Throughout this section, F is a set of q planes not through V = (0,0,0,1) which contains the plane W = 0, that is, F is a flock (possibly of only the empty cone). If the herd space of F contains at least one permutation, then F is a flock of a non-empty cone.
Proposition 8.1.1: The herd space of a flock F is determined by any three of its functions whose associated points are not collinear.
Conversely we have:
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.
Of course, there is no guarantee that the flocks constructed in the last proposition will be flocks of anything other than the empty cone unless one chooses at least one of the functions to be a permutation.
Proposition 8.1.3: The herd space of a flock F contains a constant function, if and only if, F is a star flock.
We can also characterize and determine the structure of a herd space of a linear flock.
Proposition 8.1.4: The herd space of a flock F contains at least two constant functions, if and only if, F is a linear flock. In this case, the herd space contains exactly q+1 constant functions and all the other functions are scalar multiples of the same permutation. Furthermore, in the affine plane obtained by removing the line Z = 0 from the plane W = 0, all the herd space functions that are equal to each other are associated to the points which lie on a line of the parallel class containing the common line of the flock.
We can also generalize two earlier results and phrase them in terms of herd spaces.
Proposition 8.1.5: The herd space of a flock F contains three functions based at non-collinear points that are linearly dependent over GF(q), if and only if, F is a star flock and all triples of functions based at non-collinear points are linearly dependent over GF(q).
Proposition 8.1.6: The herd space of a flock F contains three functions based at non-collinear points that are scalar multiples of each other, if and only if, F is a linear flock.
Finally, we examine some structural properties of herd spaces.
Proposition 8.1.7: In the herd space of a flock F the functions associated with three collinear points of W = 0 are linearly dependent over GF(q). In particular, if the herd space contains two functions based at points P and Q which are scalar multiples of each other, then all the functions based at points on the line PQ are scalar multiples of the same function and F is a star flock.
The last statement of the above proposition is important enough to warrant a separate statement and an alternate proof.
Theorem 8.1.1: If two functions in the herd space of a flock F are scalar multiples, then F is a star flock.