Explicit Maximal and Minimal Curves of Artin-Schreier Type from Quadratic Forms

TL;DR

This paper constructs explicit examples of maximal and minimal Artin-Schreier curves over finite fields using quadratic forms, advancing understanding of algebraic curves in finite field theory.

Contribution

It introduces a new method to explicitly construct maximal and minimal Artin-Schreier curves via quadratic forms over finite fields.

Findings

Explicit examples of maximal curves over finite fields.

Explicit examples of minimal curves over finite fields.

Construction method based on quadratic forms.

Abstract

In this work we present explicit examples of maximal and minimal curves over finite fields in odd characteristic. The curves are of Artin-Schreier type and the construction is closely related to quadratic forms from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$.