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Locally repairable codes (LRCs) have emerged as an important coding scheme indistributed storage systems (DSSs) with relatively low repair cost by accessingfewer non-failure nodes. Theoretical bounds and optimal constructions of LRCshave been widely investigated. Optimal LRCs via cyclic and constacyclic codesprovide significant benefit of elegant algebraic structure and efficientencoding procedure. In this paper, we continue to consider the constructions ofoptimal 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 $q$ iseven, and length $n=\frac{3}{2}(q+1)$ when $3 \mid (q-1)$ and $q \equiv1(\bmod~4)$, respectively. To the best of our knowledge, this is the firstconstruction of $q$-ary cyclic Singleton-optimal LRCs with length $n>q+1$ andminimum distance $d \geq 5$. On the other hand, an LRC acheiving theHamming-type bound is called a perfect LRC. By using cyclic and constacycliccodes, 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 $r=2$.

Singleton-Optimal LRCs and Perfect LRCs via Cyclic and Constacyclic Codes