Locally repairable codes, or locally recoverable codes (LRC for short), are designed for applications in distributed and cloud storage systems. Similar to classical block codes, there is an important bound called the Singleton-type bound for locally repairable codes. In this paper, an optimal locally repairable code refers to a block code achieving this Singleton-type bound. Like classical MDS codes, optimal locally repairable codes carry some very nice combinatorial structures. Since the introduction of the Singleton-type bound for locally repairable codes, people have put tremendous effort into construction of optimal locally repairable codes. There are a few constructions of optimal locally repairable codes in the literature. Most of these constructions are realized via either combinatorial or algebraic structures. In this paper, we apply automorphism group of the rational function field to construct optimal locally repairable codes by considering the group action on projective lines over finite fields. Due to various subgroups of the projective general linear group, we are able to construct optimal locally repairable codes with flexible locality as well as smaller alphabet size comparable to the code length. In particular, we produce new families of $q$ -ary locally repairable codes, including codes of length $q+1$ via cyclic groups.

中文翻译：

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通过有理函数域的自同构群构造最优局部可修复码

本地可修复代码，或本地可恢复代码（简称 LRC），专为分布式和云存储系统中的应用而设计。与经典分组码类似，对于局部可修复码有一个重要的界限，称为单例型界限。在本文中，最优局部可修复代码是指实现这种单例类型边界的块代码。像经典的 MDS 代码一样，最优的局部可修复代码带有一些非常好的组合结构。自从引入局部可修复码的单例型绑定以来，人们付出了巨大的努力来构建最优的局部可修复码。文献中有一些最优局部可修复代码的构造。大多数这些构造是通过组合或代数结构实现的。在本文中，我们通过考虑有限域上射影线上的群作用，应用有理函数域的自同构群来构造最优的局部可修复代码。由于投影一般线性群的各种子​​群，我们能够构建具有灵活局部性以及与代码长度相当的较小字母大小的最优局部可修复代码。特别是，我们产生了新的 $q$ -ary 本地可修复代码系列，包括通过循环组长度为 $q+1$ 的代码。我们能够构建具有灵活局部性以及与代码长度相当的较小字母表大小的最佳局部可修复代码。特别是，我们产生了新的 $q$ -ary 本地可修复代码系列，包括通过循环组长度为 $q+1$ 的代码。我们能够构建具有灵活局部性以及与代码长度相当的较小字母表大小的最佳局部可修复代码。特别是，我们产生了新的 $q$ -ary 本地可修复代码系列，包括通过循环组长度为 $q+1$ 的代码。