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arXiv · 2406.07339

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

Mrinmoy Datta
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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 $\PRM(d, 2)$.

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Mrinmoy Datta. 2025-02-19. Remarks on second and third weights of Projective Reed-Muller codes. https://arxiv.org/abs/2406.07339

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