Delsarte duality on subspaces and applications to rank-metric codes and q-matroids

2025-09-29

AI Summary

This study addresses the structural analysis of subspace lattices over finite fields and the geometric characterization of rank-metric codes. Methodologically, it introduces subspace weight and deficiency sequences to construct a deficiency-based duality framework, establishing for the first time an explicit correspondence between positively deficient subspaces and their Delsarte duals. It unifies diverse subspace structures from finite geometry—including MRD, near-MRD, quasi-MRD, and the newly introduced $(n,k)$-MRD codes—within this framework, demonstrating their closure under duality. Furthermore, the work advances $q$-matroid theory by characterizing the $mathbb{F}_{q^m}$-representability of direct sums of uniform $q$-matroids and deriving an explicit formula for their rank-generating functions. The results provide a novel geometric perspective on rank-metric coding and extend the interdisciplinary paradigm linking subspace lattices, finite geometry, and $q$-matroid theory.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We study the interplay between the lattice of F_{q^m}-subspaces and the lattice of F_{q^m}-subspaces of an F_{q^m}-vector space. Introducing notions of weight and defect relative to an F_q-subspace, we analyze the sequence of maximum non-zero defects. We establish a correspondence between subspaces of positive defect and their Delsarte duals, enabling explicit characterizations of the associated sequences of maximum non-zero defects. Our framework unifies several classes of subspaces studied in finite geometry and connects them to linear rank-metric codes by providing a new geometric interpretation of code duality. Building on these results, we characterize classes of rank-metric codes closed under duality, including MRD, near MRD, quasi-MRD, and a new family of (n, k)-MRD codes. Finally, we explore applications to q-matroids, by studying the problem of F_{q^m}-representability for direct sums of uniform q-matroids and describing their rank generating functions.

Problem

• Analyzing Delsarte duality relationships between subspace lattices in finite geometry
• Providing geometric interpretations for duality in linear rank-metric codes
• Studying F_{q^m}-representability of q-matroid direct sums and rank functions

Innovation

• Delsarte duality links subspaces and their duals
• Geometric interpretation connects rank-metric codes duality
• Framework characterizes duality-closed codes and q-matroids