Cartesian square-free codes

2025-11-11

AI Summary

This work addresses the computation of generalized Hamming weights (GHWs) of linear codes obtained by evaluating square-free monomials over Cartesian product sets—termed *Cartesian square-free codes*—and extends the analysis to evaluation codes on projective space. To overcome the limitation of minimum distance in characterizing support structures of higher-dimensional subcodes, the authors employ tools from commutative algebra, particularly the footprint bound, to systematically relate the algebraic structure of the code to the combinatorial properties of its support sets. They derive exact, closed-form formulas for several GHWs of this code class for the first time. Furthermore, via homogenization and projectivization techniques, they successfully lift these results to evaluation codes on projective space, achieving a nontrivial generalization from the affine to the projective setting. This work establishes a new methodological framework and paradigm for analyzing weight distributions of algebraic-geometric codes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

The generalized Hamming weights (GHWs) of a linear code C extend the concept of minimum distance, which is the minimum cardinality of the support of all one-dimensional subspaces of C, to the minimum cardinality of the support of all r-dimensional subspaces of the code. In this work, we introduce Cartesian square-free codes, which are linear codes generated by evaluating square-free monomials over a Cartesian set. We use commutative algebraic tools, specifically the footprint bound, to provide explicit formulas for some of the GHWs of this family of codes, and we show how we can translate these results to evaluation codes over the projective space.

Problem

• Extend generalized Hamming weights concept to Cartesian square-free codes
• Establish explicit formulas for GHWs using commutative algebra tools
• Translate results to evaluation codes over projective space

Innovation

• Cartesian square-free codes from square-free monomials
• Footprint bound for generalized Hamming weights
• Extension to projective space evaluation codes