Remarks on second and third weights of Projective Reed-Muller codes
Mrinmoy Datta
TL;DR
This work advances the understanding of weight distributions for projective Reed-Muller codes by providing an alternative proof of the second weight for ${PRM(d,m)}$ when ${m\ge 3}$ and ${3\le d\le \frac{q+3}{2}}$, showing the second weight is realized by hypersurfaces containing a hyperplane. It also determines the second weight for ${PRM(d,2)}$ with ${3\le d\le q-1}$ and yields an upper bound for the third weight in this case. Beyond coding-theoretic weights, the paper surveys foundational bounds on the number of ${\mathbb F}_q$-points on affine and projective hypersurfaces (Ore, Serre, Homma–Kim), and uses these results to derive structural results and classifications for plane curves attaining extremal point counts. In particular, it provides a complete description of second-highest point counts for plane curves and sharp bounds for the third-highest counts, including explicit constructions in low-degree cases. Collectively, the results connect algebraic geometry over finite fields with coding theory, offering new proofs, sharper bounds, and clearer geometric interpretations of extremal codewords.
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 $\PRM(d, 2)$.