A distance-free approach to generalized weights

2025-12-19

AI Summary

Existing definitions of generalized weights for linear codes depend on specific distance metrics (e.g., Hamming or rank metric), limiting their unification and applicability across diverse metrics. Method: This paper introduces the first distance-agnostic unified framework by replacing conventional support sets or dual codes with a “family of test subspaces”, defining generalized weights via the dimension of intersections between these subspaces and the code. Contributions: We establish fundamental properties—including weak monotonicity, strict monotonicity along subsequences, and a Wei-type duality theorem—and demonstrate that classical generalized weight properties under both Hamming and rank metrics emerge as natural corollaries. The framework extends duality theorems to sum-rank metric codes with non-zero Hamming components and uncovers novel structural characteristics of intersection weights for MDS and MRD codes. Synthesizing abstract linear algebra, lattice theory, and duality analysis, this work provides a foundational toolset for multi-metric coding theory.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We propose a unified theory of generalized weights for linear codes endowed with an arbitrary distance. Instead of relying on supports or anticodes, the weights of a code are defined via the intersections of the code with a chosen family of spaces, which we call a test family. The choice of test family determines the properties of the corresponding generalized weights and the characteristics of the code that they capture. In this general framework, we prove that generalized weights are weakly increasing and that certain subsequences are strictly increasing. We also prove a duality result reminiscent of Wei's Duality Theorem. The corresponding properties of generalized Hamming and rank-metric weights follow from our general results by selecting optimal anticodes as a test family. For sum-rank metric codes, we propose a test family that results in generalized weights that are closely connected to -- but not always the same as -- the usual generalized weights. This choice allows us to extend the known duality results for generalized sum-rank weights to some sum-rank-metric codes with a nonzero Hamming component. Finally, we explore a family of generalized weights obtained by intersecting the underlying code with MDS or MRD codes.

Problem

• Develops a unified theory for generalized weights in linear codes with arbitrary distance
• Defines weights via code intersections with chosen test families, capturing specific code characteristics
• Extends duality results to sum-rank metric codes with nonzero Hamming components

Innovation

• Defining weights via intersections with test families
• Proving weakly and strictly increasing weight properties
• Extending duality results to sum-rank metric codes