New Classes of Optimal Cyclic Codes With Hierarchical Locality

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 construct three classes of optimal cyclic H-LRCs. The minimum Hamming distance of the first class of optimal cyclic H-LRCs is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d=\delta _{1}=\delta +\mu n'+\nu +1$ </tex-math></inline-formula> while that of the second class of optimal cyclic H-LRCs is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d=\delta _{1}+1=\delta +\mu n'+\nu +2$ </tex-math></inline-formula> for some flexible integers <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta,\mu,\nu $ </tex-math></inline-formula>. The first two classes of optimal cyclic H-LRCs also have unbounded length. Although the length of the third class of optimal cyclic H-LRCs is not unbounded, it can reach large values. These classes of optimal cyclic H-LRCs by our constructions have new and flexible parameters compared with those in the literature.