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 $\text{PRM}(d,m)$ where $m\ge 3$ and $3\le d\le \frac{q+3}{2}$. 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 $\text{PRM}(d,2)$ for $3\le d\le q-1$. Furthermore, we give an upper bound for the third weight of $\text{PRM}(d,2)$.