-Flocks and -Clans

Since the most recent work on hyperovals has been intimately tied to flocks of cones in PG(3,2h) and this connection appears at present to be the most promising avenue towards a classification, we shall conclude with a brief description of this relationship. This material is taken from [Ch98].

Let F = GF(q) with q = 2h and be any automorphism of maximal order of F. Define the -cone in PG(3,q) to be the set of points,

= {(x,y,z,w) | },
together with the vertex (0,0,0,1). An -flock is a set of q planes of PG(3,q) not passing through the vertex of the -cone which do not intersect each other at a point of . Following Thas [Th87], the planes of the -flock may be expressed by
at x + bt y + ct z + w = 0, t F.

The intersection of two of these planes projected from the vertex into the plane w = 0 gives a line of the form,

(at + as)x + (bt + bs)y + (ct + cs)z = 0,    t s.

This line misses the -conic in w = 0 (and hence the planes do not intersect on the -cone) if and only if,

(at + as)x + (bt + bs)xz-1 + (ct + cs)z = 0

has no non-trivial solutions. Algebraically, this is equivalent to the statement that

,  ts,

where by trace we mean the absolute trace function over F. Define an -clan to be a set of q upper triangular matrices,

At =

such that there exists a F, with trace() = 1, so that

,   ts.

Result 3: is an -clan if and only if atx + bty + ctz + w = 0, t F is an -flock.

It is easily seen that the functions at, bt, and ct are permutations of F and we can normalize the -clan so that each of them fixes 0 and 1.

When is the Frobenius automorphism, xx2, an -flock is known in the literature simply as a flock of a quadratic cone, and the corresponding -clans are called q-clans. We shall abuse the notation somewhat and refer to q-clans as 2-clans. 2-clans have been widely studied (see [BaLuPa94], [ChPePiRo96], [FiTh79], [Pa89], [Pa92], [Pa96], [PaTh91], [PaPePi95], [PaPeRo97]).

The result that connects -clans and hyperovals is:

Result 4: If is an -clan then f(t) is an o-polynomial.

In the 2-clan case it can also be shown that g(t) is an o--polynomial. The link is strengthened by the concept of a herd which we will presently define.

Result 5: If is an -clan with respect to trace 1 element , thens F,

  is an -clan.

In light of this result we make the following definition. Given permutation polynomials f(t) and g(t) over F, with f(0) = g(0) = 0 and f(1) = g(1) = 1, if there exists an automorphism of F and an element F with trace() = 1 such that the functions

are permutations of F for each sF, then the set of q + 1 functions, {f(t)}{hs(t) | sF } will be called a herd, denoted by (f, g, ,). If all of the functions in a herd are o-polynomials, then the herd will be called a herd of ovals or an oval herd.

Result 6: If (f, g, , ) is a herd then is an -clan.

It is easily shown that all the herds which arise in the 2-clan case are oval herds, but this is not true for general .

It should be noted that all known hyperovals, with the single exception of the O'Keefe- Penttila hyperoval in PG(2,32) have projectively equivalent forms which appear in appropriate oval herds. The search for an oval herd containing this hyperoval has not yet been completed.