Generalized Hamming weights of affine Cartesian codes

Let F be any field and A(1), . . . , A(m) be finite subsets of F. We determine the maximum number of common zeroes a linearly independent family of r polynomials of degree at most d of F[x(1), . . . ,x(m)] can have in A(1) x . . . x A(m). In the case when F is a finite field, our results resolve the problem of determining the generalized Hamming weights of affine Cartesian codes. This is a generalization of the work of Heijnen and Pellikaan where these were determined for the generalized Reed-Muller codes. Finally, we determine the duals of affine Cartesian codes and compute their generalized Hamming weights as well. (C) 2018 Elsevier Inc. All rights reserved.