Locally Repairable Codes: Joint Sequential–Parallel Repair for Multiple Node Failures
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Bibliographic record
Abstract
Locally repairable codes (LRC) have been studied from two approaches to locally repair multiple failed nodes: 1) parallel approach, in which a coordinate i of an [n,k,d] linear code is said to have locality r and availability t if there exist t disjoint repair sets each of which contains at most r other coordinates that can recover the value of the i -th coordinate; 2) sequential approach, in which the erased symbols (failed nodes) are repaired, one by one, and any previously repaired node can be used to repair the remaining failed nodes. In this paper, we first consider LRC aiming at joint sequential-parallel repairing multiple failed nodes, and study the (n,k,r,t,u) -ELRCs (Exact locally repairable codes) which are [n,k] linear codes with the property that any set of failed nodes of size at most t can be simultaneously repaired in parallel mode, and each element of a set E of failed nodes of size at most u can be sequentially repaired by r (r<; k) other coordinates. We present a method by which with a given parity-check matrix of an (n,k,r,t,u) -ELRC with minimum Hamming distance d, a new ELRC with minimum Hamming distance 2d and availability t+1 is constructed that can repair each set of failed nodes E of size at most 2u+1 in sequential mode and this repair is done in at most u-t+2 steps. We construct a big family of LRCs by making use of orthogonal Latin rectangles and permutation cubes and some other combinatorial designs; the constructed codes contain the family of direct product codes; we also use m -dimensional permutation cubes to construct LRCs with short block length for each r.
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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 | 0.000 | 0.004 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it