On symmetric (78, 22, 6) designs and related self-orthogonal codes
Bibliographic record
Abstract
We describe a method for constructing 2-designs admitting a solvable automorphism group, and construct new symmetric (78,22,6) designs, Until now, only five symmetric (78,22,6) designs were known. In this paper we show that, up to isomorphism, there are at least 413 symmetric (78,22,6) designs, 412 of them having Ze as an automorphism group. Further, we show that up to isomorphism there is exactly one symmetric (78,22,6) design admitting an automorphism group isomorphic to Frob$g x Z2, namely the design constructed by Zvonimir Janko and Tran van Trung. Thus, there is no (78,22,6) difference set in the group Frobw x Zi. We study binary linear codes spanned by the incidence matrices of the constructed designs. Further, extending previous results on codes obtained from orbit matrices of 2-designs, we show that under certain conditions both fixed and nonfixed part of an orbit matrix span a self-orthogonal code over the finite field Fpn or over the ring Zm- We construct self-orthogonal codes over Z4 spanned by orbit matrices of symmetric (78,22,6) designs. © 2018 Utilitas Mathematica Publishing Incorporated. All rights reserved.
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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".