Codes with Hierarchical Locality on Artin-Schreier Surfaces

TL;DR

This paper constructs hierarchical locality codes based on Artin-Schreier surfaces, providing explicit parameters, recovery algorithms, and improved bounds on minimum distance through geometric and elementary point-counting methods.

Contribution

It introduces a new class of codes with hierarchical locality derived from Artin-Schreier surfaces, including explicit constructions, parameters, and bounds, using elementary geometric techniques.

Findings

Constructed codes with hierarchical locality on Artin-Schreier surfaces.

Provided explicit parameters and recovery algorithms for these codes.

Achieved improved bounds on minimum distance using elementary point-counting methods.

Abstract

In this article, we construct codes with hierarchical locality using natural geometric structures in Artin-Schreier surfaces of the form $y^p-y=f(x,z)$. Our main theorem describes the codes, their hierarchical structure and recovery algorithms, and gives parameters. We also develop a family of examples using codes defined over $\mathbb{F}_{p^2}$ on the surface $y^p-y=x^{p+1}z^2+x^2z^{p+1}$. We use elementary methods to count the $\mathbb{F}_{p^2}$-rational points on the surface, enabling us to provide explicit hierarchical parameters and a better bound on minimum distance for these codes. An additional example and some generalizations are also considered.