Generalized minimum distance functions

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 X is a set of projective points over a finite field and I(X) is its vanishing ideal, we show that the gmd function and the Vasconcelos function of I(X) are equal to the r-th generalized Hamming weight of the corresponding Reed-Muller-type code CX(d). We show that the r-th generalized footprint function of I(X) is a lower bound for the r-th generalized Hamming weight of CX(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.