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Locally repairable codes with locality $r$ ($r$-LRCs for short) wereintroduced by Gopalan et al. \cite{1} to recover a failed node of the code fromat most other $r$ available nodes. And then $(r,\delta)$ locally repairablecodes ($(r,\delta)$-LRCs for short) were produced by Prakash et al. \cite{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 theSingleton-type bound. It has been a great challenge to construct $q$-aryoptimal $(r,\delta)$-LRCs with length much larger than $q$. Surprisingly, Luoet al. \cite{3} presented a construction of $q$-ary optimal $r$-LRCs of minimumdistances 3 and 4 with unbounded lengths (i.e., lengths of these codes areindependent of $q$) via cyclic codes.  In this paper, inspired by the work of \cite{3}, we firstly construct twoclasses of optimal cyclic $(r,\delta)$-LRCs with unbounded lengths and minimumdistances $\delta+1$ or $\delta+2$, which generalize the results about the$\delta=2$ case given in \cite{3}. Secondly, with a slightly strongercondition, we present a construction of optimal cyclic $(r,\delta)$-LRCs withunbounded length and larger minimum distance $2\delta$. Furthermore, when$\delta=3$, we give another class of optimal cyclic $(r,3)$-LRCs with unboundedlength and minimum distance $6$.

Optimal cyclic $(r,\delta)$ locally repairable codes with unbounded length