Definitions: Flock, Star Flock, Cone, Linearized Polynomial

Theorem 11.1.3: If q = pn with p a prime > 3 then a proper star flock of any thick cone is given by F(t,g(t),0) if and only if g is a non-linear pe-linearized polynomial for some e|n with e < n.

Pf: For a star flock to be the flock of a thick cone, we must have N[q+1/2]. For a star flock to be proper, we must have N > 1. By assumption, p > 2. Thus, all except case (iii) of Theorem 11.1.2 is eliminated, and since p > 3, we have that g is a pe-linearized polynomial by that same theorem.

Now, suppose that g is a non-linear pe-linearized polynomial for some e|n with e < n. Such functions are additive (i.e., g(t+s) = g(t) + g(s)), so to calculate N we need only calculate the number of distinct values of g(t)/t for t0. Let K be the proper subfield of GF(q) of order pe. For each c in K we have g(ct) = cg(t). Thus, for c in K\{0}, g(ct)/ct = g(t)/t, and the number of distinct non-zero values of g(t)/t is at most (q-1)/(pe-1). Therefore, N(q-1)/(pe-1) + 1 < [q+1/2] since pe > 3. We obtain a star flock of a thick cone from such a g, and it is proper since g is not linear.