A non‐existence result on Cameron–Liebler line classes

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 generalized 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 article, 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. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 342–349, 2008