Optimal and Almost Optimal Locally Repairable Codes from Hyperelliptic Curves

TL;DR

This paper constructs new families of locally repairable codes using hyperelliptic curves, achieving lengths close to theoretical limits and high locality, thus advancing distributed storage solutions.

Contribution

It generalizes previous elliptic curve methods to hyperelliptic curves of genus 2, creating codes with longer lengths and higher locality than prior work.

Findings

Codes with length approaching q+4√q

Locally repairable codes with locality up to 239

Construction of multiple new code families

Abstract

Locally repairable codes are widely applicable in contemporary large-scale distributed cloud storage systems and various other areas. By making use of some algebraic structures of elliptic curves, Li et al. developed a series of $q$-ary optimal locally repairable codes with lengths that can extend to $q+2\sqrt{q}$. In this paper, we generalize their methods to hyperelliptic curves of genus $2$, resulting in the construction of several new families of $q$-ary optimal or almost optimal locally repairable codes. Our codes feature lengths that can approach $q+4\sqrt{q}$, and the locality can reach up to $239$.