On finding the largest minimum distance of locally recoverable codes: A graph theory approach
Bibliographic record
Abstract
The [ n , k , r ] -Locally recoverable codes (LRC) studied in this work are a well-studied family of [ n , k ] linear codes for which the value of each symbol can be recovered by a linear combination of at most r other symbols. In this paper, we study the LMD problem, which is to find the largest possible minimum distance of [ n , k , r ] -LRCs, denoted by D ( n , k , r ) . LMD can be approximated within an additive term of one—it is known that D ( n , k , r ) is equal to either d ⁎ or d ⁎ − 1 , where d ⁎ = n − k − ⌈ k r ⌉ + 2 . Moreover, for a range of parameters, it is known precisely whether the distance D ( n , k , r ) is d ⁎ or d ⁎ − 1 . However, the problem is still open despite a significant effort. In this work, we convert LMD to an equivalent simply-stated problem in graph theory. Using this conversion, we show that an instance of LMD is at least as hard as computing the size of a maximal graph of high girth, a hard problem in extremal graph theory. This is an evidence that LMD—although can be approximated within an additive term of one—is hard to solve in general. As a positive result, thanks to the conversion and the exiting results in extremal graph theory, we solve LMD for a range of code parameters that has not been solved before.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 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 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".