A problem of current interest, also motivated by applications to Coding theory, is to find explicit equations for 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 \(q^2+2\mathfrak {g}q+1\) 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 a 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.

当前感兴趣的一个问题，也是由编码理论的应用所激发的，是找到最大曲线的显式方程，这些方程是在有限域 \（\mathbb {F}_{q^2}\） 上定义的投影、几何上不可约的非奇异曲线，其 \（\mathbb {F}_{q^2}\） 有理点的数量达到 Hasse-Weil 上限 \（q^2+2\mathfrak {g}q+1\），其中 \（\mathfrak {g}\） 是曲线 \（\mathcal {X}\） 的属。对于伽罗瓦覆盖在埃尔米特曲线中的曲线，到目前为止，这已经临时完成，特别是在伽罗瓦群具有素数阶以及有序特征的平方的情况下。在本文中，我们获得了厄米特曲线的所有伽罗瓦覆盖的显式方程，其中伽罗瓦群为 dp，其中 p 是 \（\mathbb {F}_{q^2}\） 的特征，d 是 p 以外的素数。我们还在一些曲线的特殊 \（\mathbb {F}_{q^2}\） 有理点处计算了 Weierstrass 半群的生成器，并讨论了对 AG 码最小距离问题的一些可能的积极影响。