Definitions: Non-flat Cone, Flock, Herd

Theorem 8.1: (The General Herd Theorem) Let f, g and h be permutations of GF(q) with f(0) = g(0) = h(0) = 0. Define a point set S to consist of those points with coordinates (x, y, z, 0) such that xf(t) + yg(t) + zh(t) is a permutation of GF(q). Then {xf(t) + yg(t) + zh(t) | (x, y, z, 0) is in S} is the herd of the flock F, whose planes have the equations f(t)X + g(t)Y + h(t)Z + W = 0, of the cone with carrier S and vertex V = (0, 0, 0, 1).

Conversely, any flock of any non-empty non-flat cone admits such a representation.

Pf: Notice that the set S is non-empty, containing the points A = (1,0,0,0), B = (0,1,0,0) and C = (0,0,1,0). The q planes of F are distinct since the coordinate functions are permutations, and they do not contain the point V = (0,0,0,1). Consider the cone C(V, S). Suppose that two of these planes, say those whose equations are f(u)X + g(u)Y + h(u)Z + W = 0 and f(s)X + g(s)Y + h(s)Z + W = 0, with u not equal to s, meet at a point Z of C(V,S). Since Z is not V, it is on a unique generator of the cone, and so, there is a point (x, y, z, 0) of S so that Z = (x, y, z, ) for some in GF(q). We would then have xf(u) + yg(u) + zh(u) = xf(s) + yg(s) + zh(s) = - for u not equal to s, contradicting the definition of S. Thus, the planes of F form a flock of the (non-flat) cone C(V,S).

The converse is just Proposition 8.1.