Cyclic codes are an important class of linear codes. Bounding the minimum distance of cyclic codes is a long-standing research topic in coding theory, and several well-known and basic results have been developed on this topic. Recently, locally repairable codes (LRCs) have attracted much attention due to their repair efficiency in large-scale distributed storage systems. In this paper, by employing the singleton procedure technique, we first provide a sufficient condition for bounding the minimum distance of cyclic codes with typical defining sets. Secondly, by considering a specific case, we establish a connection between bounds for the minimum distance of cyclic codes and solutions to a system of inequalities. This connection leads to the derivation of new bounds, including some with general patterns. In particular, we provide three new bounds with general patterns, one of which serves as a generalization of the Betti-Sala bound. Finally, we present a generalized lower bound for a special case and construct several families of $(2, \delta)$-LRCs with unbounded length and minimum distance $2\delta$. It turns out that these LRCs are distance-optimal, and their parameters are new. To the best of our knowledge, this work represents the first construction of distance-optimal $(r, \delta)$-LRCs with unbounded length and minimum distance exceeding $r+\delta-1$.

翻译：循环码是一类重要的线性码。界定循环码的最小距离是编码理论中长期存在的研究课题，目前已发展出若干广为人知的基础性结果。近年来，局部可修复码因其在大规模分布式存储系统中的修复效率而备受关注。本文首先利用Singleton过程技术，为具有典型定义集的循环码最小距离的界定提供了充分条件。其次，通过考虑特殊情况，我们在循环码最小距离的界与不等式组的解之间建立了联系。这种联系导出了新的界，其中包含若干具有一般形式的界。特别地，我们给出了三个具有一般形式的新界，其中一个推广了Betti-Sala界。最后，我们给出了特殊情况下的广义下界，并构造了若干族长度无界、最小距离为$2\delta$的$(2,\delta)$-局部可修复码。结果表明，这些局部可修复码是距离最优的，且其参数是新的。据我们所知，本工作首次构造了长度无界且最小距离超过$r+\delta-1$的距离最优$(r,\delta)$-局部可修复码。