Locally repairable codes with availability have become essential components in modern large-scale distributed cloud storage systems and numerous other applications. In this paper, we focus on the construction of locally repairable codes with one or two recovering sets via elliptic function fields. Prior pioneering work by Li et al. (IEEE Trans. Inf. Theory, vol. 65, no. 1, 2019) and Ma and Xing (J. Comb. Theory Ser. A., vol. 193, 2023) employed maximal supersingular elliptic curves to obtain several optimal (classical) locally repairable codes. In contrast, we consider ordinary elliptic curves with many rational points. This approach yields several new families of \(q\)-ary optimal locally repairable codes with length \(O(q+2\sqrt{q})\) and flexible locality. Consequently, our work broadens the selection of curves available for the construction of optimal locally repairable codes. Furthermore, we present a general framework for constructing locally repairable codes with two recovering sets via automorphism groups of elliptic function fields. To realize this framework, we devise a novel construction for determining the functions \(e_i\) in the construction of locally repairable codes. By employing both supersingular and ordinary elliptic curves, we obtain several families of locally repairable codes with two recovering sets. In particular, we construct a family of \(q^2\)-ary locally repairable codes with two recovering sets, achieving length \(O(q^2+2q)\) and Singleton-defect \(O\!\left(\frac{2\ell}{q^2+2q-8\ell}\right)\), where \(\ell \mid\mid q + 2\) with \(4\ell < q\).

翻译：具备可用性的局部可修复码已成为现代大规模分布式云存储系统及众多应用中的关键组件。本文聚焦于利用椭圆函数域构造具有一至两个恢复集的局部可修复码。Li等人（IEEE Trans. Inf. Theory, vol. 65, no. 1, 2019）及Ma与Xing（J. Comb. Theory Ser. A., vol. 193, 2023）的前驱性工作采用极大超奇异椭圆曲线获得了若干最优（经典）局部可修复码。与之不同，我们考虑具有众多有理点的普通椭圆曲线。该方法构造了若干新族长度为\(O(q+2\sqrt{q})\)且灵活性可调的\(q\)元最优局部可修复码。因此，本工作拓展了可用于构造最优局部可修复码的曲线选择范围。此外，我们提出一个通过椭圆函数域自同构群构造具有两个恢复集的局部可修复码的通用框架。为实现该框架，我们设计了一种新型构造方法，用于确定局部可修复码构造中的函数\(e_i\)。通过同时采用超奇异与普通椭圆曲线，我们获得了若干族具有两个恢复集的局部可修复码。特别地，我们构造了一族\(q^2\)元局部可修复码（含两个恢复集），其长度可达\(O(q^2+2q)\)，Singleton缺陷为\(O\!\left(\frac{2\ell}{q^2+2q-8\ell}\right)\)，其中\(\ell \mid\mid q + 2\)且\(4\ell < q\)。