Hierarchical Locally Recoverable Codes on surfaces

TL;DR

This paper constructs hierarchical locally recoverable codes from algebraic surfaces with fibrations, using geometric and arithmetic properties to determine code parameters and count rational points.

Contribution

It introduces a novel method of creating hierarchical locally recoverable codes from algebraic surfaces with specific fibrations, linking geometry with coding theory.

Findings

Codes with explicit parameters derived from surfaces

Estimates and counts of rational points on certain surfaces

New connections between algebraic geometry and coding theory

Abstract

We construct locally recoverable codes with hierarchy from surfaces in $\mathbb{A}^3$ admitting a fibration by curves of Artin-Schreier or Kummer type. We derive the parameters of our codes by leveraging the geometry and arithmetic of the fibration, which is obtained by projection onto one of the coordinates. As a byproduct, we obtain estimates for (and in one case an explicit count of) the number of rational points in certain families of surfaces.