On the study of semi-involutory and semi-orthogonal matrices
Bibliographic record
Abstract
This paper investigates semi-involutory and semi-orthogonal matrices, presenting two algorithms for verifying these properties for n × n matrices over F p m ⁎ . The algorithms significantly reduce computational complexity by avoiding the need for non-singular diagonal matrices. The structure of circulant matrices with these properties is also explored, including the derivation of the exact form of their corresponding diagonal matrices. We further investigate MDS matrices with semi-involutory and semi-orthogonal properties. We prove that semi-involutory circulant matrices of order n ≥ 3 over F 2 m , where gcd ( n , 2 m − 1 ) = 1 , cannot be MDS matrices. Additionally, we show that semi-orthogonal circulant matrices of order 2 n over F 2 m cannot be MDS. Finally, explicit formulas for counting 3 × 3 semi-involutory and semi-orthogonal MDS matrices over F 2 m are derived, and exact counts for 4 × 4 semi-involutory and semi-orthogonal MDS matrices over F 2 m for m = 3 and 4 are provided.
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 | 0.000 | 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 | 0.000 | 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".