Unitals in the Desarguesian projective plane of order 16

Journal of Statistical Planning and Inference 144 (2014) 110–122

Contents lists available at ScienceDirect

Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi

Unitals in the Desarguesian projective plane of order 16 John Bamberg a, Anton Betten b,n, Cheryl E. Praeger a, Alfred Wassermann c a

Centre for Mathematics of Symmetry and Computation, The University of Western Australia (M019), 35 Stirling Highway, Crawley, WA 6009, Australia b Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA c ~ Bayreuth, D-95440 Bayreuth, Germany Mathematisches Institut, Universitat

a r t i c l e in f o

abstract

Available online 23 October 2012

Using computer, we classify the unitals in the Desarguesian projective plane of order 16. We use computational methods based on analysis involving tactical decompositions to break symmetry, making a computer search feasible. We prove that all unitals in PGð2,16Þ are known, namely, up to isomorphism, there are exactly two Buekenhout– Metz unitals (one of them the Hermitian curve). & 2012 Elsevier B.V. All rights reserved.

Keywords: Projective plane Unital Buekenhout-Metz

1. Introduction and statement of results Let p ¼ PGð2,q2 Þ be the Desarguesian projective plane of order q2. Up to isomorphism, the Hermitian curve Hð2,q2 Þ in p qþ1 is the set of points ða, b, gÞ satisfying the equation aq þ 1 þ b þ gq þ 1 ¼ 0. This is an example of a unital, i.e., a set of q3 þ1 points U such that for any line ‘ of p, the intersection ‘ \ U is either a single point or consists of exactly q þ 1 points. In particular, the intersection sets ‘ \ U of size q þ 1 form the blocks of a design. The Hermitian curve is called the classical unital, and there are others, most notably the examples due to Buekenhout and Metz. For more details about unitals we refer to Barwick and Ebert (2008). By a result of Penttila and Royle (1995), there are exactly two unitals in PGð2,9Þ, both Buekenhout–Metz (one of them the Hermitian curve). For q¼4, the situation is similar. It is known that there are exactly two Buekenhout–Metz unitals in PGð2,16Þ (see Ebert, 1992), but the existence of a different kind of unital could not be ruled out so far. That there are no others in PGð2,16Þ is a consequence of our result: Theorem 1. In PGð2,16Þ, all unitals are known. They are the two Buekenhout–Metz unitals (one of them being the Hermitian curve). The remainder of this paper is devoted to the proof of this result. 2. Tactical decompositions and decomposition stacks Let us recall the notion of a tactical decomposition of an incidence structure (c.f. Moore, 1896a, 1896b; Dembowski, 1997). Consider a point-line incidence geometry ðP,LÞ. For simplicity, we identify the elements of L with subsets of P. For a point P, let ðPÞ be the set of lines through P. For a set of lines L, let ½L be the set of points incident with at least one member S of L, so that ½L ¼ ‘2L ‘. A linear space is a point-line incidence geometry with the following two properties: each pair of points determines a unique line and each line has at least two points. Examples of linear spaces are projective planes, in particular the Desarguesian planes PGð2,qÞ. n

Corresponding author. Fax: þ 1 970 491 2161. E-mail address: [email protected] (A. Betten).

