On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators
Bibliographic record
Abstract
Uniformity in binary tuples of various lengths in a pseudorandom sequence is an important randomness property. We consider ideal \begin{document}$ t $\end{document} -tuple distribution of a filtering de Bruijn generator consisting of a de Bruijn sequence of period \begin{document}$ 2^n $\end{document} and a filtering function in \begin{document}$ m $\end{document} variables. We restrict ourselves to the family of orthogonal functions, that correspond to binary sequences with ideal 2-level autocorrelation, used as filtering functions. After the twenty years of discovery of Welch-Gong (WG) transformations, there are no much significant results on randomness of WG transformation sequences. In this article, we present new results on uniformity of the WG transform of orthogonal functions on de Bruijn sequences. First, we introduce a new property, called invariant under the WG transform, of Boolean functions. We have found that there are only two classes of orthogonal functions whose WG transformations preserve \begin{document}$ t $\end{document} -tuple uniformity in output sequences, up to \begin{document}$ t = (n-m+1) $\end{document} . The conjecture of Mandal et al. in [ 29 ] about the ideal tuple distribution on the WG transformation is proved. It is also shown that the Gold functions and quadratic functions can guarantee \begin{document}$ (n-m+1) $\end{document} -tuple distributions. A connection between the ideal tuple distribution and the invariance under WG transform property is established.
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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 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".