Maximal quadrics over finite fields and minimal codewords of projective Reed-Muller codes
This provides a complete classification of minimal codewords for a specific family of codes, which is an incremental advance in coding theory.
The authors classify minimal codewords of projective Reed-Muller codes of order 2 by characterizing quadrics over finite fields whose rational points are maximal under inclusion. They prove that, except for one case over F2, any two absolutely irreducible quadrics with nested point sets are equal, yielding a precise characterization and exact counts for each weight.
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.