A non‐existence result on Cameron–Liebler line classes

Abstract

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, q 2 − 1, q 2 , q 2 + 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 .

Substructures in finite classical polar spaces

“…Such a line must then have at least 1 + p ≥ 3 points of B and thus the line must be a line contained in the quadric. Using this observation, De Beule et al [9] could show that PG(3, q) does not have a Cameron-Liebler line class of parameter x with 2 < x < q/2. The bound x < q/2 results from the requirement that the corresponding blocking set has to be small.…”

Tight sets, weighted m-covers, weighted m-ovoids, and minihypers

“…We prove that F is the union of x pairwise disjoint subspaces PG(r, q). This is already known for Q + (5, q), see [8]. We rely on Theorem 2.2.…”

“…Recently, many results in finite geometry were obtained by applying characterization results on minihypers [8,17,18]. In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles.…”

“…Although minihypers were first introduced to study the problem of linear codes meeting the Griesmer bound [20,21], characterization results on minihypers can be used to solve problems in finite geometry, see [8], [18] and [4] for applications on substructures of finite projective spaces and generalized quadrangles. We refer to [33] for a survey on the use of minihypers in the study of linear codes meeting the Griesmer bound and in the study of geometrical problems.…”

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

“…Later on it was found that these line classes have many connections to other geometric and combinatorial objects, such as blocking sets of PG(2, q), projective two-intersection sets in PG (5, q), two-weight linear codes, and strongly regular graphs. In the last few years, Cameron-Liebler line classes have received considerable attention from researchers in both finite geometry and algebraic combinatorics; see, for example, [7,20,21,26,11,10]. In [6], the authors gave several equivalent conditions for a set of lines of PG (3, q) to be a Cameron-Liebler line class; Penttila [23] gave a few more of such characterizations.…”