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.

翻译：我们研究了二阶射影Reed-Muller码中极小码字的分类问题。该问题等价于在有限域上识别其有理点集在包含关系下为极大的二次曲面。我们证明，除在$\mathbb{F}_2$上的一种特殊情况外，任意两个绝对不可约二次曲面，若其有理点集相互包含，则作为射影簇必然相等。由此我们推导出二阶射影Reed-Muller码极小码字的精确刻画，并进一步给出了每个可能权重下极小码字的确切数目。