**Sections**

## -Flocks and -Clans

Since the most recent work on hyperovals has been intimately tied
to flocks of cones in
PG(3,2^{h}) 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 = 2*^{h} 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
a_{t} x + b_{t} y + c_{t} 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,

(a_{t} + a_{s})x + (b_{t} + b_{s})y + (c_{t} + c_{s})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,

(a_{t} + a_{s})x + (b_{t} + b_{s})xz^{-1} + (c_{t} + c_{s})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,

A_{t} =
such that there exists a **F**, with
trace() = 1, so that

,
ts.
** Result 3:** *
is an -clan if and only if a*_{t}x + b_{t}y + c_{t}z +
w = 0, t **F** is an
-flock.

It is easily seen that the functions a_{t}, b_{t}, and c_{t}
are permutations of **F** and
we can normalize the -clan so that each of them fixes 0
and 1.

When is the Frobenius automorphism, xx^{2}, 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 s**F**, then the
set of *q + 1* functions,
{f(t)}{h_{s}(t) | s**F** } 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.

**Sections**