  We generalize to any q a theorem about covering radius of linear codes proved
by Helleseth, Klove and Mykkelvit. Then we determine the covering radius of
first order generalized Reed-Muller codes in second order generalized
Reed-Muller codes. Using these results, we are able to give bounds for the
covering radius of first order generalized Reed-Muller codes. Finaly, using
Magma, we get some improvements for q=3.
