AbstractIn this paper we construct two classes of translation hyperovals in any Hall plane of even orderq2 ≥ 16. Two hyperovals constructed in the same Hall plane are equivalent under the action of the automorphism group of that Hall plane iff they are in the same class

O'Keefe, Christine M.

Pascasio, Arlene A.

Penttila, Tim

Elsevier - Publisher Connector 

Europ. J. Combinatorics (1992) W, 195-199 
Hyperovals in Hall Planes 
CHRISTINE M. O’KEEFE, ARLENE A. PASCASIO AND TIM PENT~ILA 
In this paper we construct two classes of translation hyperovals in any Hall plane of even 
order q* P 16. Two hyperovals constructed in the same Hall plane are equivalent under the 
action of the automorphism group of that Hall plane iff they are in the same class. 
1. INTRODUCTION AND PRELIMINARIES 
A k-arc in a (finite) projective plane n,, of order n is a set of k points, no three of 
which are collinear. The maximum k for which there exist k-arcs in Ed, is n + 1 if n is 
odd, and n + 2 if n is even. An (n + l)-arc is an oval and an (n + 2)-arc is a hyperoval. 
For more details on these and other definitions and results needed, see [4] or [6], but 
note that these references use the term ‘oval’ for a hyperoval. 
In the following we denote the automorphism group of n,, by Aut(n,) and the 
stabiliser of a k-arc % in Aut(n,J by Aut(%!). If nq = PG(2, q), the desarguesian 
projective plane of order q, where q =p” and p is prime, then Aut(PG(2, q)) is 
PTL(3, 4). 
Let I be a line of n,,. An elation of n,, with axis 1 is an automorphism of n, which 
fixes 1 pointwise and fixes a point C E I linewise. A translation n-arc SY in n,, is an n-arc 
the stabiliser of which contains a group of elations acting transitively on its points. (A 
permutation group G acts transitively on a set X if whenever x1, x2 E X, there exists an 
element g E G such that gxl = x2.) 
Furthermore, a frurr.rlation hyperoval of n” is a hyperoval which contains a 
translation n-arc. Let 9Z be a translation n-arc with group T of elations, and suppose 
that X is contained in a (translation) hyperoval %. Then the two points P and Q of 
Z?5% lie on the axis of each non-identity element of T. This is because % is fixed by 
each element of T, and hence X is fixed by each element of T (for two hyperovals meet 
in at most half their number of points). Thus the two points P and Q are fixed or 
interchanged by each element of T, and the centre of each element of T must lie on 
PQ. Since all elements of T cannot have the same centre, as T is transitive on the 
points of %, it follows that PQ is the axis of each element of T. Thus PQ is the axis of 
T. 
In PG(2, q), q even, the translation hyperovals are exactly those hyperovals which 
are images under an element of PI’L(3, q) of a hyperoval 
9(a) = {(I, t, t”): t E GF(q)) U ((0, 1, (0, (6% l)}, 
where (Y is a generator of Aut(GF(q)) (see [12]). The regular hyperovals are the 
images of the hyperoval g(2). A regular hyperoval therefore consists of the points of a 
non-degenerate conic together with its nucleus N, which is often also called the nucleus 
of the hyperoval. 
The Hall plane of order q2, for q 3 3, is constructed from the desarguesian plane of 
order q2 as follows. Let I, be a line of PG(2, q’), and let AG(2, q2) denote the affine 
plane PG(2, q’)\l,. Let 9 be a derivation set for AG(2, q2); that is, a set of q + 1 
points of I, such that, for every pair of points X, Y of AG(2, q’) for which the line XY 
meets 1, in 9, there is a Baer subplane of PG(2, q2) containing X, Y and 9. We define 
195 
0195-6698/92/03Ot95 + 05 $02.00/O 0 1992 Academic Press Limited 
196 C. M. O’Keefe, A. A. Pascasio and T. Penttila 
a new incidence structure aAG(2, 4’): a point of sAG(2, q*) is a point of AG(2, q’); 
a line of gAG(2, q*) is either a line of AG(2, q*) the ideal point of which is not in 9, 
or is the set of affine points of a Baer subplane of PG(2, q*) which contains 9; and 
incidence is the natural containment relation. The incidence structure 9AG(2, q*) is an 
