  Explicit bases for the subfield subcodes of projective Reed-Muller codes over
the projective plane and their duals are obtained. In particular, we provide a
formula for the dimension of these codes. For the general case over the
projective space, we generalize the necessary tools to deal with this case as
well: we obtain a universal Gr\"obner basis for the vanishing ideal of the set
of standard representatives of the projective space and we show how to reduce
any monomial with respect to this Gr\"obner basis. With respect to the
parameters of these codes, by considering subfield subcodes of projective
Reed-Muller codes we obtain long linear codes with good parameters over a small
finite field.
