Delsarte duality on subspaces and applications to rank-metric codes and q-matroids
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912614833979392 |
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| author | Borello, Martino Polverino, Olga Zullo, Ferdinando |
| author_facet | Borello, Martino Polverino, Olga Zullo, Ferdinando |
| contents | We study the interplay between the lattice of F_{q^m}-subspaces and the lattice of F_{q^m}-subspaces of an F_{q^m}-vector space. Introducing notions of weight and defect relative to an F_q-subspace, we analyze the sequence of maximum non-zero defects. We establish a correspondence between subspaces of positive defect and their Delsarte duals, enabling explicit characterizations of the associated sequences of maximum non-zero defects. Our framework unifies several classes of subspaces studied in finite geometry and connects them to linear rank-metric codes by providing a new geometric interpretation of code duality. Building on these results, we characterize classes of rank-metric codes closed under duality, including MRD, near MRD, quasi-MRD, and a new family of (n, k)-MRD codes. Finally, we explore applications to q-matroids, by studying the problem of F_{q^m}-representability for direct sums of uniform q-matroids and describing their rank generating functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Delsarte duality on subspaces and applications to rank-metric codes and q-matroids Borello, Martino Polverino, Olga Zullo, Ferdinando Combinatorics Information Theory We study the interplay between the lattice of F_{q^m}-subspaces and the lattice of F_{q^m}-subspaces of an F_{q^m}-vector space. Introducing notions of weight and defect relative to an F_q-subspace, we analyze the sequence of maximum non-zero defects. We establish a correspondence between subspaces of positive defect and their Delsarte duals, enabling explicit characterizations of the associated sequences of maximum non-zero defects. Our framework unifies several classes of subspaces studied in finite geometry and connects them to linear rank-metric codes by providing a new geometric interpretation of code duality. Building on these results, we characterize classes of rank-metric codes closed under duality, including MRD, near MRD, quasi-MRD, and a new family of (n, k)-MRD codes. Finally, we explore applications to q-matroids, by studying the problem of F_{q^m}-representability for direct sums of uniform q-matroids and describing their rank generating functions. |
| title | Delsarte duality on subspaces and applications to rank-metric codes and q-matroids |
| topic | Combinatorics Information Theory |
| url | https://arxiv.org/abs/2509.24409 |