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,
The intersection of two of these planes projected from the vertex into the plane w = 0 gives a line of the form,
This line misses the -conic in w = 0 (and hence the planes do not intersect on the -cone) if and only if,
has no non-trivial solutions. Algebraically, this is equivalent to the statement that
where by trace we mean the absolute trace function over F. Define an -clan to be a set of q upper triangular matrices,
such that there exists a F, with trace() = 1, so that
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,
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
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.