  We study affine cartesian codes, which are a Reed-Muller type of evaluation
codes, where polynomials are evaluated at the cartesian product of n subsets of
a finite field F_q. These codes appeared recently in a work by H. Lopez, C.
Renteria-Marquez and R. Villareal and, in a generalized form, in a work by O.
Geil and C. Thomsen. Using methods from Gr\"obner basis theory we determine the
second Hamming weight (also called next-to-minimal weight) for particular cases
of affine cartesian codes and also some higher Hamming weights of this type of
code.
