Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal \((2, \delta )\)-LRCs over \(\mathbb {F}_q\) with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a \((2, \delta )\)-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe–Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes.

本地可修复代码（LRC）由于其在分布式存储系统中的应用而引起了广泛的关注。在本文中，我们提供了具有灵活参数的基于\(\mathbb {F}_q\)的最优\((2, \delta )\) -LRC 的新结构。首先，采用有限几何技术，我们引入一个简单但有用的条件来确保穿孔单纯形码成为\((2, \delta )\) -LRC。值得注意的是，该条件仅对打孔集的大小提出要求。其次，利用有限域上的字符和和 Krawtchouk 多项式，我们通过新结构的删截集来确定更多删截单纯形码的参数。推导了具有新参数的LRC 的无限族。我们所有的新 LRC 对于广义 Cadambe-Mazumdar 界都是最优的，其中一些也是 Griesmer 码或距离最优码。