We study the $r$-th generalized minimum distance function (gmd function for short) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. If $\mathbb{X}$ is a set of projective points over a finite field and $I(\mathbb{X})$ is its vanishing ideal, we show that the gmd function and the Vasconcelos function of $I(\mathbb{X})$ are equal to the $r$-th generalized Hamming weight of the corresponding Reed-Muller-type code $C_\mathbb{X}(d)$. We show that the $r$-th generalized footprint function of $I(\mathbb{X})$ is a lower bound for the $r$-th generalized Hamming weight of $C_\mathbb{X}(d)$. As an application to coding theory we show an explicit formula and a combinatorial formula for the second generalized Hamming weight of an affine cartesian code. 查看全文>>