Constructions of Optimal Cyclic Codes With h-Level Hierarchical Locality

Abstract

In order to correct different numbers of erasures in distributed storage systems, the design of locally repairable codes with hierarchical locality (H-LRCs) is crucial. In this paper, we study h-level H-LRCs where h is not limited to 2. We present six classes of cyclic h-level H-LRCs which are optimal with respect to the generalized Singleton-like bound. The first five classes of cyclic h-level H-LRCs have length $ln_{1}$ such that $\gcd (l,q)=1$ , and $n_{1}|(q-1)$ or $n_{1}|(q+1)$ . The last class of cyclic h-level H-LRCs has length $n|(q+1)$ . The minimum Hamming distances of them are $d=\delta _{1}+i$ where i can take 0, 2, $\delta _{h}$ , and so on. Furthermore, these six classes of cyclic h-level H-LRCs have new parameters which are not covered in the literature.