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.

翻译：本文基于有限几何中的组合结构，提出了新的$q$元Singleton最优局部可修复码（LRCs）构造，其最小距离为$d=6$，局部性为$r=3$。通过利用已知的$q$阶完全相互正交拉丁方（MOLS）集与仿射平面$\mathrm{AG}(2,q)$之间的对应关系，我们系统地在射影平面$\mathrm{PG}(2,q)$中构造了不相交的4-弧族，使得任意两个不同4-弧的并集构成一个8-弧。这些4-弧构成了我们所谓的4-局部弧，它们的存在性等价于目标码的存在性。对于任意素幂$q\ge 7$，我们的构造可生成长度为$n = 2q$、$2q-2$或$2q-6$的码，分别对应于$q$为偶数、$q\equiv 3 \pmod{4}$或$q\equiv 1 \pmod{4}$的情况。