  We give a recursive construction for projective Reed-Muller codes in terms of
affine Reed-Muller codes and projective Reed-Muller codes in fewer variables.
From this construction, we obtain the dimension of the subfield subcodes of
projective Reed-Muller codes for some particular degrees that give codes with
good parameters. Moreover, from this recursive construction we derive a lower
bound for the generalized Hamming weights of projective Reed-Muller codes which
is sharp in most of the cases we have checked.
