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| Natura: | Preprint |
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2021
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| Accesso online: | https://arxiv.org/abs/2110.09330 |
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| _version_ | 1866910349126533120 |
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| author | De Beule, Jan Mannaert, Jonathan |
| author_facet | De Beule, Jan Mannaert, Jonathan |
| contents | In this article we study Cameron-Liebler line classes in PG$(n,q)$ and AG$(n,q)$, objects also known as boolean degree one functions. A Cameron-Liebler line class $\mathcal{L}$ is known to have a parameter $x$ that depends on the size of $\mathcal{L}$. One of the main questions on Cameron-Liebler line classes is the (non)-existence of these sets for certain parameters $x$. In particularly it is proven in [12] for $n=3$, that the parameter $x$ should satisfy a modular equality. This equality excludes about half of the possible parameters. We generalize this result to a modular equality for Cameron-Liebler line classes in PG$(n,q)$, and AG$(n,q)$ respectively. Since it is known that a Cameron-Liebler line class in AG$(n,q)$ is also a Cameron-Liebler line class in its projective closure, we end this paper with proving that the modular equality in AG$(n,q)$ is a stronger condition than the condition for the projective case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_09330 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A modular equality for Cameron-Liebler line classes in projective and affine spaces of odd dimension De Beule, Jan Mannaert, Jonathan Combinatorics In this article we study Cameron-Liebler line classes in PG$(n,q)$ and AG$(n,q)$, objects also known as boolean degree one functions. A Cameron-Liebler line class $\mathcal{L}$ is known to have a parameter $x$ that depends on the size of $\mathcal{L}$. One of the main questions on Cameron-Liebler line classes is the (non)-existence of these sets for certain parameters $x$. In particularly it is proven in [12] for $n=3$, that the parameter $x$ should satisfy a modular equality. This equality excludes about half of the possible parameters. We generalize this result to a modular equality for Cameron-Liebler line classes in PG$(n,q)$, and AG$(n,q)$ respectively. Since it is known that a Cameron-Liebler line class in AG$(n,q)$ is also a Cameron-Liebler line class in its projective closure, we end this paper with proving that the modular equality in AG$(n,q)$ is a stronger condition than the condition for the projective case. |
| title | A modular equality for Cameron-Liebler line classes in projective and affine spaces of odd dimension |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2110.09330 |