  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 generalization 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.
