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We study affine cartesian codes, which are a Reed-Muller type of evaluationcodes, where polynomials are evaluated at the cartesian product of n subsets ofa 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 thesecond Hamming weight (also called next-to-minimal weight) for particular casesof affine cartesian codes and also some higher Hamming weights of this type ofcode.

On the second Hamming weight of some Reed-Muller type codes