A Novel Framework for Relating Quasi-Cyclic Codes and Quasi-Twisted Codes
Bibliographic record
Abstract
In this paper, we aim to analyze the algebraic structure of repeated-root quasi-cyclic codes of length$p^{k}{n}\ell$and index$\ell$over the finite field$\mathbb{F}_{q}$, where$k$is a positive integer,$q=p^{\alpha}$and$(n,p)=1$. For this purpose, a quasi-cyclic code over$\mathbb{F}_{q}$is regarded as a linear code over an auxiliary ring. By introducing a ring isomorphism, we provide a one-to-one correspondence between this class of quasi-cyclic codes and nonrepeated-root$(1-u)$. quasi-twisted codes of length$n\ell$and index$\ell$over the chain ring$\mathbb{F}_{q}+u\mathbb{F}_{q}+\cdots+u^{p^{k}-1}\mathbb{F}_{q}$, where$u^{p^{k}}=0$. Our approach enables us to extend the results regarding non-repeated-root quasi-twisted codes over rings to repeated-root quasi-cyclic codes over finite fields. To illustrate the effectiveness of our method, we provide examples that demonstrate how it simplifies the structure of this class of codes.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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".