Footprints or generalized Bezout's theorem

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