A geometric invariant of linear rank-metric codes
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908578515779584 |
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| author | Astore, Valentina Borello, Martino Calderini, Marco Salizzoni, Flavio |
| author_facet | Astore, Valentina Borello, Martino Calderini, Marco Salizzoni, Flavio |
| contents | Rank-metric codes have been a central topic in coding theory due to their theoretical and practical significance, with applications in network coding, distributed storage, crisscross error correction, and post-quantum cryptography. Recent research has focused on constructing new families of rank-metric codes with distinct algebraic structures, emphasizing the importance of invariants for distinguishing these codes from known families and from random ones. In this paper, we introduce a novel geometric invariant for linear rank-metric codes, inspired by the Schur product used in the Hamming metric. By examining the sequence of dimensions of Schur powers of the extended Hamming code associated with a linear code, we demonstrate its ability to differentiate Gabidulin codes from random ones. From a geometric perspective, this approach investigates the vanishing ideal of the linear set corresponding to the rank-metric code. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19087 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A geometric invariant of linear rank-metric codes Astore, Valentina Borello, Martino Calderini, Marco Salizzoni, Flavio Information Theory Combinatorics 11T71, 51E20, 94B27 Rank-metric codes have been a central topic in coding theory due to their theoretical and practical significance, with applications in network coding, distributed storage, crisscross error correction, and post-quantum cryptography. Recent research has focused on constructing new families of rank-metric codes with distinct algebraic structures, emphasizing the importance of invariants for distinguishing these codes from known families and from random ones. In this paper, we introduce a novel geometric invariant for linear rank-metric codes, inspired by the Schur product used in the Hamming metric. By examining the sequence of dimensions of Schur powers of the extended Hamming code associated with a linear code, we demonstrate its ability to differentiate Gabidulin codes from random ones. From a geometric perspective, this approach investigates the vanishing ideal of the linear set corresponding to the rank-metric code. |
| title | A geometric invariant of linear rank-metric codes |
| topic | Information Theory Combinatorics 11T71, 51E20, 94B27 |
| url | https://arxiv.org/abs/2411.19087 |