Maximal quadrics over finite fields and minimal codewords of projective Reed-Muller codes

Alain Couvreur; Rati Ludhani

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arXiv · 2604.16676

Maximal quadrics over finite fields and minimal codewords of projective Reed-Muller codes

Alain Couvreur·Rati Ludhani

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Abstract

We study the classification of minimal codewords of projective Reed-Muller codes of order $2$. This problem is equivalent to identifying quadrics over finite fields whose set of rational points is maximal with respect to the inclusion. We prove that except one particular case over $\mathbb{F}_2$, any two absolutely irreducible quadrics whose sets of rational points are contained within one another should be equal as projective varieties. We deduce a precise characterisation of the minimal codewords of projective Reed-Muller codes of order $2$ and further give their exact number for each possible weight.

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Alain Couvreur, Rati Ludhani. 2026-04-17. Maximal quadrics over finite fields and minimal codewords of projective Reed-Muller codes. https://arxiv.org/abs/2604.16676

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