  We study the footprint function, with respect to a monomial order, of
complete intersection graded ideals in a polynomial ring with coefficients in a
field. For graded ideals of dimension one, whose initial ideal is a complete
intersection, we give a formula for the footprint function and a sharp lower
bound for the corresponding minimum distance function. This allows us to
recover a formula for the minimum distance of an affine cartesian code and the
fact that in this case the minimum distance and the footprint functions
coincide. Then we present an extension of a result of Alon and F\"uredi, about
coverings of the cube $\{0,1\}^n$ by affine hyperplanes, in terms of the
regularity of a vanishing ideal.
