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arXiv · 2605.19691

The geometry of rank-metric codes

Gianira N. Alfarano·Martino Borello·Alessandro Neri
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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.

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Gianira N. Alfarano, Martino Borello, Alessandro Neri. 2026-05-19. The geometry of rank-metric codes. https://arxiv.org/abs/2605.19691

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