Locally repairable codes with locality r (r-LRCs for short) were introduced by Gopalan et al. [1] to recover a failed node of the code from at most other r available nodes. And then $(r,\delta )$-locally repairable codes ($(r,\delta )$-LRCs for short) were produced by Prakash et al. [2] for tolerating multiple failed nodes. An r-LRC can be viewed as an $(r,2)$-LRC. An $(r,\delta )$-LRC is called optimal if it achieves the Singleton-type bound. It has been a great challenge to construct q-ary optimal $(r,\delta )$-LRCs with length much larger than q. Surprisingly, Luo et al. [3] presented a construction of q-ary optimal r-LRCs of minimum distances 3 and 4 with unbounded lengths (i.e., lengths of these codes are independent of q) via cyclic codes.

In this paper, inspired by the work of [3], we firstly construct two classes of optimal cyclic $(r,\delta )$-LRCs with unbounded lengths and minimum distances $\delta +1$ or $\delta +2$, which generalize the results about the $\delta =2$ case given in [3]. Secondly, with a slightly stronger condition, we present a construction of optimal cyclic $(r,\delta )$-LRCs with unbounded length and larger minimum distance 2δ. Furthermore, when $\delta =3$, we give another class of optimal cyclic $(r,3)$-LRCs with unbounded length and minimum distance 6.

Gopalan等人介绍了局部性为r的局部可修复代码（r -LRCs）。[1]从最多其他r个可用节点中恢复代码的失败节点。接着$（\mathrm{[R}，\delta ）$-本地可修复代码（$（\mathrm{[R}，\delta ）$-LRCs）由Prakash等人生产。[2]用于容忍多个故障节点。一个ř -LRC可以被看作是一个$（\mathrm{[R}，2）$-LRC。一个$（\mathrm{[R}，\delta ）$如果达到单例类型界限，则-LRC被称为最优。构造q元最优值一直是一个巨大的挑战$（\mathrm{[R}，\delta ）$长度远大于q的-LRC 。令人惊讶的是，Luo等。文献[3]通过循环码提出了最小距离为3和4的q元最优r -LRC的构造，其无限长度（即，这些码的长度与q无关）。

在本文中，受[3]的启发，我们首先构造了两类最优循环 $（\mathrm{[R}，\delta ）$-LRC具有无限制的长度和最小距离 $\delta +\mathrm{1个}$ 要么 $\delta +2$，将关于 $\delta =2$在[3]中给出的情况。其次，在条件稍强的情况下，我们提出了最优周期的构造$（\mathrm{[R}，\delta ）$-LRCs与无界的长度和较大的最小距离2 δ。此外，当$\delta =3$，我们给出了另一类最优循环 $（\mathrm{[R}，3）$-无限制长度和最小距离的LRC 6。