affine plane of order q*, and is uniquely completable to a projective plane of the same 
order, by the addition of an ideal line. This is the Hall plane HalI of order q’ (see 
]7,101). 
The construction of ovals in Hall planes of odd order has been considered in 
[l, 8,9,11]. Korchmaros [9] also gives a family of parabolas in AG(2, 2*“), each of 
which is still an arc in Hall(22”) and, in fact, is completable to a hyperoval in 
Hall(22”) (see the remarks following Theorem 2.2 below). This construction seems to 
have been discovered independently of the earlier result by Crismale [3]. Cherowitzo 
[2] conducted a complete computer search for hyperovals in all translation planes of 
order 16. The hyperovals that he found in Hall(16) fall into 15 equivalence classes of 
translation hyperovals, each with two points on the ideal line and with one of the three 
possible abstract automorphism groups (see also [5]). 
We will give two constructions of translation hyperovals in any Hall plane Hali 
of even order q* 3 16. In both cases the hyperovals constructed have two points on the 
ideal line of the Hall plane, and the collection of hyperovals arising by the second 
construction includes the examples of Crismale [3] and Korchmaros [9]. 
2. THE CONSTRUCIYONS 
Throughout this section we assume that q 3 4. The affine points of each hyperoval 
that we will construct in a Hall plane of even order is the set of affine points of a 
regular hyperoval in the corresponding desarguesian plane. This restricts the position 
of the nucleus of the hyperoval, as we now demonstrate. 
Let c& be a conic of PG(2, q*) with nucleus N and let 1, be a secant of the regular 
hyperoval %= % U {N}. Let AG(2, q*) = PG(2, q*)\l, and let X= %X,. Let $.% be a 
derivation set contained in 1,. If X is a translation q*-arc in 9AG(2, q*) then N E 1, in 
PG(2, q*). To see why this is so, note that AG(2, q*) and gAG(2, q*) have the same 
translation group (with axis I,) (see [7] or [lo]), so that the translation q*-arc X in 
BAG(2, q*) is also a translation q2-arc in AG(2, q*). But a translation q*-arc contained 
in the regular hyperoval % of PG(2, q2) cannot contain N, for Aut( X) = Aut( %‘) fixes 
N, while the stabiliser of a translation q*-arc is transitive on its points. Thus N E 1,. 
THEOREM 2.1. Let 9 be a derivation set contained in a line 1, of PG(2, q*), where 
q 3 4 is even. Let % be a conic in PG(2, q*) which contains a point of 1,\9 and the 
nucleus N of which is contained in 9. Let 2 = V U {N} and let 38 = %‘U,. Then X is a 
translation q*-arc in BAG(2, q*) which k uniquely completable to a translation 
hyperoval in Hall(q*). Furthermore, any two hyperovals in HalI arising from this 
construction are equivalent under the action of Aut(Hall(q*)). 
PROOF. First we show that X is a q*-arc in Hall(q*). The ideal line of Hall(q*) 
contains no point of X, so we consider the lines of the corresponding affine plane 
9AG(2, q2). A line of BAG(2, q*) which is a line of AG(2, q2) the ideal point of 
which is not in 9 contains at most two points of X, for X is an arc in AG(2, q*). Now 
let 1 be a line of BAG(2, q2) which is the set of points of an affine Baer subplane B of 
AG(2, q*), and suppose that 1 contains three points of X’. Now the projective 
completion a of B contains three points and the nucleus of 2, and since a regular 
hyperoval is determined by three of its points plus a nucleus, so %‘rl a is a hyperoval 
of B. Since a hyperoval has only O-secants and 2-secants, 9 n 1, = 9 must be a 
Hyperovals in Hall planes 197 
2-secant of % n @, which contradicts the hypothesis that % contains a point of l,\ZB. 
Hence no line of Hall(q’) contains three points of X, so that X is a q*-arc of Hall(q*). 
Let P = 5% rl 1, in PG(2, q*). Each line in the parallel class of lines in AG(2, q*) with 
ideal point P is a l-secant of X in AG(2, q*). But these lines form a parallel class of 
lines in BAG(2, q*) with ideal point P’, say. Thus XU {P’} is a (q* + 1)-arc in 
