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

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 been widely investigated. Optimal via cyclic constacyclic provide significant benefits elegant algebraic structure efficient encoding procedure. In this paper, we continue to consider the long code lengths. Specifically, first obtain two classes q-ary Singleton-optimal (n,k,d=6;r=2)-LRCs length n=3(q+1) when 3|(q−1) q is even, n=32(q+1) q≡1(mod4), respectively. To best our knowledge, construction n>q+1 minimum distance d≥5. On other hand, LRC achieving Hamming-type bound called a perfect LRC. By using codes, construct new families n=qm−1q−1, d=5 locality r=2.