Remarks on second and third weights of projective Reed–Muller codes

Determining the weight distributions of the projective Reed–Muller codes is a very hard problem and has been studied extensively in the literature. In this paper, we provide an alternative proof of the second weight of the projective Reed–Muller codes PRM[Formula: see text] (d, m) where [Formula: see text] and [Formula: see text]. We show that the second weight is attained by codewords that correspond to hypersurfaces containing a hyperplane under the hypothesis on [Formula: see text]. Furthermore, we compute the second weight of PRM[Formula: see text](d, 2) for [Formula: see text]. Furthermore, we give an upper bound for the third weight of PRM[Formula: see text](d, 2).