Codes with Hierarchical Locality on Artin-Schreier Surfaces

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 $y^p-y=x^{p+1}{z}^2+x^2{z}^{p+1}$. 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.