Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair cost by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefit of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code length. Specifically, we first obtain two classes of $q$-ary cyclic Singleton-optimal $(n, k, d=6;r=2)$-LRCs with length $n=3(q+1)$ when $3 \mid (q-1)$ and $q$ is even, and length $n=\frac{3}{2}(q+1)$ when $3 \mid (q-1)$ and $q \equiv 1(\bmod~4)$, respectively. To the best of our knowledge, this is the first construction of $q$-ary cyclic Singleton-optimal LRCs with length $n>q+1$ and minimum distance $d \geq 5$. On the other hand, an LRC acheiving the Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic codes, we construct two new families of $q$-ary perfect LRCs with length $n=\frac{q^m-1}{q-1}$, minimum distance $d=5$ and locality $r=2$.

翻译：局部修复码作为分布式存储系统中重要的编码方案，可通过访问较少非故障节点实现低修复成本。理论界值与最优LRCs的构造已得到广泛研究。基于循环码和常循环码的最优LRCs兼具简洁代数结构与高效编码流程的优势。本文继续研究基于循环码和常循环码的长码长最优LRCs构造。具体而言，我们首先获得两类$q$元循环Singleton-最优$(n, k, d=6;r=2)$-LRCs：当$3 \mid (q-1)$且$q$为偶数时，码长$n=3(q+1)$；当$3 \mid (q-1)$且$q \equiv 1(\bmod~4)$时，码长$n=\frac{3}{2}(q+1)$。据我们所知，这是首个码长$n>q+1$且最小距离$d \geq 5$的$q$元循环Singleton-最优LRCs构造。另一方面，达到Hamming界值的LRC被称为完美LRC。通过使用循环码和常循环码，我们构造了两个新的$q$元完美LRCs族，其码长$n=\frac{q^m-1}{q-1}$，最小距离$d=5$，局部性$r=2$。