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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v less-than-or-equal-to 72"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>72</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">v \leq 72</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using our methods, while only lengths <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v less-than-or-equal-to 34"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>34</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">v \leq 34</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v less-than-or-equal-to 72"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>72</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">v \leq 72</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v greater-than 100"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>></mml:mo> <mml:mn>100</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">v>100</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We demonstrate the usefulness of our algorithms by providing the first ever examples of periodic Golay pairs of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v equals 90"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>=</mml:mo> <mml:mn>90</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">v = 90</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The smallest length for which the existence of periodic Golay pairs is undecided is now <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="106"> <mml:semantics> <mml:mn>106</mml:mn> <mml:annotation encoding="application/x-tex">106</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 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 | 0.003 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| 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".