In this paper we introduce a new type of code, called projective nested cartesian code. It is obtained by the evaluation of homogeneous polynomials of a fixed degree on a certain subset of $mathbb{P}^n(mathbb{F}_q)$, and they may be seen as a general
ization of the so-called projective Reed-Muller codes. We calculate the length and the dimension of such codes, a lower bound for the minimum distance and the exact minimum distance in a special case (which includes the projective Reed-Muller codes). At the end we show some relations between the parameters of these codes and those of the affine cartesian codes.
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 a
nd R. Villareal and, in a generalized form, in a work by O. Geil and C. Thomsen. Using methods from Grobner 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.
