The second weight of generalized Reed-Muller codes in most cases

The second weight of the Generalized Reed-Muller code of order $d$ over the finite field with $q$ elements is now known for $d<q$ and $d>(n-1)(q-1)$. In this paper, we determine the second weight for the other values of $d$ which are not multiple of $q-1$ plus 1. For the special case $d=a(q-1)+1$ we give an estimate.