A new proof of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords of generalized Reed–Muller codes

We give a new proof of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords of generalized Reed–Muller codes published in 1970. To prove this theorem, we consider the intersection of the support of minimal weight codewords with affine hyperplanes and we proceed by recursion.