In this paper we consider the Hermitian codes defined as the dual codes of one-point evaluation codes on the Hermitian curve $\mathcal{H}$ over the finite field ${\mathbb{F}}_{q^2}$. We focus on those with distance $d\ge q^2-q$ and give a geometric description of the support of their minimum-weight codewords. We consider the unique writing $\mu q+\lambda (q+1)$ of the distance d with $\mu ,\lambda$ non negative integers, and $\mu \le q$, and consider all the curves $\mathcal{X}$ of the affine plane ${\mathbb{A}}_{{\mathbb{F}}_{q^2}}^2$ of degree $\mu +\lambda$ defined by polynomials with $x^{\mu }{y}^{\lambda }$ as leading monomial with respect to the DegRevLex term ordering (with $y>x$). We prove that a zero-dimensional subscheme Z of ${\mathbb{A}}_{{\mathbb{F}}_{q^2}}^2$ is the support of a minimum-weight codeword of the Hermitian code with distance d if and only if it is made of d simple ${\mathbb{F}}_{q^2}$-points and there is a curve $\mathcal{X}$ such that Z coincides with the scheme theoretic intersection $\mathcal{H}\cap \mathcal{X}$ (namely, as a cycle, $Z=\mathcal{H}\cdot \mathcal{X}$). Finally, exploiting this geometric characterization, we propose an algorithm to compute the number of minimum weight codewords and we present comparison tables between our algorithm and MAGMA command MinimumWords.

在本文中，我们将Hermitian码定义为Hermitian曲线上的单点评估码的对偶码 $\mathcal{H}$ 在有限域上 ${\mathbb{F}}_{q^2}$。我们专注于有距离的人$d\ge q^2-q$并给出其最小权重码字支持的几何描述。我们认为独特的写作$\mu q+\lambda （q+\mathrm{1个}）$的距离d与$\mu ，\lambda$ 非负整数，以及 $\mu \le q$，并考虑所有曲线 $\mathcal{X}$ 仿射平面 ${\mathbb{一种}}_{{\mathbb{F}}_{q^2}}^2$ 度 $\mu +\lambda$ 由多项式定义 $X^{\mu }{ÿ}^{\lambda }$作为DegRevLex术语排序方面的领先单项式（带有$ÿ>X$）。我们证明了零维subscheme ž的${\mathbb{一种}}_{{\mathbb{F}}_{q^2}}^2$是一个最小的码字重量随距离的厄密码的支持d当且仅当它是由d简单${\mathbb{F}}_{q^2}$点，有一条曲线 $\mathcal{X}$使得Z与方案理论交点重合$\mathcal{H}\cap \mathcal{X}$ （即，作为一个周期， $ž=\mathcal{H}\cdot \mathcal{X}$）。最后，利用这种几何特征，我们提出了一种算法来计算最小权重码字的数量，并给出了算法与MAGMA命令MinimumWords之间的比较表。