# Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus

## TL;DR

This paper studies algebraic geometry codes from abelian surfaces over finite fields, providing bounds on their minimum distance and exploring conditions that improve these bounds by excluding low genus curves.

## Contribution

It introduces new bounds on code minimum distance based on the geometric properties of abelian surfaces, especially those lacking low genus curves.

## Key findings

- Established a general bound on the minimum distance of the codes.
- Showed that excluding low genus curves can sharpen the bounds.
- Explored Weil restrictions and non-principally polarized abelian surfaces.

## Abstract

We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain low genus curves. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization.