Designed for distributed storage systems, locally repairable codes (LRCs) can reduce the repair bandwidth and disk I/O complexity during the storage node repair process. A code with locality $(r,\delta)$ (also called an $(r,\delta)$-LRC) can repair up to $\delta-1$ symbols in a codeword simultaneously by accessing at most other $r$ symbols in the codeword. An optimal $(r,\delta)$-LRC is a code that achieves the Singleton-type bound on $r,\delta$, the code length, the code size and the minimum distance. Constructing optimal LRCs receives wide attention recently. In this paper, we give a new method to analyze the $(r,\delta)$-locality of cyclic codes. Via the characterization of locality, we obtain several classes of optimal $(r,\delta)$-LRCs. When the minimum distance $d$ is greater than or equal to $2\delta+1$, we present a class of $q$-ary optimal $(r,\delta)$-LRCs whose code lengths are unbounded. In contrast, the existing works only show the existence of $q$-ary optimal $(r,\delta)$-LRCs of code lengths $O(q^{1+\frac{\delta}{2}})$. When $d$ is in between $\delta$ and $2\delta$, we present five classes of optimal $(r,\delta)$-LRCs whose lengths are unbounded. Compared with the existing constructions of optimal $(r,\delta)$-LRCs with unbounded code lengths, three classes of our LRCs do not have the restriction of the distance $d\leq q$. In other words, our optimal $(r,\delta)$-LRCs can have large distance over a relatively small field, which are desired by practical requirement of high repair capability and low computation cost. Besides, for the case of the minimal value $2$ of $\delta$, we find out all the optimal cyclic $(r,2)$-LRCs of prime power lengths.

翻译：针对分布式存储系统设计的局部修复码（LRC）可在存储节点修复过程中降低修复带宽和磁盘I/O复杂度。具有$(r,\delta)$局部性的码（亦称$(r,\delta)$-LRC）能够通过访问码字中至多$r$个其他符号，同时修复一个码字中至多$\delta-1$个符号。优$(r,\delta)$-LRC是指达到关于$r,\delta$、码长、码规模与最小距离的Singleton型界的码。优LRC的构建近来受到广泛关注。本文提出一种分析循环码$(r,\delta)$局部性的新方法。通过局部性的刻画，我们获得若干类优$(r,\delta)$-LRC。当最小距离$d$满足$d\ge 2\delta+1$时，我们呈现一类$q$元优$(r,\delta)$-LRC，其码长无界；而现有工作仅证明存在码长为$O(q^{1+\frac{\delta}{2}})$的$q$元优$(r,\delta)$-LRC。当$d$介于$\delta$与$2\delta$之间时，我们给出五类码长无界的优$(r,\delta)$-LRC。与现有码长无界优$(r,\delta)$-LRC构造相比，我们有三类LRC不受距离$d\le q$的限制。换言之，我们的优$(r,\delta)$-LRC可在相对较小的域上实现大距离，这符合高修复能力与低计算开销的实际需求。此外，针对$\delta$最小取值$2$的情形，我们找出了所有具有素数幂长度的最优循环$(r,2)$-LRC。