A non-existence result on Cameron-Liebler line classes
نویسندگان
چکیده
Cameron-Liebler line classes are sets of lines in PG(3, q) that contain a fixed number x of lines of every spread. Cameron and Liebler classified Cameron-Liebler line classes for x ∈ {0, 1, 2, q2 − 1, q2, q2 + 1} and conjectured that no others exist. This conjecture was disproven by Drudge for q = 3 [8] and his counterexample was generalised to a counterexample for any odd q by Bruen and Drudge [4]. A counterexample for q even was found by Govaerts and Penttila [9]. Non-existence results on Cameron-Liebler line classes were found for different values of x. In this paper, we improve the non-existence results on Cameron-Liebler line classes of Govaerts and Storme [11], for q not a prime. We prove the non-existence of Cameron-Liebler line classes for 3 ≤ x < q 2 .
منابع مشابه
On Cameron–Liebler line classes
Cameron–Liebler line classes are sets of lines in PGð3; qÞ that contain a fixed number x of lines of every spread. Cameron and Liebler classified them for x A f0; 1; 2; q 1; q; q þ 1g and conjectured that no others exist. This conjecture was disproven by Drudge and his counterexample was generalised to a counterexample for any odd q by Bruen and Drudge. Nonexistence of Cameron–Liebler line clas...
متن کاملCameron-Liebler line classes in PG(n, 4)
We derive a new existence condition for Cameron – Liebler line classes in PG(3, q). As an application, we obtain the characterization of Cameron – Liebler line classes in PG(n, 4), n ≥ 3.
متن کاملA new family of tight sets in Q+(5, q)
In this paper, we describe a new infinite family of q −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 −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...
متن کاملA modular equality for Cameron-Liebler line classes
In this paper we prove that a Cameron-Liebler line class L in PG(3, q) with parameter x has the property that ( x 2 ) +n(n−x) ≡ 0 mod q+1 for the number n of lines of L in any plane of PG(3, q). It follows that the modular equation ( x 2 ) + n(n − x) ≡ 0 mod q + 1 has an integer solution in n. This result rules out roughly at least one half of all possible parameters x. As an application of our...
متن کاملCameron-Liebler line classes
New examples of Cameron-Liebler line classes in PG(3,q) are given with parameter 1 2 (q 2− 1). These examples have been constructed for many odd values of q using a computer search, by forming a union of line orbits from a cyclic collineation group acting on the space. While there are many equivalent characterizations of these objects, perhaps the most significant is that a set of lines L in PG...
متن کامل