New Constructions of Optimal Cyclic (r,\delta) Locally Repairable Codes from Their Zeros

TL;DR

This paper introduces new methods to construct optimal cyclic (r, δ)-locally repairable codes using their zeros, expanding the range of parameters and simplifying the construction process for distributed storage systems.

Contribution

It characterizes (r, δ)-locality of cyclic codes via zeros and provides flexible, comprehensive constructions of optimal codes for lengths dividing (q-1) or (q+1).

Findings

Constructs new classes of optimal cyclic (r, δ)-LRCs.

Includes all previously proposed optimal codes as special cases.

Allows for codes with parameters where (r+δ-1) does not divide n.

Abstract

An $(r, \delta)$-locally repairable code ($(r, \delta)$-LRC for short) was introduced by Prakash et al. \cite{Prakash2012} for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$-LRCs produced by Gopalan et al. \cite{Gopalan2012}. An $(r, \delta)$-LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. \cite{Chen2018} generalized the construction of cyclic $r$-LRCs proposed by Tamo et al. \cite{Tamo2015,Tamo2016} and constructed several classes of optimal $(r, \delta)$-LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$, respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of \cite{Chen2018,Chen2019}, this paper first characterizes $(r, \delta)$-locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic $(r, \delta)$-LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$, respectively from the product of two sets of zeros. Our constructions include all optimal cyclic $(r,\delta)$-LRCs proposed in \cite{Chen2018,Chen2019}, and our method seems more convenient to obtain optimal cyclic $(r, \delta)$-LRCs with flexible parameters. Moreover, many optimal cyclic $(r,\delta)$-LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$, respectively such that $(r+\delta-1)\nmid n$ can be obtained from our method.