A new family of tight sets in ${\mathcal{Q}}^{+}(5,q)$

Abstract

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 $q \equiv 5 \mbox{ or } 9 \bmod{12}$. Under the Klein correspondence, these correspond to Cameron--Liebler line classes of $\mathrm{PG}(3,q)$ 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 $\mathrm{PG}(3,q)$ for all odd $q$. The study of Cameron--Liebler line classes is closely related to the study of symmetric tactical decompositions of $\mathrm{PG}(3,q)$ (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 $q\equiv 9\mathrm{mod}12$ (so $q=3^{2e}$ for some positive integer $e$), providing an infinite family of counterexamples to a conjecture made by Cameron and Liebler in 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 $\mathrm{AG}(2,3^{2e})$ for all positive integers $e$. This proves a conjecture made by Rodgers in his PhD thesis.

27 pages

References in corpus (2)

• Cameron-Liebler line classes
• Cameron-Liebler line classes with parameter $x=\frac{q^2-1}{2}$

Cited by in corpus (17)

• Distance-regular graphs
• Derivation of Cameron-Liebler line classes
• Cameron-Liebler sets of generators in finite classical polar spaces
• Cameron-Liebler Line Classes with parameter $x=\frac{(q+1{)}^2}{3}$
• The Cameron-Liebler problem for sets
• Cameron-Liebler line classes in AG(3,q)
• Cameron-Liebler line classes in $\mathrm{PG}(3,5)$
• Cameron-Liebler k-sets in subspaces and non-existence conditions
• Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets
• A modular equality for Cameron-Liebler line classes in projective and affine spaces of odd dimension
• Cameron-Liebler line classes with parameter $x=\frac{q^2-1}{2}$
• Construction of strongly regular Cayley graphs based on three-valued Gauss periods
• A New Infinite Family of Hemisystems of the Hermitian Surface
• Conference matrices with maximum excess and two-intersection sets
• Packings of partial difference sets
• Cyclic arcs of Singer type and strongly regular Cayley graphs over finite fields
• The degree of functions in the Johnson and q-Johnson schemes