Codes from Jacobian surfaces

TL;DR

This paper investigates algebraic geometry codes derived from Jacobian surfaces of genus 2 curves, establishing bounds on their minimum distance based on rational point counts on related curves over finite fields.

Contribution

It introduces a new lower bound for the minimum distance of these codes using Weil-type bounds for rational points on curves on abelian surfaces.

Findings

Derived a lower bound for code minimum distance

Connected rational point counts to code parameters

Applicable to curves on possibly singular or non-absolutely irreducible surfaces

Abstract

This paper is concerned with some Algebraic Geometry codes on Jacobians of genus 2 curves. We derive a lower bound for the minimum distance of these codes from an upper "Weil type" bound for the number of rational points on irreducible (possibly singular or non-absolutely irreducible) curves lying on an abelian surface over a finite field.