Remarks on second and third weights of Projective Reed-Muller codes

TL;DR

This paper provides new proofs and calculations for the second and third weights of projective Reed-Muller codes, revealing geometric structures related to hypersurfaces and hyperplanes.

Contribution

It offers an alternative proof for the second weight of $ ext{PRM}(d,m)$ and computes the second weight for specific cases, also providing an upper bound for the third weight.

Findings

Second weight attained by hypersurfaces containing a hyperplane

Computed second weight for $ ext{PRM}(d,2)$ when $3 \\le d \\le q-1$

Provided an upper bound for the third weight of $ ext{PRM}(d,2)$

Abstract

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 $\PRM (d, m)$ 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 $\PRM (d, 2)$ for $3 \le d \le q-1$. Furthermore, we give an upper bound for the third weight of $\PRM(d, 2)$.