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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.27689 |
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| _version_ | 1866911551285362688 |
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| author | Alderson, Tim Ball, Simeon |
| author_facet | Alderson, Tim Ball, Simeon |
| contents | Let $\mathcal{X}$ be a set of $(h-1)$-dimensional subspaces of $\mathrm{PG}(kh-1,q)$ with the property that every hyperplane contains at most $t$ elements of $\mathcal{X}$. We prove the upper bound $|\mathcal{X}| \leq (t-k+2)q^h + t$, and characterise the structure of $\mathcal{X}$ in the case of equality. We call sets attaining this bound \emph{length-maximal}. For $k=3$, such sets are known as maximal arcs and have been well-studied. They are known to exist for $t<q^h$ if and only if $q$ is even and $t$ divides $q^h$. For $k=4$ and $q>2$, we show that any length-maximal set must satisfy $t = q^h+1$ and that every hyperplane is either a $t$-secant or a $1$-secant. For $k \geq 5$ and $q>2$, no length-maximal set exists. In the language of additive codes, these results assert that additive two-weight codes over $\mathbb{F}_{q^h}$ attaining the natural Griesmer-type bound do not exist when the code dimension is $5$ or more and $q>2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27689 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sets of subspaces with restricted hyperplane intersection numbers Alderson, Tim Ball, Simeon Combinatorics 51E20 (primary), 94B05, 11T71 (secondary) Let $\mathcal{X}$ be a set of $(h-1)$-dimensional subspaces of $\mathrm{PG}(kh-1,q)$ with the property that every hyperplane contains at most $t$ elements of $\mathcal{X}$. We prove the upper bound $|\mathcal{X}| \leq (t-k+2)q^h + t$, and characterise the structure of $\mathcal{X}$ in the case of equality. We call sets attaining this bound \emph{length-maximal}. For $k=3$, such sets are known as maximal arcs and have been well-studied. They are known to exist for $t<q^h$ if and only if $q$ is even and $t$ divides $q^h$. For $k=4$ and $q>2$, we show that any length-maximal set must satisfy $t = q^h+1$ and that every hyperplane is either a $t$-secant or a $1$-secant. For $k \geq 5$ and $q>2$, no length-maximal set exists. In the language of additive codes, these results assert that additive two-weight codes over $\mathbb{F}_{q^h}$ attaining the natural Griesmer-type bound do not exist when the code dimension is $5$ or more and $q>2$. |
| title | Sets of subspaces with restricted hyperplane intersection numbers |
| topic | Combinatorics 51E20 (primary), 94B05, 11T71 (secondary) |
| url | https://arxiv.org/abs/2603.27689 |