Optimal and Almost Optimal Locally Repairable Codes from Hyperelliptic Curves
Junjie Huang, Chang-An Zhao
TL;DR
This work develops an automorphism-group–based framework to construct locally repairable codes from genus-2 hyperelliptic curves, extending prior elliptic-curve methods to achieve longer code lengths approaching $q+4\sqrt{q}$ and locality up to $239$. By exploiting fixed fields $F^{\mathcal{G}}$ that are rational and carefully chosen subspaces $V$ of Riemann–Roch spaces, the authors produce $q$-ary LRCs with parameters $[n, k, d]_q$ where $n=\ell(r+1)$ and $k=rt-(r-1)$, with $d$ bounded below by $n-\deg G$; many constructions yield either optimal or almost-optimal codes. The paper demonstrates numerous parameter sets, including cases where $n>q$ and even exceeds the $q+2\sqrt{q}$ bound, by leveraging maximal curves and explicit automorphism groups such as $\tilde S_4$, $\tilde S_5$, and $C_{10}$. These results significantly broaden the catalog of long, high-rate LRCs suitable for distributed storage, with practical implications for efficient local repair and repair bandwidth.
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 $239$.