On the Reed-Muller codes

AbstractWe give a brief but complete account of all the essential facts concerning the Reed-Muller and punctured Reed-Muller codes. The treatment is new and includes an easy, direct proof of the fact that the punctured Reed-Muller codes are the codes of the projective geometries over the binary field. We also establish the existence of two short exact sequences that lead to new proofs that the minimum-weight vectors of the Reed-Muller and punctured Reed–Muller codes are the incidence vectors of the appropriate geometric objects