Hermitian codes and complete intersections

Abstract

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.