Bibliographic record
Abstract
In this paper, we provide algorithmic methods for conducting exhaustive searches for periodic Golay pairs. Our methods enumerate several lengths beyond the currently known state-of-the-art available searches: we conducted exhaustive searches for periodic Golay pairs of all lengths v ≤ 72 v \leq 72 using our methods, while only lengths v ≤ 34 v \leq 34 had previously been exhaustively enumerated. Our methods are applicable to periodic complementary sequences in general. We utilize sequence compression, a method of sequence generation derived in 2013 by Ðoković and Kotsireas. We also introduce and implement a new method of “multi-level” compression, where sequences are uncompressed in several steps. This method allowed us to exhaustively search all lengths v ≤ 72 v \leq 72 using less than 10 compute years. For cases of complementary sequences where uncompression is not possible, we introduce some new methods of sequence generation inspired by the isomorph-free exhaustive generation algorithm of orderly generation. Finally, we pose a conjecture regarding the structure of periodic Golay pairs and prove it holds in many lengths, including all lengths v > 100 v>100 . We demonstrate the usefulness of our algorithms by providing the first ever examples of periodic Golay pairs of length v = 90 v = 90 . The smallest length for which the existence of periodic Golay pairs is undecided is now 106 106 .
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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 | 0.004 | 0.038 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.008 | 0.007 |
| Science and technology studies | 0.004 | 0.003 |
| Scholarly communication | 0.005 | 0.014 |
| Open science | 0.004 | 0.006 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.042 | 0.010 |
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".