Abstract
This paper presents a new alphabet-dependent bound for codes with hierarchical locality. Then, the complete list of possible localities is derived for a class of codes obtained by deleting specific columns from a Simplex code. This list is used to show that these codes are optimal codes with hierarchical locality.
Highlights
In modern distributed storage systems (DSSs) failures happen frequently, whence decreasing the number of connections required for node repair is crucial
We extend the alphabet-dependent bound to H-Locally repairable codes (LRCs) with h-level hierarchical locality
The objective is to describe the locality parameters for δ > 2 and all dimensions. We show that these codes are locally repairable codes for every dimension implying a complete optimization of the number of nodes contacted for repair according to the number of failures
Summary
In modern distributed storage systems (DSSs) failures happen frequently, whence decreasing the number of connections required for node repair is crucial. A Singleton-type bound for codes with hierarchical locality was derived in [15] and constructions attaining the bound were provided in [2,15]. We describe the restrictions of dimension k − 1 and prove the locality for this dimension using combinatorial techniques These results allow us to derive the weight enumerator of the constructed codes. Our main contribution is the complete list of possible localities for these codes This shows that the constructed codes are alphabet-optimal H-LRCs. since a special case of this construction leads to the Reed–Muller codes RM(1, m), we obtain as a corollary to our result that the Reed–Muller codes RM(1, m) are H-LRCs and we derive their locality parameters
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