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 article, we provide an alternative proof of the second weight of the projective Reed-Muller codes (d, m) where m ≥ 3 and 3 ≤ d ≤ q+3/2. We show that the second weight is attained by codewords that correspond to hypersurfaces containing a hyperplane under the hypothesis on d. Furthermore, we compute the second weight of (d, 2) for 3 ≤ d ≤ q-1. Furthermore, we give an upper bound for the third weight of (d, 2).