The geometry of rank-metric codes
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910236285075456 |
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| author | Alfarano, Gianira N. Borello, Martino Neri, Alessandro |
| author_facet | Alfarano, Gianira N. Borello, Martino Neri, Alessandro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19691 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The geometry of rank-metric codes Alfarano, Gianira N. Borello, Martino Neri, Alessandro Combinatorics Information Theory Rings and Algebras 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. |
| title | The geometry of rank-metric codes |
| topic | Combinatorics Information Theory Rings and Algebras |
| url | https://arxiv.org/abs/2605.19691 |