repeated block
Bibliographic record
Abstract
Purpose: To construct Hadamard matrices by using Goethals — Seidel difference families having a repeated block, generalizingthe so called propus construction. In particular we construct the first examples of symmetric Hadamard matrices of order 236.Methods: The main ingredient of the propus construction is a difference family in a finite abelian group of order v consisting offour blocks (X1, X2, X3, X4) where X1 is symmetric and X2 X3. The parameters (v; k1, k2, k3, k4; λ) of such family must satisfythe additional condition ki λ v. We modify this construction by imposing different symmetry conditions on some of theblocks and construct many examples of Hadamard matrices of this kind. In this paper we work with the cyclic group Zv of order v.For larger values of v we build the blocks Xi by using the orbits of a suitable small cyclic subgroup of the automorphism groupof Zv. Results: We continue the systematic search for symmetric Hadamard matrices of order 4v by using the propus construction.Such searches were carried out previously for odd v 51. We extend it to cover the case v53. Moreover we construct thefirst examples of symmetric Hadamard matrices of order 236. A wide collection of symmetric and skew-symmetric Hadamardmatrices was obtained and the corresponding difference families tabulated by using the symmetry properties of their blocks.Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, compression and masking ofvideo information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in themathematical network Internet together with executable on line algorithms.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 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) | 0.003 | 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".