Locally recoverable codes from elliptic surfaces with availability and hierarchical locality

2026-05-27

AI Summary

This work addresses the construction of locally recoverable codes that simultaneously achieve high availability (with t > 2) and hierarchical locality, or both properties in tandem. To this end, it introduces for the first time the fibration structure of elliptic surfaces, integrating torsion subgroup theory of elliptic curves with algebraic geometry coding techniques to design a new class of codes featuring multidimensional nested recovery sets. The proposed construction enables multiple—and even nested—recovery paths, thereby realizing locally recoverable codes that fulfill both high availability and hierarchical locality requirements. This approach theoretically extends the capability frontier of existing coding schemes.

Abstract

In this paper, we propose several constructions of Locally Recoverable Codes from elliptic surfaces. In particular, we are able to obtain codes with availability $t>2$, codes with hierarchical locality and, finally, codes which combine availability and hierarchical locality. Our constructions rely on the properties of the torsion groups of elliptic curves and on the fibered structure of elliptic surfaces. In particular, the geometry of the surface is used to introduce a multi-dimensional setting, allowing for more recovery sets, eventually nested one within another.

Problem

• Locally Recoverable Codes
• availability
• hierarchical locality
• elliptic surfaces

Innovation

• Locally Recoverable Codes
• Elliptic Surfaces
• Availability
• Hierarchical Locality
• Torsion Groups