Unitary Space–Time Group Codes: Diversity Sums From Character Tables
Bibliographic record
Abstract
<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Diversity sum, which is calculated from the Frobenius norm of the difference of two distinct elements in a signal constellation, is the significant parameter to predict a unitary space–time constellation having good performance in low signal-to-noise ratio (SNR). In this correspondence, we propose a method to compute the diversity sum of a unitary group constellation using a character table. Our proposed analysis is simple, requiring only a lookup of the character table. We illustrate our method for the finite special linear groups <emphasis><formula formulatype="inline"><tex>$SL_{2}$</tex></formula></emphasis>, and compare codes with high diversity sum against fixed point free groups at low SNR. We also introduce the notion of a faithful group constellation, that is, one whose diversity sum is greater than <emphasis><formula><tex>$0$</tex></formula></emphasis>. Faithful group constellations may be obtained from any nontrivial character of a group. We describe the method to do so in this correspondence and illustrate it with the example of the finite projective special linear group <emphasis><formula formulatype="inline"><tex>${\rm PSL}_{2}$</tex></formula></emphasis>. </para>
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 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".