Bibliographic record
Abstract
Introduction: There are just a few known methods for the construction of symmetric C-matrices, due to the lack of a universal structure for them. This obstruction is fundamental, in addition, the structure of C-matrices with a double border is incompletely described in literature, which makes its study especially relevant. The purpose: To describe the two-border two-circulant construction in detail with the proposal of the concept of C-pairs Legendre. Results: The paper deals with C-matrices of order n=2v+2 with two borders and extends the so called generalized Legendre pairs, v odd, to a wider class of Legendre C-pairs with even and odd v, defined on a finite abelian group G of order v. Such a pair consists of two functions a, b: G→Z, whose values are +1 or −1 except that a(e)=0, where e is the identity element of G and Z is the ring of integers. To characterize the Legendre C-pairs we use the subsets X={xÎG: a(x)=–1} and Y={xÎG: b(x)=–1} of G. We show that a(x−1)=(−1)v a(x) for all x. For odd v we show that X and Y form a difference family, which is not true for even v. These difference families are precisely the so called Szekeres difference sets, used originally for the construction of skew-Hadamard matrices. We introduce the subclass of the special Legendre C-pairs and prove that they exist whenever 2v+1 is a prime power. In the last two sections of the paper we list examples of special cyclic Legendre C-pairs for lengths v<70. Practical relevance: C-matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for search of conference 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 | 0.001 |
| 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".