A new family of maximal curves

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 ${\mathcal{X}}_n$.