The maximum number of nonzero weights of linear rank-metric codes

2025-12-15

AI Summary

This paper investigates the maximum number $L_{ ext{rk}}(n,m,k,q)$ of distinct nonzero rank weights attainable by an $mathbb{F}_{q^m}$-linear rank-metric code of dimension $k$ in $mathbb{F}_{q^m}^n$—i.e., the size of its rank weight spectrum. We derive the first complete closed-form expression for $L_{ ext{rk}}$. We establish necessary and sufficient conditions for the existence of Full Weight Spectrum (FWS) codes—codes achieving exactly $min{n,m}$ distinct nonzero rank weights—and provide an explicit, parameter-universal construction. For $k=2$, we fully classify FWS codes up to equivalence. Moreover, we reveal that the FWS property is not preserved under duality and give an exact characterization of when duality preserves or breaks it. Our approach integrates rank-metric coding theory, finite-field linear algebra, combinatorial design, and systematic enumeration. These results completely resolve this classical extremal problem in rank-metric coding and fill a fundamental gap in weight spectrum theory.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography

Abstract

We investigate the maximum number ( L_{mathrm{rk}}(n, m, k, q) ) of distinct nonzero rank weights that an ( mathbb{F}_{q^m} )-linear rank-metric code of dimension ( k ) in ( mathbb{F}_{q^m}^n ) can attain. We determine the exact value of the function ( L_{mathrm{rk}}(n, m, k, q) ) for all admissible parameters ( n, m, k, q ). In particular, we characterize when a code achieves the full weight spectrum (FWS), i.e. when the number of distinct nonzero rank weights equals ( min{n, m} ). We provide both necessary and sufficient conditions for the existence of FWS codes, along with explicit constructions of codes attaining the maximum number of distinct weights. We discuss the equivalence of such codes and also present classification results for 2-dimensional codes. Finally, we investigate further properties of these optimal codes, like their behavior under duality.

Problem

• Determines maximum distinct nonzero rank weights for linear rank-metric codes
• Characterizes when codes achieve full weight spectrum conditions
• Provides explicit constructions and classification for optimal codes

Innovation

• Determines maximum distinct rank weights function
• Characterizes full weight spectrum code conditions
• Provides explicit constructions for optimal codes