A gap result for Cameron-Liebler k-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 x2k+1kq subspaces of dimension k in PG(2k+1,q) where k≥1 and x≥0 are integers such that every regular k-spread of PG(2k+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≤qlogq−q and 3≤x≤q∕23. 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.