Polynomials, Divided Differences, and Codes
Bibliographic record
Abstract
Multiplicity codes (Kopparty et al., J. ACM 2014) are multivariate polynomial codes where the codewords are described by evaluations of polynomials (with a degree bound) and their derivatives up to some order (the multiplicity parameter), on a suitably chosen affine set of points. While efficient decoding algorithms were known in some special cases of point sets, by a reduction to univariate multiplicity codes, a general algorithm for list decoding up to the distance of the code when the point set is an arbitrary finite grid, was obtained only recently (Bhandari et al., IEEE TIT 2023). This required the characteristic of the field to be zero or larger than the degree bound, which is somewhat necessary since list decoding up to distance with small output list size is not possible when the characteristic is significantly smaller than the degree. In this work, we present an alternative construction based on divided differences of polynomials, that conceptually resembles the classical multiplicity codes but has good list decodability "insensitive to the field characteristic". We obtain a simple algorithm that list decodes this code up to distance for arbitrary finite grids over all finite fields. Our construction can also be interpreted as a folded Reed-Muller code, which may be of independent interest.
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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 | 0.001 | 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 | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| 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".