Singleton-Optimal LRCs and Perfect LRCs via Cyclic Codes

Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair cost by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic codes provide significant benefit of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic codes with longer code length. Specifically, we first obtain two classes of Singleton-optimal cyclic LRCs with length
$n=3(q+1)$
when
$3\vert (q-1)$
and
$q$
is even, and length
$n=\frac{3}{2}(q+1)$
when
$3\vert (q-1)$
and
$q$
is odd, respectively. To the best of our knowledge, this is the first construction of q-ary cyclic Singleton-optimal LRCs with length
$n > q+1$
and minimum distance
$d\geq 5$
. By using cyclic codes as well, we construct a new family of perfect LRCs with
$d=5$
, which generalize the result of Goparaju and Calderbank.