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

2026-04-17

AI Summary

This study addresses the classification of minimal codewords in second-order projective Reed–Muller codes, a problem equivalent to characterizing maximal quadratic hypersurfaces over finite fields under inclusion of their sets of rational points. By analyzing inclusion relations among the rational point sets of absolutely irreducible quadratic hypersurfaces, the authors establish a one-to-one correspondence between such hypersurfaces and minimal codewords. The key technical contribution is the proof that, except for a single exceptional case over $\mathbb{F}_2$, if the rational point sets of two absolutely irreducible quadratic hypersurfaces are mutually inclusive, then the hypersurfaces must be identical. Leveraging this result, the paper provides a complete classification of all minimal codewords and precisely enumerates their number for each possible weight.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• minimal codewords
• projective Reed-Muller codes
• quadrics over finite fields
• maximal rational points
• absolutely irreducible quadrics

Innovation

• minimal codewords
• projective Reed-Muller codes
• quadrics over finite fields
• maximal rational point sets
• absolutely irreducible quadrics