Most $q$-matroids are not representable
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915638400778240 |
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| author | Degen, Sebastian Kühne, Lukas |
| author_facet | Degen, Sebastian Kühne, Lukas |
| contents | A $q$-matroid is the analogue of a matroid which arises by replacing the finite ground set of a matroid with a finite-dimensional vector space over a finite field. These $q$-matroids are motivated by coding theory as the representable $q$-matroids are the ones that stem from rank-metric codes. In this note, we establish a $q$-analogue of Nelson's theorem in matroid theory by proving that asymptotically almost all $q$-matroids are not representable. This answers a question about representable $q$-matroids by Jurrius and Pellikaan strongly in the negative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06795 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Most $q$-matroids are not representable Degen, Sebastian Kühne, Lukas Combinatorics A $q$-matroid is the analogue of a matroid which arises by replacing the finite ground set of a matroid with a finite-dimensional vector space over a finite field. These $q$-matroids are motivated by coding theory as the representable $q$-matroids are the ones that stem from rank-metric codes. In this note, we establish a $q$-analogue of Nelson's theorem in matroid theory by proving that asymptotically almost all $q$-matroids are not representable. This answers a question about representable $q$-matroids by Jurrius and Pellikaan strongly in the negative. |
| title | Most $q$-matroids are not representable |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.06795 |