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

Abstract

In this paper, we describe a new infinite family of q 2 −1 2 -tight sets in the hyperbolic quadrics Q + (5, q), for q ≡ 5 or 9 mod 12. Under the Klein correspondence, these correspond to Cameron-Liebler line classes of PG(3, q) having parameter 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 q 2 +1 2 in 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 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 ≡ 9 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 1 2 (3 2e − 3 e ), 1 2 (3 2e + 3 e ) in the affine plane AG(2, 3 2e ) for all positive integers e. This proves a conjecture made by Rodgers in his PhD thesis.

A family of m-ovoids of parabolic quadrics

“…Nevertheless, we include a short section on q+1 2 -ovoids of Q(4, q) since the construction is direct and is done using the same idea as in the construction of q−1 2 -ovoids of Q(4, q). Our approach to the construction of these m-ovoids is similar to the one used previously in [13], [10], and [16]: we prescribe an automorphism group for the m-ovoids that we intend to construct, and then take unions of orbits of the point set of Q(4, q) under the action of the prescribed automorphism group. Of course this approach had been used previously in the constructions of m-ovoids, tight sets, and other geometric objects.…”

Cameron–Liebler line classes with parameter x=q2−12

“…Furthermore we constructed the first infinite family of sets of type (m, n) in the affine plane AG(2, q), where q is an even power of 3. It would be interesting to come up with a general construction of Cameron-Liebler line classes in PG(3, q) when q is even since there are some known examples in this case in the thesis [25] of Rodgers. We close this paper by referring the reader to a paper [8] by De Beule, Demeyer, Metsch, and Rodgers. Immediately after we finished a draft of this manuscript, we became aware that De Beule, Demeyer, Metsch and Rodgers [8] also obtained the same result on Cameron-Liebler line classes with parameter x = q 2 −1 2 at almost the same time.…”

“…It would be interesting to come up with a general construction of Cameron-Liebler line classes in PG(3, q) when q is even since there are some known examples in this case in the thesis [25] of Rodgers. We close this paper by referring the reader to a paper [8] by De Beule, Demeyer, Metsch, and Rodgers. Immediately after we finished a draft of this manuscript, we became aware that De Beule, Demeyer, Metsch and Rodgers [8] also obtained the same result on Cameron-Liebler line classes with parameter x = q 2 −1 2 at almost the same time. The approaches for proving the main result are comparable but different enough to justify that we write two separate papers; our approach is more algebraic and the approach taken by De Beule, Demeyer, Metsch and Rodgers is more geometric.…”

“…In [26], Rodgers developed a method to obtain new affine two-intersection sets in AG(2, 3 2e ) by establishing certain tactical decompositions of the points and lines of PG(3, 3 2e ) induced by Cameron-Liebler line classes with some nice properties. He was thus able to rediscover an example of sets of type (3,6) in AG (2,9), and obtain a new example of affine two-intersection sets in AG (2,81). In his thesis [25], Rodgers made the following conjecture.…”

“…It was once conjectured [6, p. 97] that the above trivial examples and their complements are all of the Cameron-Liebler line classes. The first counterexample to this conjecture was given by Drudge [9] in PG (3,3), and it has parameter x = 5. Later Bruen and Drudge [4] generalized Drudge's example into an infinite family with parameter x = q 2 +1 2 for all odd q.…”