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Monomial opolynomials
A special class of hyperovals, containing most of the known
examples, consists of those
which are projectively equivalent to a hyperoval having a
monomial opolynomial. Such an
opolynomial must be of the form f(x) = x^{k}. We define
(h) = {k  x^{k} is an opolynomial over
GF(2^{h})}.
As has been observed by numerous authors ([Ch88],[Gl83],[Hi79],[Pa71]):
Result 1: If k (h) then 1/k, 1k, 1/(1k), k/(k1) and
(k  1)/k (h)
where these numbers are taken
modulo 2^{h}  1. These six opolynomials give projectively
equivalent hyperovals.
We give a brief description of what is known to be in
(h).
 2 (h)h.
 These are
the hyperconics, the only
family of monomial hyperovals to be found in all planes under
consideration.
 2^{i}(h) if and only if (i,h)= 1.
 These hyperovals were
determined by Segre [Se57] in 1957. They are called
translation hyperovals since they
admit as an automorphism group a group of translations which is
transitive on the affine
points of the hyperoval. When i1, or h1, these hyperovals
are not equivalent to hyperconics. Payne [Pa71] has shown that these are the only additive
opolynomials. An examination
of the Euler totient function reveals that when h = 5 or h 7, the translation
hyperovals provide examples of hyperovals which are not complete
conics (usually referred to
as irregular hyperovals). The situation for h = 1,2,3 and
4 is well known, all hyperovals in these planes are hyperconics,
but for h = 6 no examples other than the hyperconics were
known. The question of
whether or not irregular hyperovals exist in the Desarguesian
plane of order 64, open since
1957, has finally been settled in the affirmative ( Hyperovals in PG(2,64)).
 6 (h) for h odd.

Discovered by Segre [Se62] in 1962, but most of
the proofs appear in the 1971 treatment by Segre and Bartocci
[SeBa71].
 + and 3+ 4 (h) for
h odd, where and are
automorphisms such that ^{4}^{2}2 mod (2^{h}1).

These two families were
discovered by Glynn [Gl83] in 1982 (a few of the initial members of
these families in small planes were already known). Alternate versions of the proofs that these are hyperovals can be found in [Ch98].
Glynn implemented a fast algorithm for determining membership in
(h) and
searched all values of h up to and including 19 as a prelude to
the abovementioned result. He
has since extended this search and found no new hyperovals. We
record this as:
Result 2 (Glynn [Gl89]): The sets (h)
are completely determined for h28.
The collineation groups stabilizing the monomial opolynomial
hyperovals have been studied
by O'Keefe and Penttila [OKPe94].
Another approach to classifying the monomial opolynomials,
inspired by work on flocks of cones (see below), is concerned
with the number of nonzero bits in the binary representation of
the exponent of the monomial. The one bit exponents correspond to
the translation opolynomials. The two bit exponents have been
classified by Cherowitzo and Storme [ChSt98]. The three bit exponent classification is being worked on.
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