Clubs in projective spaces and three-weight rank-metric codes
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| Format: | Preprint |
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2025
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| _version_ | 1866912515034710016 |
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| author | Mannaert, Jonathan Santonastaso, Paolo Zullo, Ferdinando |
| author_facet | Mannaert, Jonathan Santonastaso, Paolo Zullo, Ferdinando |
| contents | Linear sets over finite fields are central objects in finite geometry and coding theory, with deep connections to structures such as semifields, blocking sets, KM-arcs, and rank-metric codes. Among them, $i$-clubs, a class of linear sets where all but one point (which has weight $i$) have weight one, have been extensively studied in the projective line but remain poorly understood in higher-dimensional projective spaces. In this paper, we investigate the geometry and algebraic structure of $i$-clubs in projective spaces. We establish upper bounds on their rank by associating them with rank-metric codes and analyzing their parameters via MacWilliams identities. We also provide explicit constructions of $i$-clubs that attain the maximum rank for $i \geq m/2$, and we demonstrate the existence of non-equivalent constructions when $i \leq m-2$. The special case $i = m-1$ is fully classified. Furthermore, we explore the rich geometry of three-weight rank-metric codes, offering new constructions from clubs and partial classification results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_00502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Clubs in projective spaces and three-weight rank-metric codes Mannaert, Jonathan Santonastaso, Paolo Zullo, Ferdinando Combinatorics Information Theory Linear sets over finite fields are central objects in finite geometry and coding theory, with deep connections to structures such as semifields, blocking sets, KM-arcs, and rank-metric codes. Among them, $i$-clubs, a class of linear sets where all but one point (which has weight $i$) have weight one, have been extensively studied in the projective line but remain poorly understood in higher-dimensional projective spaces. In this paper, we investigate the geometry and algebraic structure of $i$-clubs in projective spaces. We establish upper bounds on their rank by associating them with rank-metric codes and analyzing their parameters via MacWilliams identities. We also provide explicit constructions of $i$-clubs that attain the maximum rank for $i \geq m/2$, and we demonstrate the existence of non-equivalent constructions when $i \leq m-2$. The special case $i = m-1$ is fully classified. Furthermore, we explore the rich geometry of three-weight rank-metric codes, offering new constructions from clubs and partial classification results. |
| title | Clubs in projective spaces and three-weight rank-metric codes |
| topic | Combinatorics Information Theory |
| url | https://arxiv.org/abs/2508.00502 |