The geometry of rank-metric codes

2026-05-19

AI Summary

This work investigates the structure and metric properties of rank-metric codes from a geometric perspective. By introducing generator tensors and their slice spaces, the authors develop a unified geometric framework that translates the metric characteristics of codes into geometric conditions concerning hyperplane intersections, establishing a correspondence with equivalence classes of associated systems. They further propose a generalized notion of weights based on evasive systems, revealing novel connections between rank-metric codes and additive Hamming-metric codes, and enabling a unified derivation of several classical bounds and semifield-theoretic results. This framework effectively links rank distributions to the underlying geometric properties of systems, not only recovering known results—such as those for MRD and constant-rank codes—but also extending them, thereby demonstrating its efficacy in preserving and characterizing the intrinsic metric structure.

Abstract

In this paper, we develop a geometric framework for matrix rank-metric codes based on generator tensors and their slice spaces. To every nondegenerate matrix rank-metric code, we associate two systems, which translate metric properties of the code into geometric conditions involving intersections with hyperplanes. This leads to a correspondence between equivalence classes of nondegenerate matrix rank-metric codes and equivalence classes of systems, as well as to Delsarte-type incidence identities relating the rank distribution of a code over a finite field to those of its associated systems. As an application, we introduce generalized weights through the notion of evasive systems, study faithful and one-weight codes over finite fields, and recover known bounds and results from the theory of semifields. Finally, we use this framework to associate additive Hamming-metric codes with matrix rank-metric codes and show that several metric properties are preserved under this correspondence.

Problem

• rank-metric codes
• geometry
• generator tensors
• evasive systems
• semifields

Innovation

• rank-metric codes
• generator tensors
• slice spaces
• evasive systems
• Delsarte-type identities