Structure of weighted projective Reed-Muller codes

2026-03-25

AI Summary

This work investigates the structure of weighted projective Reed–Muller codes and presents systematic constructions of their duals and subfield subcodes. Under specific weight conditions, a recursive construction framework is introduced, enabling the first comprehensive characterization of the dual code structure and yielding an upper bound on generalized Hamming weights. In the low-degree case, the dual codes are explicitly realized as evaluation codes, revealing a non-degenerate Schur product structure. The main contributions include an efficient recursive construction for these codes, explicit methods for building their duals and subfield subcodes, and an evaluation-code representation for low-degree duals, thereby enriching the application of algebraic coding theory in weighted projective spaces.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Applications of cryptography

Abstract

We provide a comprehensive overview of the fundamental structural properties of weighted projective Reed-Muller codes. We give a recursive construction for these codes, under some conditions for the weights, and we use it to derive bounds on the generalized Hamming weights and to obtain a recursive construction for their subfield subcodes and their dual codes. The dual codes are further studied in more generality, where the recursive constructions may not apply, obtaining a description as an evaluation code when the degree is low. We also provide insights into the Schur products of these codes when they are not degenerate.

Problem

• weighted projective Reed-Muller codes
• generalized Hamming weights
• dual codes
• subfield subcodes
• Schur products

Innovation

• weighted projective Reed-Muller codes
• recursive construction
• generalized Hamming weights
• dual codes
• Schur products