Hall(q*). Now a (q* + 1) arc in a projective plane of order q* is uniquely completable 
to a (q* + 2)-arc [6], so there exists a unique point Q’ on the ideal line of HalI such 
that X u {P’, Q’} is a hyperoval in Hall(q*). This is the unique hyperoval in Hall(q’) 
containing X, as two hyperovals have at most half their number of points in common. 
By the above arguments, since X is a translation hyperoval of PG(2, q*), it is a 
translation hyperoval of Hall(q*). 
Next we show that any two hyperovals arising by this construction are equivalent 
under the action of Aut(Hall(q’)). First, PGL(3, q2) is a subgroup of PTL(3, q*) and is 
transitive on the set of tonics [6, Theorem 7.2.11. Also, the stabiliser of a conic in 
PGL(3, q*) is transitive on the tangents to that conic [6, Theorem 7.2.3, Corollary 81, 
and the nucleus of a conic is fixed by the stabiliser of that conic. Let Ce be a conic, let f 
be a line tangent to U: at a point P, and let N denote the nucleus of %. We now show 
that the stabiliser PGL(3, q*)& of V and t in PGL(3, q*), in its action on t, is 
transitive on derivation sets on t containing N but not containing P. 
First, IPGL(3, q*)%,,l = q*(q* - l), and there are q* elations with axis t fixing %, so 
IPGL(3, q*)&l = q* - 1. Let G denote PGL(3, q’)l&. We use the orbit-stabiliser 
theorem to show that G is transitive on the q* - 1 derivation sets containing N but not 
P, by showing that the stabiliser of such a derivation set in G is trivial. Now G is cyclic 
oforderq*- 1 and any non-trivial element g E G fixes only N and P. Thus all orbits of 
G on t\{N, P} have the same length, which must be a divisor of q* - 1. So any union 
of orbits of G on t\{N, P} has length not coprime to q’ - 1. 
Let 9 be a derivation set on t containing N but not containing P. Then G, = G9,1NI, 
for any three points of t determine a derivation set. Since 19\(N)] = q, which is 
coprime to q* - 1, it follows that 9\(N) is not a union of orbits of G on t\{N, P}, and 
hence is not stabilised by any non-trivial element of G. Thus G9,(NI = 1, so that 
G9 := 1. 
We have shown that PGL(3, q*) is transitive on the set of configurations of 
PG(2, q’) formed by a conic % with a tangent t at a point P, and a derivation set on t 
containing the nucleus N of % but not containing P. So, PGL(3, q2)a is transitive on 
the set of hyperovals of PG(2, q*) described in the statement of the theorem, and 
hence Aut(Hall(q*)) is transitive on the set of q*-arcs arising from them, by [7] or [lb]. 
Therefore, Aut(Hall(q2)) is transitive on the set of hyperovals constructed by the 
method described in this theorem (by the uniqueness of the completion of the 
constructed q*-arcs to hyperovals of Hall(q)). -I 
THEOREM 2.2. Let 9 be a derivation set contained in a line 1, of PG(2, q2), where 
q 2 4 is even. Let % be a conic in PG(2, q*) which contains a point of 9 and the nucleus 
of which is contained in l,\g. Let 5? = %’ U {N} and let X = al,. Then X is u 
translation q*-arc in gAG(2, q*) which is uniquely completable to a translation 
hyperoval in Hall(q*). Furthermore, any two hyperovals in Hall(q*) arising from this 
construction are equivalent under the action of Aut(Hall(q2)). 
PKOOF. The proof is similar to the proof of Theorem 2.1. The only major difference 
is in the proof that a line 1 of BAG(2, q*) which is the set of points of a Baer subplane 
9? of AG(2, q*) contains at ‘most two points of the q*-arc 2. Suppose that 1 contains 
three points of X, so that the projective completion a of 5% contains four points of 5%’ 
and has 1, n .%I = CJ as a tangent line. But four points plus a tangent line determine a 
198 C. M. O’Keefe, A. A. Pam&o and T. Penttila 
regular hyperoval, so 8 II %’ is a hyperoval of 3. This means that ?J has two points of 
%‘, contrary to the hypothesis. The rest of the argument follows, interchanging the 
roles of N and P where necessary. 0 
Each member of the family of parabolas in AG(2, 22m) given in [9] and [3] is still an 
