Constructions of Optimal Cyclic $(r,\delta)$ Locally Repairable Codes

A code is said to be a $r$-local locally repairable code (LRC) if each of its coordinates can be repaired by accessing at most $r$ other coordinates. When some of the $r$ coordinates are also erased, the $r$-local LRC can not accomplish the local repair, which leads to the concept of $(r,\delta)$-locality. A $q$-ary $[n, k]$ linear code $\cC$ is said to have $(r, \delta)$-locality ($\delta\ge 2$) if for each coordinate $i$, there exists a punctured subcode of $\cC$ with support containing $i$, whose length is at most $r + \delta - 1$, and whose minimum distance is at least $\delta$. The $(r, \delta)$-LRC can tolerate $\delta-1$ erasures in total, which degenerates to a $r$-local LRC when $\delta=2$. A $q$-ary $(r,\delta)$ LRC is called optimal if it meets the Singleton-like bound for $(r,\delta)$-LRCs. A class of optimal $q$-ary cyclic $r$-local LRCs with lengths $n\mid q-1$ were constructed by Tamo, Barg, Goparaju and Calderbank based on the $q$-ary Reed-Solomon codes. In this paper, we construct a class of optimal $q$-ary cyclic $(r,\delta)$-LRCs ($\delta\ge 2$) with length $n\mid q-1$, which generalizes the results of Tamo \emph{et al.} Moreover, we construct a new class of optimal $q$-ary cyclic $r$-local LRCs with lengths $n\mid q+1$ and a new class of optimal $q$-ary cyclic $(r,\delta)$-LRCs ($\delta\ge 2$) with lengths $n\mid q+1$. The constructed optimal LRCs with length $n=q+1$ have the best-known length $q+1$ for the given finite field with size $q$ when the minimum distance is larger than $4$.