A Novel Construction Technique for Some Classes of Quasi-Cyclic Codes
Bibliographic record
Abstract
Let R be an arbitrary finite commutative chain ring, γ be a fixed generator of the maximal ideal of R, and F<inf>q</inf> = R/〈γ〉. In this paper, using the Generalized Discrete Fourier Transform, we derive the generator matrix of repeated-root quasi-cyclic codes of length n = ℓm, where q = 2<sup>t</sup>, m = 2m<sup>'</sup>, (m<sup>'</sup>, 2) = 1 and $or{d_{m'}}\left( {{2^t}} \right) = m' - 1$, over R. Then, we specialize this result to derive the generator matrix for simple-root quasi-cyclic codes of length n = ℓm where ord<inf>m</inf>(q) = m − 1. Our work presented in this paper can be used to find all classes of quasi-cyclic codes whose co-index satisfies these conditions. Finally, this method is used to list some binary quasi-cyclic codes with index 2 and find optimal and self-dual codes among them.
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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 | 0.000 | 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".