Denniston partial difference sets exist in the odd prime case
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917794590752768 |
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| author | Davis, James A. Huczynska, Sophie Johnson, Laura Polhill, John |
| author_facet | Davis, James A. Huczynska, Sophie Johnson, Laura Polhill, John |
| contents | Denniston constructed partial difference sets (PDSs) with the parameters $(2^{3m}, (2^{m+r} - 2^m + 2^r)(2^m-1), 2^m-2^r+(2^{m+r}-2^m+2^r)(2^r-2), (2^{m+r}-2^m+2^r)(2^r-1))$ in elementary abelian groups of order $2^{3m}$ for all $m \geq 2, 1 \leq r < m$. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters $(p^{3m}, (p^{m+r} - p^m + p^r)(p^m-1), p^m-p^r+(p^{m+r}-p^m+p^r)(p^r-2), (p^{m+r}-p^m+p^r)(p^r-1))$ exist in all elementary abelian groups of order $p^{3m}$ for all $m \geq 2, r \in \{1, m-1\}$ where $p$ is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_00512 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Denniston partial difference sets exist in the odd prime case Davis, James A. Huczynska, Sophie Johnson, Laura Polhill, John Combinatorics 05E30, 05B10, 11TT22 Denniston constructed partial difference sets (PDSs) with the parameters $(2^{3m}, (2^{m+r} - 2^m + 2^r)(2^m-1), 2^m-2^r+(2^{m+r}-2^m+2^r)(2^r-2), (2^{m+r}-2^m+2^r)(2^r-1))$ in elementary abelian groups of order $2^{3m}$ for all $m \geq 2, 1 \leq r < m$. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters $(p^{3m}, (p^{m+r} - p^m + p^r)(p^m-1), p^m-p^r+(p^{m+r}-p^m+p^r)(p^r-2), (p^{m+r}-p^m+p^r)(p^r-1))$ exist in all elementary abelian groups of order $p^{3m}$ for all $m \geq 2, r \in \{1, m-1\}$ where $p$ is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes. |
| title | Denniston partial difference sets exist in the odd prime case |
| topic | Combinatorics 05E30, 05B10, 11TT22 |
| url | https://arxiv.org/abs/2311.00512 |