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.

翻译：局部可修复码（LRCs）因其在分布式存储系统中的应用而备受关注。本文在$\mathbb{F}_q$上构造了具有灵活参数的最优$(2, \delta)$-LRCs。首先，利用有限几何技术，我们引入了一个简单而有效的条件，以确保穿孔单纯形码成为一个$(2, \delta)$-LRC。值得注意的是，该条件仅对穿孔集的大小提出了要求。其次，利用有限域上的特征和与Krawtchouk多项式，我们确定了具有新型结构穿孔集的更多穿孔单纯形码的参数。由此推导出多个具有新参数的LRC无限族。我们提出的所有新LRCs均关于广义Cadambe-Mazumdar界是最优的，其中部分码同时也是Griesmer码或距离最优码。