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

TL;DR

This paper classifies minimal codewords of projective Reed-Muller codes of order 2 by characterizing quadrics over finite fields with maximal rational points, providing a detailed enumeration of these codewords.

Contribution

It offers a complete classification of minimal codewords in these codes and characterizes quadrics with maximal rational points, filling a gap in coding theory.

Findings

Except one case over , absolutely irreducible quadrics with nested rational points are equal.

Provides a precise characterization of minimal codewords.

Gives exact counts of minimal codewords for each weight.

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.