Essential dimension and error-correcting codes
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Bibliographic record
Abstract
One of the important open problems in the theory of central simple algebras is to compute the essential dimension of GLn /μm, i.e., the essential dimension of a generic division algebra of degree n and exponent dividing m. In this paper we study the essential dimension of groups of the form G = (GLn1 × · · · ×GLnr )/C , where C is a central subgroup of GLn1 × · · · ×GLnr . Equivalently, we are interested in the essential dimension of a generic r-tuple (A1, . . . , Ar) of central simple algebras such that deg(Ai) = ni and the Brauer classes of A1, . . . , Ar satisfy a system of homogeneous linear equations in the Brauer group. The equations depend on the choice of C via the error-correcting code Code(C) which we naturally associate to C. We focus on the case where n1, . . . , nr are powers of the same prime. The upper and lower bounds on ed(G) we obtain are expressed in terms of coding-theoretic parameters of Code(C), such as its weight distribution. Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when r ≥ 3; in some cases we even compute the exact value. The Appendix by Athena Nguyen contains an explicit description of the Galois cohomology of groups of the form (GLn1 × · · · ×GLnr )/C. This description and its corollaries are used throughout the paper.
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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 | 0.001 | 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 it