On the Average Locality of Locally Repairable Codes
Bibliographic record
Abstract
A linear block code with dimension $k$, length $n$, and minimum distance $d$ is called a locally repairable code (LRC) with locality $r$ if it can retrieve any coded symbol by at most $r$ other coded symbols. LRCs have been recently proposed and used in practice in distributed storage systems (DSSs) such as Windows Azure storage and Facebook HDFS-RAID. Theoretical bounds on the maximum locality of LRCs ($r$) have been established. The \textit{average} locality of an LRC ($\overline{r}$) directly affects the costly repair bandwidth, disk I/O, and number of nodes involved in the repair process of a missing data block. There is a gap in the literature studying $\overline{r}$. In this paper, we establish a lower bound on $\overline{r}$ of arbitrary $(n,k,d)$ LRCs. Furthermore, we obtain a tight lower bound on $\overline{r}$ for a practical case where the code rate $(R=\frac{k}{n})$ is greater than $(1-\frac{1}{\sqrt{n}})^2$. Finally, we design three classes of LRCs that achieve the obtained bounds on $\overline{r}$. Comparing with the existing LRCs, our proposed codes improve the average locality without sacrificing such crucial parameters as the code rate or minimum distance.
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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 | 0.002 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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".