New Optimal Linear Codes With Hierarchical Locality

Abstract

Locally repairable codes with hierarchical locality (H-LRCs) are designed to correct different numbers of erasures, which play a crucial role in large-scale distributed storage systems. In this paper, we construct three classes of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary optimal H-LRCs by employing matrix product codes, concatenated codes and cyclic codes, respectively. The first two constructions are based on the idea of constructing new codes from old, which produces several new classes of optimal H-LRCs whose lengths can reach up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q^{2}+q$ </tex-math></inline-formula> or unbounded. The final construction generates a class of new optimal cyclic H-LRCs whose lengths divide <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q-1$ </tex-math></inline-formula> . Compared with the previously known ones, our constructions are new in the sense that their parameters are not covered by the codes available in the literature.