Galois subcovers of the Hermitian curve in characteristic $p$ with respect to subgroups of order $dp$ with $d\not=p$ prime

TL;DR

This paper derives explicit equations for Galois subcovers of the Hermitian curve with specific group orders, advancing the understanding of maximal curves and their applications in coding theory.

Contribution

It provides explicit equations for all Galois covers of the Hermitian curve with Galois group order $dp$, where $p$ is the characteristic and $d$ is a prime not equal to $p$, filling a gap in the existing literature.

Findings

Explicit equations for Galois covers with group order $dp$ are obtained.

Generators of the Weierstrass semigroup at certain points are computed.

Potential improvements in algebraic geometry code minimum distance bounds are discussed.

Abstract

A problem of current interest, also motivated by applications to Coding theory, is to find explicit equations for \textit{maximal} curves, that are projective, geometrically irreducible, non-singular curves defined over a finite field $\mathbb{F}_{q^2}$ whose number of $\mathbb{F}_{q^2}$-rational points attains the Hasse-Weil upper bound of $q^2+2\mathfrak{g}q+1$ where $\mathfrak{g}$ is the genus of the curve $\mathcal{X}$. For curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order and also when has order the square of the characteristic. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order $dp$ where $p$ is the characteristic of $\mathbb{F}_{q^2}$ and $d$ is prime other than $p$. We also compute the generators of the Weierstrass semigroup at a special $\mathbb{F}_{q^2}$-rational point of some of the curves, and discuss some possible positive impacts on the minimum distance problems of AG-codes.