Stopping Redundancy Hierarchy Beyond the Minimum Distance
Bibliographic record
Abstract
Stopping sets play a crucial role in failure events of iterative decoders over a binary erasure channel (BEC). The ℓth stopping redundancy is the minimum number of rows in the parity-check matrix of a code, which contains no stopping sets of size up to ℓ. In this paper, a notion of coverable stopping sets is defined. In order to achieve maximum-likelihood performance under iterative decoding over the BEC, the parity-check matrix should contain no coverable stopping sets of size ℓ, for 1 ≤ ℓ ≤ n-k, where n is the code length, k is the code dimension. By estimating the number of coverable stopping sets, we obtain upper bounds on the ℓth stopping redundancy, 1 ≤ ℓ ≤ n-k. The bounds are derived for both specific codes and code ensembles. In the range 1 ≤ ℓ ≤ d-1, for specific codes, the new bounds improve on the results in the literature. Numerical calculations are also presented.
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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 | 0.002 | 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 | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".