Footprints or generalized Bezout's theorem

In two recent papers the first by Feng, Rao, Berg, and Zhu and the second by Feng, Zhu, Shi, and Rao, the authors use a generalization of Bezout's theorem to estimate the minimum distance and generalized Hamming weights for a class of error-correcting codes obtained by evaluation of polynomials in points of an algebraic curve. The main aim of this note is to show that instead of using this rather complex method the same results and some improvements can be obtained by using the so-called footprint from Grobner basis theory. We also develop the theory further such that the minimum distance and the generalized Hamming weights not only can be estimated but also can actually be determined.