The geometry of rank-metric codes

TL;DR

This paper introduces a geometric framework for matrix rank-metric codes using generator tensors, establishing new correspondences and identities that connect metric properties with geometric conditions.

Contribution

It develops a novel geometric approach to analyze rank-metric codes, linking their properties to systems of hyperplanes and introducing generalized weights and code correspondences.

Findings

Established a geometric correspondence between codes and systems.

Derived Delsarte-type incidence identities relating rank distributions.

Connected rank-metric codes with additive Hamming-metric codes.

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.