A Cameron-Liebler line class is a set L of lines in PG(3, q) for which there exists a number x such that IL n SI = x for all spreads S. There are many equivalent properties: Theorem 1 lists eight. This paper classifies Cameron-Liebler line classes with x < 4 (with two exceptions). It is also shown that the study of Cameron-Liebler line classes is equivalent to the study of weighted sets of points in PG(3, q) with two weights on lines.
Throughout this paper, ~ denotes the set of points, .~¢ the set of lines and H the set of planes of PG(3, q). The incidence map of PG(3, q) is the linear map
is the linear map with P(f~*)= Zp~e Ef for all P e ~, f e Q~. With respect to the usual bases of Q~ and Q~', ~t is represented by an incidence matrix of the point-line structure of PG(3, q), and ~t* is represented by the transpose of one.
Note that Q-~= ima ~)kera*, and that this sum is orthogonal with respect to the usual form (f, g) = ~ df dg.
Throughout e denotes the constant function with value 1. Sometimes e will have domain ~; at other times it will have domain Lf. This should be clear from the context.
Since G = PGL(4, q) preserves incidence, (e), (e) ± n im a and ker a* are all modules over QG. Since G has rank 3 on lines, they are all irreducible.
Finally, for P ~, star(P) = {d e Lf: P E f}; for rr e H, ~ = {f e Lf: d _~ rr}; and if P e re, pencil(P, rr) = {E e LP: P ~ d ___ rr}.
The first result gives many equivalent conditions on a set of lines in PG(3, q). Parts (i) to (v) were proved equivalent in I-2]. THEOREM 1. Let L be a set of lines of PG(3, q). The following are equivalent: (i) XL e im a. (ii) XL ~ (ker a*)±.