A Class of Binary Locally Repairable Codes
Bibliographic record
Abstract
An (n, k) erasure code that can recover any coded symbol by at most r other coded symbols is called a locally repairable code (LRC) with locality r. LRCs have been recently implemented in distributed storage systems. Coding complexity reduction can be significantly decreased by using binary LRCs (BLRCs) as they eliminate costly multiplication calculation. In this paper, motivated by the recently erasure codes with d = 4 used in practice, we propose BLRCs when (r + 1) | n and d = 4. We prove that our proposed binary codes are optimal for r ∈ {1, 3}, meaning that neither their locality nor their minimum distance can be improved by non-binary codes. For r ≥ 4, our proposed binary codes offer near-optimal code rate, with a rate gap of O(log r/n) compared with optimal nonbinary codes. While keeping the bulk of code structure binary, we eliminate this rate gap by using fields with sizes as small as r + 2 for only two redundant symbols. These non-binary codes still eliminate the need for costly multiplications in many operations including a single failure repair (a dominant repair scenario). Using the construction of spanning BLRC with d = 4 as a backbone, we also construct LRCs with minimum distance d ≥ 6. Furthermore, we obtain a closed-form equation for the mean-time to data-loss of arbitrary erasure codes.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 source (direct Gemma or distilled Codex), 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".