Bibliographic record
Abstract
Introduction: It is conjectured that the cyclic Legendre pairs of odd lengths >1 always exist. Such a pair consists of two functions a, b: G→Z, whose values are +1 or −1, and whose periodic autocorrelation function adds up to the constant value −2 (except at the origin). Here G is a finite cyclic group and Z is the ring of integers. These conditions are fundamental and the closely related structure of Hadamard matrices having a two circulant core and double border is incompletely described in literature, which makes its study especially relevant. Purpose: To describe the two-border two-circulant-core construction for Legendre pairs having three new lengths. Results: To construct new Legendre pairs we use the subsets X={x∈G: a(x)=–1} and Y={x∈G: b(x)=–1} of G. There are 20 odd integers v less than 200 for which the existence of Legendre pairs of length v is undecided. The smallest among them is v=77. We have constructed Legendre pairs of lengths 91, 93 and 123 reducing thereby the number of undecided cases to 17. In the last section of the paper we list some new examples of cyclic Legendre pairs for lengths v≤123. Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, and compression and masking of video information. Programs for search of Hadamard matrices and a library of constructed matrices are used in the mathematical network “mathscinet.ru” together with executable on-line algorithms
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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 | 0.000 | 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 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".