arc in Ha11(22”) and completes to a translation hyperoval of Ha11(22”), since it is an 
instance of the construction described in Theorem 2.2. The tonics used by Korchmaros 
and C&male have homogeneous equations of the form %: x2 + sxz + yz = 0, where 
s E GF(22m)\GF(2m), and the derivation set used is ((0, 1, 0)) U ((1, n, 0): n E 
GF(2”)). (In the notation of Crismale [3] these are the curves with g = 1 and hence 
r = 2.) The derivation set ??J is on the line z = 0, which is tangent to %. Also, % 
contains a point (0, 1,0) of 58 and has nucleus (1, -s, 0) which is not contained in 9. 
THEOREM 2.3. A hyperoval in Hall(q2) constructed as in Theorem 2.1 b inequivalent 
under the action of Aut(Hall(q2)) to a hyperoval constructed in the same plane by the 
method of Theorem 2.2. 
PROOF. Let ,%I and g2 be hyperovals of Hall(q2), constructed as in Theorem 2.1 
and Theorem 2.2, respectively. Let Xi = %i\l,. Suppose that %I and g2 are equivalent 
under the action of Aut(Hall(q’)), so there exists an element g E Aut(Hall(q2)) such 
that g%, = %, and g%‘, = X2. Since every element of Aut(Hall(q2)) is inherited from 
an element of PTL(3, q2)9, it follows that there exists an element h E PTL(3, q2), such 
that hXI = X2, and so h& = % since h& and & are contained in unique hyperovals 
,%I and g2 of PG(2, q2). But the nucleus of %I is contained in 9 and the nucleus of $& 
is not contained in 58, while h preserves 9. Since a regular hyperoval has a unique 
nucleus, this is a contradiction, and the result is proved. q 
EXAMPLE 2.4. Let I, be the line z = 0 of PG(2, 16), and let 9 = ((0, 1, 0)) U 
((1, x, 0): x E GF(4)) be a derivation set on 1,. Here we give two tonics in PG(2, 16) 
through the same point P E 1,\9 and with the same nucleus in 9 which, under the 
construction in Theorem 2.1, have only one common point on the ideal line of the 
corresponding Hall plane Hall( 16). 
Let w be a primitive element for GF(16) satisfying w4 + o + 1 = 0. The tonics are: 
ce,: wx2+ w3y2+z2+yz =o and %$e,: 02x2 + 6J4y2 + z2 + yz = 0. 
Each conic contains the point P = (w, 1,0) and has nucleus N = (1, 0,O). By Theorem 
2.1, these give rise to hyperovals X1 and %$ in Ha11(16), with a common point P’ on 
the ideal line. However, the other ideal point of X’, is distinct from the other ideal 
point of &. 
ACKNOWLEDGEMENT 
The second author acknowledges the support of the International Development 
Program of Australian Universities and Colleges. 
REFERENCES 
1. A. A. Bruen and J. C. Fisher, Arcs and ovals in derivable planes, Math. Z., 125 (1972). 122-128. 
2. W. Cherowitzo, Hyperovals in the translation planes of order 16, JCMCC, 9 (1991) 39-55. 
3. M. Crismale, (4’ + q + I)-insiemi di tipo (0, 1,2, q + 1) e ovali nel piano di Hall di ordine pari q’, Nor. 
Mat., 1 (1981), 127-136. 
4. P. Dembowski, Finite Geomern’es, Springer-Verlag, Berlin, 1968. 
Hyperovals in Hall planes 199 
5. R. H. F. Denniston, Some non-Desarguesian translation ovals, Ars Combin., 7 (1979), 221-222. 
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S, G. Korchmaros, Ovali nei piani di Hall di ordine dispari, Afti Accad. Naz. Lincei Rend., (8) 56 (1974). 
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11. C. M. O’Keefe and A. A. Pascasio, Images of tonics under derivation, submitted. 
12. S. E. Payne, A complete determination of translation ovoids in finite Desarguian planes, Arfi Accad. 
Naz. Lincei Rend., 51 (1971) 328-331. 
Received 11 March 1991 and accepted in revised form II December 1991 
CHRISTINE M. O’KEEFE 
Department of Pure Mathematics, 
The Universiry of Adelaide, 
G. P. 0. Box 498, Adelaide, 
S.A. 5001, Australia 
ARLENE A. PASCASI~.) 
Mathematics Departmenr, 
De La Salle University, 
P.O. Box 3819. 
Manila 1004, Philippines 
TIM PENTTILL\ 
Department of Mathematics. 
The University of Western Australia, 
Nedlands, W.A. 6009, Australia 


Hyperovals in Hall planes 

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Hyperovals in Hall planes

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