A gap result for Cameron–Liebler [formula omitted]-classes.

The notion of Cameron–Liebler line classes was generalized in Rodgers et al. (0000) to Cameron–Liebler k -classes, where k = 1 corresponds to the line classes. Such a set consists of x 2 k + 1 k q subspaces of dimension k in PG ( 2 k + 1 , q ) where k ≥ 1 and x ≥ 0 are integers such that every regular k -spread of PG ( 2 k + 1 , q ) contains exactly x subspaces from the set. Examples are known for x ≤ 2 . The authors of Rodgers et al. (0000) show that there are no Cameron–Liebler k -classes when k = 2 and 3 ≤ x ≤ q , or when 3 ≤ k ≤ q log q − q and 3 ≤ x ≤ q ∕ 2 3 . We improve these results by weakening the condition on the upper bound for x to a bound that is linear in q . For this, we use a technique that was originally used to extend nets to affine planes. [ABSTRACT FROM AUTHOR]