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## Herd Spaces and Herds

Let V = (0,0,0,1) and let F be any set of q planes, including the plane W = 0, which do not pass through V. Note that F is a flock of, at least, the empty cone. For each point P = (a,b,c,0) of W = 0, consider the line VP. The points of this line, other than V have coordinates of the form (a,b,c,-), where ranges over the values of GF(q). As we have previously done (), to each plane of F we associate a unique element of the field GF(q), subject only to the condition that 0 is associated to the plane W = 0. With this parameterization of the planes of F, and since the planes do not pass through V, there exist functions f, g and h : GF(q)GF(q), so that the planes with t running through GF(q), have equations:
: f(t)X + g(t)Y + h(t)Z + W = 0.
As the planes of F do not pass through V, each of them must intersect each of the lines VP at a point. Thus, for each point P = (a,b,c,0) we can define a function fP:GF(q)GF(q), by fP(t) = af(t) + bg(t) + ch(t). Note that if the plane meets the line VP at the point (a,b,c,-) then fP(t) = . The set of ordered pairs, {(P,fP) | P is a point of W = 0} is called the Herd Space of F. We shall usually be less formal and refer to the elements of a herd space as being functions (i.e, second coordinates) with associated points (i.e., first coordinates).

The subset of the herd space of F consisting of all of those functions which are permutations is called the Herd of F. Note that a herd may be the empty set. Also note that while the actual functions in a herd space depend upon the parameterization of F, whether or not they are permutations does not, thus the set of points of W = 0 associated to the herd of F is independent of the parameterization of F.

Let C = C(V, S) be a cone and let PG(3,q) be coordinatized in the standard way (). If F is a flock of C, then for each point P in S, since no two planes of F can meet on VP, fP is a permutation of GF(q). The set of these permutations, one for each point of S, is called the Herd of F relative to C. The herd of a flock F relative to cone C is thus a subset of the herd of F.

It is immediate, from our normalization, that:

Proposition 8.1: If F is a flock of a non-flat cone, then the coordinate functions of F are permutations of GF(q).

[In the cases of point or flat cones, we can only say that g, or f and g, respectively, are permutations.]

By reparametrizing the planes of the flock, we can arrange to have any coordinate function be any permutation we like (for non-empty non-flat cones). We shall normally then take g(t) = t (for all non-empty cones). Other normalizations are at times useful, in particular, for quadratic cones in even characteristic, the normalization g(t) = t1/2 brings to light some additional structural properties of the herd [forward reference: oval herds].

Given a flock F, let S be the set of points of W = 0 associated to the herd of F. Clearly, the cone C(V,S) has F as a flock, and is the critical cone of that flock. Rephrasing this observation (without reference to herd spaces) gives a fundamental result, which has been alluded to in earlier work and has been proved in many special cases:

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.

The herd concept has evolved over time and there are several variants in the literature. [more reference: concerning herds]

Herds have proved to be very useful in the study of flocks, simplifying many proofs and opening a new direction of investigation. They can also be used to classify flocks for small q .

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