Linear rank-metric intersecting codes
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908429289783296 |
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| author | Bartoli, Daniele Borello, Martino Marino, Giuseppe Scotti, Martin |
| author_facet | Bartoli, Daniele Borello, Martino Marino, Giuseppe Scotti, Martin |
| contents | In this paper we introduce and investigate rank-metric intersecting codes, a new class of linear codes in the rank-metric context, inspired by the well-studied notion of intersecting codes in the Hamming metric. A rank-metric code is said to be intersecting if any two nonzero codewords have supports intersecting non trivially. We explore this class from both a coding-theoretic and geometric perspective, highlighting its relationship with minimal codes, MRD codes, and Hamming-metric intersecting codes. We derive structural properties, sufficient conditions based on minimum distance, and geometric characterizations in terms of 2-spannable $q$-systems. We establish upper and lower bounds on code parameters and show some constructions, which leave a range of unexplored parameters. Finally, we connect rank-intersecting codes to other combinatorial structures such as $(2,1)$-separating systems and frameproof codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear rank-metric intersecting codes Bartoli, Daniele Borello, Martino Marino, Giuseppe Scotti, Martin Combinatorics Information Theory In this paper we introduce and investigate rank-metric intersecting codes, a new class of linear codes in the rank-metric context, inspired by the well-studied notion of intersecting codes in the Hamming metric. A rank-metric code is said to be intersecting if any two nonzero codewords have supports intersecting non trivially. We explore this class from both a coding-theoretic and geometric perspective, highlighting its relationship with minimal codes, MRD codes, and Hamming-metric intersecting codes. We derive structural properties, sufficient conditions based on minimum distance, and geometric characterizations in terms of 2-spannable $q$-systems. We establish upper and lower bounds on code parameters and show some constructions, which leave a range of unexplored parameters. Finally, we connect rank-intersecting codes to other combinatorial structures such as $(2,1)$-separating systems and frameproof codes. |
| title | Linear rank-metric intersecting codes |
| topic | Combinatorics Information Theory |
| url | https://arxiv.org/abs/2507.00569 |