Bases of Minimum-Weight Vectors for Codes from Designs

An explicit basis of incidence vectors for thep-ary code of the design of points and hyperplanes of the affine geometryAGm(Fp) for any primepand any integerm≥2 is obtained, which, as a corollary, gives a new elementary proof that this code is a generalized Reed–Muller code. In the proof a class of nonsingular matrices related to Vandermonde matrices is introduced.