Several Classes of Optimal Cyclic Locally Repairable Codes of Unbounded Code Lengths

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.