关于大局部弧构造的建设
On the construction of large local arcs
AI总结：
本文提出局部弧概念，用于构造具有更优性能的局部可修复码。
摘要：
受最优局部可修复码构造的启发，我们引入了一个新的有限几何概念——局部弧，其定义为在PG(2,q)中的一组不相交点集S_i，使得S_i ∪ S_j构成一条弧。我们专注于最大k-均匀局部弧大小的上界和下界。对于q=p^m（p为素数），我们在PG(2,q)中构造出大小为Ω(q^d)的k-均匀局部弧，其中d介于1.1167到1.25之间，仅取决于m。对于k=4，这表明存在最小距离为6、局部性为3且修复组互不相同的最优局部可修复码（LRCs），其长度为q的超线性函数——这在之前已知的O(q)构造基础上有了显著改进。
英文摘要：
Motivated by the construction of optimal locally repairable codes, we introduce the new finite geometric concept of a \emph{local arc} which is defined as a collection $\mathcal{S}$ of disjoint point sets $S_{i}$ in $\mathrm{PG}(2,q)$ such that $S_{i} \cup S_{j}$ is an arc for any $S_{i}, S_{j} \in \mathcal{S}$. We focus on the upper and lower bounds on the sizes of maximum $k$-uniform local arcs. For $q=p^m$ with $p$ prime, we construct $k$-uniform local arcs in $\mathrm{PG}(2,q)$ of size $Ω(q^{d})$ where $d$ is between $1.1167$ and $1.25$ depending only on $m$. For $k=4$, this implies the existence of optimal locally repairable codes (LRCs) with minimum distance 6, locality 3, and disjoint repair groups, whose length is superlinear in $q$--a significant improvement over the previously known $O(q)$ constructions for such LRCs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
