Rank-metric codes over arbitrary fields: Bounds and constructions

2026-01-21

AI Summary

This study investigates the parameter bounds and constructions of rank-metric codes over arbitrary fields, including non-finite fields such as algebraically closed fields and the real numbers, with a focus on the tightness of Singleton-type bounds and the existence of maximum rank distance (MRD) codes. By integrating tools from algebraic coding theory, Galois theory, linear algebra, and subspace geometry, the work systematically analyzes the theoretical foundations and upper bounds for rank-metric codes over general fields, and develops novel MRD code constructions over fields admitting cyclic Galois extensions. Key contributions include uncovering the failure mechanism of the Singleton bound in non-finite settings, advancing explicit constructions of MRD codes over several classes of fields, establishing connections between rank-metric codes and algebraic geometry, topology, and measure theory, and proposing new conjectures and research directions regarding the existence of MRD codes.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web search models and ranking
• Security and Privacy: Large-scale security measurements

Abstract

Rank-metric codes, defined as sets of matrices over a finite field with the rank distance, have gained significant attention due to their applications in network coding and connections to diverse mathematical areas. Initially studied by Delsarte in 1978 and later rediscovered by Gabidulin, these codes have become a central topic in coding theory. This paper surveys the development and mathematical foundations, in particular, regarding bounds and constructions of rank-metric codes, emphasizing their extension beyond finite fields to more general settings. We examine Singleton-like bounds on code parameters, demonstrating their sharpness in finite field cases and contrasting this with contexts where the bounds are not tight. Furthermore, we discuss constructions of Maximum Rank Distance (MRD) codes over fields with cyclic Galois extensions and the relationship between linear rank-metric codes with systems and evasive subspaces. The paper also reviews results for algebraically closed fields and real numbers, previously appearing in the context of topology and measure theory. We conclude by proposing future research directions, including conjectures on MRD code existence and the exploration of rank-metric codes over various field extensions.

Problem

• rank-metric codes
• arbitrary fields
• Singleton bound
• MRD codes
• field extensions

Innovation

• rank-metric codes
• arbitrary fields
• Maximum Rank Distance (MRD)
• Singleton bound
• Galois extensions