Polynomial Invariants of q-Matroids and Rank-Metric Codes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916970767581184 |
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| author | Gluesing-Luerssen, Heide Jany, Benjamin |
| author_facet | Gluesing-Luerssen, Heide Jany, Benjamin |
| contents | It is shown that the Whitney function of a representable q-matroid and the collection of all higher weight enumerators of any representing rank-metric code determine each other via a monomial substitution. Moreover, the q-matroid itself and the collection of all higher support enumerators of the code determine each other. Next, it is proven that the Whitney function of a q-matroid and the Whitney function of its projectivization determine each other via a monomial substitution. Finally, q-matroids with isomorphic projectivizations are studied. It is shown that the projectivizations are isomorphic iff the q-matroids admit a dimension-preserving lattice isomorphism between their lattices of flats. Such q-matroids are called weakly isomorphic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_21618 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial Invariants of q-Matroids and Rank-Metric Codes Gluesing-Luerssen, Heide Jany, Benjamin Combinatorics Information Theory It is shown that the Whitney function of a representable q-matroid and the collection of all higher weight enumerators of any representing rank-metric code determine each other via a monomial substitution. Moreover, the q-matroid itself and the collection of all higher support enumerators of the code determine each other. Next, it is proven that the Whitney function of a q-matroid and the Whitney function of its projectivization determine each other via a monomial substitution. Finally, q-matroids with isomorphic projectivizations are studied. It is shown that the projectivizations are isomorphic iff the q-matroids admit a dimension-preserving lattice isomorphism between their lattices of flats. Such q-matroids are called weakly isomorphic. |
| title | Polynomial Invariants of q-Matroids and Rank-Metric Codes |
| topic | Combinatorics Information Theory |
| url | https://arxiv.org/abs/2509.21618 |