Codes with Hierarchical Locality on Artin-Schreier Surfaces

In this article, we construct codes with hierarchical locality using natural geometric structures in Artin-Schreier surfaces of the form yᵖ-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 Fp² on the surface yᵖ-y=xᵖ⁺¹z²+x²zᵖ⁺¹. We count the Fp²-rational points on the surface, a topic of more general number theoretic interest, and provide more explicit parameters a better bound on minimum distance for these codes. An additional example and some generalizations are also considered.