On the stability of periodic binary sequences with zone restriction
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Bibliographic record
Abstract
Abstract Traditional global stability measure for sequences is hard to determine because of large search space. We propose the k -error linear complexity with a zone restriction for measuring the local stability of sequences. For several classes of sequences, we demonstrate that the k -error linear complexity is identical to the k -error linear complexity within a zone, while the length of a zone is much smaller than the whole period when the k -error linear complexity is large. These sequences have periods $$2^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , or $$2^v r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>v</mml:mi> </mml:msup> <mml:mi>r</mml:mi> </mml:mrow> </mml:math> ( r odd prime and 2 is primitive modulo r ), or $$2^v p_1^{s_1} \cdots p_n^{s_n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>v</mml:mi> </mml:msup> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:msubsup> <mml:mo>⋯</mml:mo> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:msubsup> </mml:mrow> </mml:math> ( $$p_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> is an odd prime and 2 is primitive modulo $$p_i^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> , where $$1\le i \le n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>i</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> ) respectively. In particular, we completely determine the spectrum of 1-error linear complexity with any zone length for an arbitrary $$2^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:math> -periodic binary sequence.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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 it