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

A code is said to be an $r$ -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 cannot accomplish the local repair, which leads to the concept of $(r,\delta)$ -locality. A $q$ -ary $[n, k]$ linear code $\mathcal {C}$ is said to have $(r, \delta)$ -locality ( $\delta \ge 2$ ) if for each coordinate $i$ , there exists a punctured subcode of $\mathcal {C}$ 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 every local code (i.e., punctured subcode), which degenerates to an $r$ -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 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 for a given finite field with size $q$ when the minimum distance is larger than 4.