Singleton-optimal LRCs and perfect LRCs via cyclic and constacyclic codes

Abstract

Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair costs by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefits of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code lengths. Specifically, we first obtain two classes of q-ary cyclic Singleton-optimal (n,k,d=6;r=2)-LRCs with length n=3(q+1) when 3|(q−1) and q is even, and length n=32(q+1) when 3|(q−1) and q≡1(mod4), 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≥5. On the other hand, an LRC achieving the Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic codes, we construct two new families of q-ary perfect LRCs with length n=qm−1q−1, minimum distance d=5 and locality r=2.