Optimal (r, δ)-Locally Repairable Codes From Simplex Code and Cap Code

Abstract

Locally repairable codes (LRCs) are implemented in distributed storage systems (DSSs) due to their low repair overhead. A linear code $\mathcal {C}$ is said to have $(r,\delta)$ -locality 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 $ . An LRC is called optimal if its minimum distance attains Singleton-type bound was proposed. In this letter, optimal LRCs are considered. We first determine $(r,\delta)$ -locality of three dimensional Simplex code, then using anticode strategy, a class of $[{3q,3,2q-1}]_{q}$ LRCs with $(2,q)$ locality are derived for general $q$ . Finally, using an ovoid in $PG(3, q)$ , we construct $[q^{2}+1,4,q(q-1)]_{q}$ and $[{4q-4,4,3q-5}]_{q}$ LRCs with $r=3$ and $\delta =q-1$ . All LRCs constructed in this letter attain the Singleton-type bound.