The constructions of Singleton-optimal locally repairable codes with minimum distance 6 and locality 3
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912925948575744 |
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| author | Xiong, Yanzhen Lu, Jianbing |
| author_facet | Xiong, Yanzhen Lu, Jianbing |
| contents | In this paper, we present new constructions of $q$-ary Singleton-optimal locally repairable codes (LRCs) with minimum distance $d=6$ and locality $r=3$, based on combinatorial structures from finite geometry. By exploiting the well-known correspondence between a complete set of mutually orthogonal Latin squares (MOLS) of order $q$ and the affine plane $\mathrm{AG}(2,q)$, We systematically construct families of disjoint 4-arcs in the projective plane $\mathrm{PG}(2,q)$, such that the union of any two distinct 4-arcs forms an 8-arc. These 4-arcs form what we call 4-local arcs, and their existence is equivalent to that of the desired codes. For any prime power $q\ge 7$, our construction yields codes of length $n = 2q$, $2q-2$, or $2q-6$ depending on whether $q$ is even, $q\equiv 3 \pmod{4}$, or $q\equiv 1 \pmod{4}$, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21494 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The constructions of Singleton-optimal locally repairable codes with minimum distance 6 and locality 3 Xiong, Yanzhen Lu, Jianbing Information Theory Combinatorics In this paper, we present new constructions of $q$-ary Singleton-optimal locally repairable codes (LRCs) with minimum distance $d=6$ and locality $r=3$, based on combinatorial structures from finite geometry. By exploiting the well-known correspondence between a complete set of mutually orthogonal Latin squares (MOLS) of order $q$ and the affine plane $\mathrm{AG}(2,q)$, We systematically construct families of disjoint 4-arcs in the projective plane $\mathrm{PG}(2,q)$, such that the union of any two distinct 4-arcs forms an 8-arc. These 4-arcs form what we call 4-local arcs, and their existence is equivalent to that of the desired codes. For any prime power $q\ge 7$, our construction yields codes of length $n = 2q$, $2q-2$, or $2q-6$ depending on whether $q$ is even, $q\equiv 3 \pmod{4}$, or $q\equiv 1 \pmod{4}$, respectively. |
| title | The constructions of Singleton-optimal locally repairable codes with minimum distance 6 and locality 3 |
| topic | Information Theory Combinatorics |
| url | https://arxiv.org/abs/2602.21494 |