Optimal (r, δ)-Locally Repairable Codes From Simplex Code and Cap Code
Qiang Fu, Ruihu Li, Sen Yang · IEEE Access · 2020
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.