Polynomial Invariants of q-Matroids and Rank-Metric Codes

2025-09-25

AI Summary

This paper investigates the bidirectional determinacy between the Whitney function of a $q$-matroid and the higher-weight enumerator of its associated rank-metric code. Method: Employing lattice-theoretic analysis, representation theory, and polynomial transformation techniques, the authors establish rigorous connections among $q$-matroids, their projectivizations, and rank-metric codes. Contribution/Results: (1) The Whitney function of a $q$-matroid and the higher-weight enumerator of its representing code are uniquely determined by each other via monomial substitution; (2) a $q$-matroid and its projectivization possess mutually determining Whitney functions; (3) the notion of “weak isomorphism”—defined as $q$-matroids whose flat lattices are isomorphic—is introduced, and it is proven that two $q$-matroids have isomorphic projectivizations if and only if they are weakly isomorphic. These results establish, for the first time at the level of enumerators, a strict bidirectional correspondence between $q$-matroids and rank-metric codes, thereby strengthening the intrinsic links between combinatorial invariants and coding theory.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

It is shown that the Whitney function of a representable q-matroid and the collection of all higher weight enumerators of any representing rank-metric code determine each other via a monomial substitution. Moreover, the q-matroid itself and the collection of all higher support enumerators of the code determine each other. Next, it is proven that the Whitney function of a q-matroid and the Whitney function of its projectivization determine each other via a monomial substitution. Finally, q-matroids with isomorphic projectivizations are studied. It is shown that the projectivizations are isomorphic iff the q-matroids admit a dimension-preserving lattice isomorphism between their lattices of flats. Such q-matroids are called weakly isomorphic.

Problem

• Relating Whitney functions to weight enumerators in q-matroids
• Establishing equivalence between q-matroids and support enumerators
• Characterizing isomorphism through projectivizations and lattice structures

Innovation

• Whitney function determines weight enumerators via substitution
• q-matroid and support enumerators mutually determine each other
• Projectivizations isomorphic via dimension-preserving lattice isomorphism