Definitions: Flock, Herd Space

Proposition 8.1.2: Given any three functions f, g and h : GF(q) GF(q), with f(0) = g(0) = h(0) = 0, such that there are no pairs of distinct elements s and t of GF(q) for which f(t) = f(s), g(t) = g(s) and h(t) = h(s), then there exists a flock F whose herd space contains these three functions.

Pf: We associate to each function one point out of any fixed triple of non-collinear points of W = 0. For the sake of this argument we will choose to associate f with (1,0,0,0), g with (0,1,0,0) and h with (0,0,1,0). For each t in GF(q), we consider the triple of points of PG(3,q) given by (1,0,0,-f(t)), (0,1,0,-g(t)) and (0,0,1,-h(t)). These points are not collinear and so they determine a unique plane, . The set of planes constructed in this way will be a flock provided there are exactly q of them (i.e., no two of the constructed planes are the same.) If the plane constructed with index s, st, was , then we would have f(s) = f(t), g(s) = g(t) and h(s) = h(t), contradicting the assumption about the functions.