A new family of tight sets in \({\mathcal {Q}}^{+}(5,q)\)
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- 作者：Jan De Beule ; Jeroen Demeyer ; Klaus Metsch…
- 关键词：Tight sets ; Cameron–Liebler line classes ; Sets of type $$(m ; n)$$ ( m ; n ) ; Tactical decompositions ; 51E20 ; 05B25
- 刊名：Designs, Codes and Cryptography
- 出版年：2016
- 出版时间：March 2016
- 年：2016
- 卷：78
- 期：3
- 页码：655-678
- 全文大小：556 KB
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25.Tee G.: Eigenvectors of block circulant and alternating circulant matrices. N. Z. J. Math. 36, 195–211 (2007). - 作者单位：Jan De Beule (1)
Jeroen Demeyer (1)
Klaus Metsch (1) (2)
Morgan Rodgers (1)
1. Department of Mathematics, Ghent University, Krijgslaan 281, S22, 9000, Ghent, Belgium
2. Mathematisches Institut, Universität Gießen, Arndtstraße 2, 35392, Giessen, Germany - 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Combinatorics
Coding and Information Theory
Data Structures, Cryptology and Information Theory
Data Encryption
Discrete Mathematics in Computer Science
Information, Communication and Circuits - 出版者：Springer Netherlands
- ISSN：1573-7586
文摘
In this paper, we describe a new infinite family of \(\frac{q^{2}-1}{2}\)-tight sets in the hyperbolic quadrics \({\mathcal {Q}}^{+}(5,q)\), for \(q \equiv 5 \text{ or } 9 \,\hbox {mod}\,{12}\). Under the Klein correspondence, these correspond to Cameron–Liebler line classes of \(\mathop {\mathrm{PG}}(3,q)\) having parameter \(\frac{q^{2}-1}{2}\). This is the second known infinite family of nontrivial Cameron–Liebler line classes, the first family having been described by Bruen and Drudge with parameter \(\frac{q^{2}+1}{2}\) in \(\mathop {\mathrm{PG}}(3,q)\) for all odd \(q\). The study of Cameron–Liebler line classes is closely related to the study of symmetric tactical decompositions of \(\mathop {\mathrm{PG}}(3,q)\) (those having the same number of point classes as line classes). We show that our new examples occur as line classes in such a tactical decomposition when \(q \equiv 9 \,\hbox {mod}\,12\) (so \(q = 3^{2e}\) for some positive integer \(e\)), providing an infinite family of counterexamples to a conjecture made by Cameron and Liebler (in Linear Algebra Appl 46, 91–102, 1982); the nature of these decompositions allows us to also prove the existence of a set of type \(\left( \frac{1}{2}(3^{2e}-3^{e}), \frac{1}{2}(3^{2e}+3^{e}) \right) \) in the affine plane \(\mathop {\mathrm{AG}}(2,3^{2e})\) for all positive integers \(e\). This proves a conjecture made by Rodgers in his Ph.D. thesis. Keywords Tight sets Cameron–Liebler line classes Sets of type \((m, n)\) Tactical decompositions