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

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.