Geometric construction of k-optimal locally repairable codes
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866916065341079552 |
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| author | Fu, Yi Shan, Xiuling |
| author_facet | Fu, Yi Shan, Xiuling |
| contents | A linear code is referred to as a locally repairable code (LRC) with locality r if any erased code symbol can be recovered by accessing at most r other code symbols. LRCs are highly desirable for distributed storage systems to enhance repair efficiency. In this paper, we investigate LRCs with disjoint repair sets via the parity-check matrix method. Firstly, we propose a novel concept of the s-Pasch configuration and present a geometric characterization for the existence of LRCs with minimum distance 5 and locality 3. Subsequently, we construct k-optimal LRCs by exploiting the point-line relationship in PG(2,q). Finally, a family of q-ary k-optimal LRCs with minimum distance 6 and general locality r is constructed using partial r-spreads. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_31214 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric construction of k-optimal locally repairable codes Fu, Yi Shan, Xiuling Information Theory Combinatorics A linear code is referred to as a locally repairable code (LRC) with locality r if any erased code symbol can be recovered by accessing at most r other code symbols. LRCs are highly desirable for distributed storage systems to enhance repair efficiency. In this paper, we investigate LRCs with disjoint repair sets via the parity-check matrix method. Firstly, we propose a novel concept of the s-Pasch configuration and present a geometric characterization for the existence of LRCs with minimum distance 5 and locality 3. Subsequently, we construct k-optimal LRCs by exploiting the point-line relationship in PG(2,q). Finally, a family of q-ary k-optimal LRCs with minimum distance 6 and general locality r is constructed using partial r-spreads. |
| title | Geometric construction of k-optimal locally repairable codes |
| topic | Information Theory Combinatorics |
| url | https://arxiv.org/abs/2605.31214 |