A new family of maximal curves

TL;DR

This paper introduces a new family of maximal curves over finite fields, generalizing known curves, and provides detailed automorphism group calculations and genus spectrum analysis.

Contribution

It constructs a novel family of maximal curves for any prime power q and odd n ≥ 5, expanding the understanding of maximal curve structures.

Findings

The curves have exactly q(q^2-1)(q^n+1) automorphisms.

Unless q=2, the curves are not Galois subcovers of the Hermitian curve.

New values of the genus spectrum for maximal curves are identified.

Abstract

In this article we construct for any prime power $q$ and odd $n \ge 5$, a new $\mathbb{F}_{q^{2n}}$-maximal curve $\mathcal X_n$. Like the Garcia--G\" uneri--Stichtenoth maximal curves, our curves generalize the Giulietti--Korchm\'aros maximal curve, though in a different way. We compute the full automorphism group of $\mathcal X_n$, yielding that it has precisely $q(q^2-1)(q^n+1)$ automorphisms. Further, we show that unless $q=2$, the curve $\mathcal{X}_n$ is not a Galois subcover of the Hermitian curve. Finally, we find new values of the genus spectrum of $\mathbb{F}_{q^{2n}}$-maximal curves, by considering some Galois subcovers of $\mathcal X_n$.