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Hyperovals in Desarguesian Planes of Small Order

The hyperovals in the Desarguesian planes of orders 2, 4 and 8 are all hyperconics [more] (see [Hi79]), so we shall only examine the planes of orders 16, 32 and 64.

PG(2,16):

In 1958, Lunelli and Sce [LuSc58], carried out a computer search for complete arcs in small order planes at the suggestion of B. Segre. In PG(2,16) they found a number of hyperovals which were not hyperconics. In 1975, M. Hall Jr. [Ha75] showed, also with considerable aid from a computer, that there were only two classes of projectively inequivalent hyperovals in this plane, the hyperconics and the hyperovals found by Lunelli and Sce. Out of the 2040 o-polynomials which give the Lunelli-Sce hyperoval, we display only one:

f(x) = x12 + x10 + ß11x8 + x6 + ß2x4 + ß9x2,

where ß is a primitive element of GF(16) satisfying ß4 = ß + 1.

In [Ha75], Hall described a number of collineations of the plane which stabilized the Lunelli-Sce hyperoval, but did not show that they generated the full automorphism group of this hyperoval. In 1978, Payne and Conklin [PaCo78] using properties of a related generalized quadrangle showed that the automorphism group could be no larger than the group given by Hall. Also in 1978, Korchmáros [Ko78] independently gave a constructive proof of this result and showed that the Lunelli-Sce hyperoval is the unique irregular hyperoval admitting a transitive automorphism group (and that the only hyperconics admitting such a group are those of orders 2 and 4).

O'Keefe and Penttila [OKPe91] have reproved Hall's classification result without the use of a computer. Their argument consists of finding an upper bound on the number of o-polynomials defined over GF(16) and then, by examining the possible automorphism groups of hyperovals in this plane, showing that if a hyperoval other than the known ones existed in this plane then the upper bound would be exceeded. Brown and Cherowitzo [BrCh99] have provided a group-theoretic construction of the Lunelli-Sce hyperoval as the union of orbits of the group generated by the elations of PGU(3,4) considered as a subgroup of PGL(3,16). Also included in this paper is a discussion of some remarkable properties concerning the intersections of Lunelli-Sce hyperovals and hyperconics. In [ChPePiRo96] it is shown that the Lunelli-Sce hyperoval is the first non-trivial member of the Subiaco family (see also [BrCh99].), and in [ChOKPe??] it is shown to be the first non-trivial member of the Adelaide family.

PG(2,32):

Since h = 5 is odd, a number of monomial hyperovals are present in this plane. Each of the types mentioned above has a representative, but due to the small size of the plane there are some spurious equivalences, in fact, each of the Glynn type hyperovals is projectively equivalent to a translation hyperoval, and the Payne hyperoval is projectively equivalent to the Subiaco hyperoval (this does not occur in larger planes). Specifically, there are three classes of monomial type hyperovals, the hyperconics (k = 2), proper translation hyperovals (k = 4) and the Segre hyperovals (k = 6). There are also classes corresponding to the Payne hyperovals and the second family (for more details see [Ch88]). In [OKPePr91], O'Keefe, Penttila and Praeger have determined the collineation groups stabilizing each of these hyperovals. In the original determination of the collineation group for the Payne hyperovals the case of q = 32 had to be treated separately and relied heavily on computer results. In [OKPePr91] an alternative version of the proof is given which does not depend on computer computations.

In 1991, O'Keefe and Penttila [OKPe92] by means of a detailed investigation of the divisibility properties of the orders of automorphism groups of hypothetical hyperovals in this plane, discovered a new hyperoval. Its o-polynomial is given by:

f(x) = x4 + x16 + x28 + ß11(x6 + x10 + x14 + x18 + x22 + x26) + ß20(x8 + x20) + ß6(x12 + x24),
where ß is a primitive root of GF(32) satisfying ß5 = ß2 + 1. The full automorphism group of this hyperoval has order 3.

In 1992, Penttila and Royle [PeRo94] were able to cleverly structure an exhaustive computer search for all hyperovals in this plane. The result was that the above listing is complete, there are just six classes of hyperovals in PG(2,32).

PG(2,64):

Until recently this would have been a very short section as the only known hyperoval in this plane was the hyperconic. By extending the ideas in [OKPe92] to PG(2,64), Penttila and Pinneri [PePi94] were able to search for hyperovals whose automorphism group admitted a collineation of order 5. They found two and showed that no other hyperoval exists in this plane that has such an automorphism. The hyperovals are:

f(x) = x8 + x12 + x20 + x22 + x42 + x52 + ß21(x4+x10+x14+x16+x30+x38+x44+x48+x54+x56+x58+x60+x62) + ß42(x2 + x6 + x26 + x28 + x32 + x36 + x40),

which has an automorphism group of order 15, and

f(x) = x24 + x30 + x62 + ß21(x4+x8+x10+x14+x16+x34+x38 +x40+x44+x46+x52+x54+x58+x60) + ß42(x6+ x12+ x18+ x20+ x26+ x32+ x36+ x42+ x48+x50),

which has an automorphism group of order 60, where ß is a primitive element of GF(64) satisfying ß6 = ß + 1. In [ChPePiRo96] it is shown that these are Subiaco hyperovals.

By refining the computer search program, Penttila and Royle [PeRo94] were able to extend the search to hyperovals admitting an automorphism of order 3, and found the hyperoval:

f(x) = x4 + x8 + x14 + x34 + x42 + x48 + x6221(x6+x16+x26+x28+x30+x32+x40+x58) + ß42(x10 + x18 + x24 + x36 + x44 + x50 + x52+ x60),

which has an automorphism group of order 12 (ß is a primitive element of GF(64) as above). This hyperoval is the first distinct Adelaide hyperoval. These examples provide the affirmative answer to Segre's question concerning the existence of irregular hyperovals in this plane. It should be noted that in all three examples the coefficients of the o-polynomials lie in the subfield of order 4 of GF(64). Computer searches have been made for o-polynomials over GF(64) with coefficients in the subfield of order 2 ([Gl89]) showing that only the hyperconics have such an o-polynomial, and with coefficients in the subfield of order 4 producing no hyperovals other than the three above ([PeRo94]). Similar searches over different fields are reported in [OKPe91a].

Penttila and Royle [PeRo95] have shown that any other hyperoval in this plane would have to have a trivial automorphism group. This would mean that there would be many projectively equivalent copies of such a hyperoval, but general searches by the author and others have found none, giving credence to the conjecture that there are no others in this plane.


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