Bibliographic record
Abstract
We call the polynomial [Formula: see text] a Barker polynomial of degree n-1 if each aj ∈{-1, 1} and [Formula: see text] Properties of Barker polynomials were studied by Turyn and Storer thoroughly in the early sixties, and by Saffari in the late eighties. In the last few years P. Borwein and his collaborators revived interest in the study of Barker polynomials (Barker codes, Barker sequences). In this paper we give a new proof of the fact that there is no Barker polynomial of even degree greater than 12, and hence Barker sequences of odd length greater than 13 do not exist. This is intimately tied to irreducibility questions and proved as a consequence of the following new result. Theorem.Ifn ≔ 2m + 1 > 13and[Formula: see text]where eachbj ∈{-1, 0, 1}for even values of j, each bj is an integer divisible by 4 for odd values of j, then there is no polynomial[Formula: see text]such that[Formula: see text], where[Formula: see text]and[Formula: see text]denotes the collection of all polynomials of degree 2m with each of their coefficients in {-1, 1}. A clever usage of Newton's identities plays a central role in our elegant proof.
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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 | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.006 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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".