0378-3758/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jspi.2012.10.006

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

111

A partition F of a set X is a collection of disjoint non-empty subsets, called classes, that cover X. We are only interested in partitions of finite sets, so that the number of classes is finite also. We write 9F9 for the number of classes of F. A partition C is said to be a refinement of F, indicated as C!F, if each class of F is a union of classes of C. Consider the partition Cð0Þ of X ¼ P [ L whose classes are the two sets P and L. A decomposition of ðP,LÞ is a partition C with C!Cð0Þ . To every decomposition C we can associate a partition VC ¼ ðV 1 , . . . ,V m Þ of the point set P and another partition BC ¼ ðB1 , . . . ,Bn Þ of the line set L. For a point P, and for B 2 BC , let r P,B ¼ 9ðPÞ \ B9 For a line ‘, and for V 2 VC , let k‘,V ¼ 9‘ \ V9 Define r P ¼ ðr P,B1 , . . . ,r P,Bn Þ, and k‘ ¼ ðk‘,V 1 , . . . ,k‘,V m Þ. The decomposition C is point-tactical (or row-tactical) if rP ¼rQ whenever P,Q 2 V i for some i. In this case, the assignment P/r P induces a function on the point classes VC , and we can define the induced mapping r : VC -Nn ,V/r V :¼ r P where P is a representative of V 2 VC . Likewise, the decomposition C is line-tactical (or column-tactical) if k‘ ¼ k‘0 whenever ‘,‘0 2 Bj for some j. In this case, assignment ‘/k‘ induces a function on the point classes VC , and we can define the induced mapping k : BC -Nm ,

B/kB :¼ k‘

where ‘ is a representative of B 2 BC . Often, one records these maps in the form of two matrices. The rows of this matrix correspond to the classes in VC , and the columns correspond to the classes in BC . The entry in row Vi and column Bj is r P,Bj in the first matrix (P 2 V i ) and it is k‘,V i in the second (‘ 2 Bj ). These matrices are the decomposition scheme. Depending on whether the decomposition is point-tactical or line-tactical, the decomposition scheme consists of either the first or the second of these matrices. A decomposition that is both point-tactical and line-tactical is simply called tactical. If a decomposition is tactical, the decomposition scheme consists of both matrices. We will append to each matrix an additional row on the top and an additional column on the left. The top row indicates the sizes of the line-classes, while the column on the left shows the sizes of the point-classes. In the top left corner, we use the symbol ‘-’ to indicate that the decomposition is point-tactical, and we use the symbol ‘k’ to indicate that the decomposition is line-tactical. Tactical decompositions arise frequently: let A be a group of automorphisms of ðP,LÞ. It is well-known that the orbits of A on points and on lines form a tactical decomposition. This decomposition is the tactical decomposition by automorphisms. The combinatorial structure of a unital is as follows: a line ‘ with 9U \ ‘9 ¼ q þ 1 is called a secant line. A line ‘ with 9U \ ‘9 ¼ 1 is called a tangent line. It is well-known that these are the only two possibilities for lines. The intersection sets U \ ‘, where ‘ runs through all secants, form the set of blocks of a 2ðq3 þ1,q þ1,1Þ design, called a unital design. Simple counting arguments show that there are q2 ðq2 qþ 1Þ secant lines and q3 þ 1 tangent lines. Moreover, a point on the unital is on q2 secants and on exactly one tangent line. A point off the unital is on q2 q secants and on q þ 1 tangents. We summarize this in the form of the following two decomposition schemes:

The set of tangent lines form a unital in the dual plane, called the dual unital. The dual unital may or may not be isomorphic to the unital. We use the notion of tactical decomposition stacks to describe the parameters of geometries. A decomposition stack is a sequence of decompositions Cð0Þ , Cð1Þ , . . . , CðhÞ with Cðu þ 1Þ !CðuÞ for u ¼ 0, . . . ,h1. The integer h is known as the depth of the decomposition stack. The use of decomposition stacks for the study of linear spaces has been suggested by Betten and Braun (1992). If C and F are decompositions we say that F is the coarsest row tactical refinement of C if 1. F!C, 2. F is row-tactical, 3. whenever X is row-tactical and X!C then X!F. The coarsest column-tactical refinement is defined in a similar way, requiring column-tacticality instead. A decomposition stack Cð0Þ , Cð1Þ , . . . , CðhÞ is called TDO (short for tactical decomposition by ordering) if 1. Fðu þ 1Þ is the coarsest row-tactical refinement of FðuÞ if u is even and

112

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

2. Fðu þ 1Þ is the coarsest column-tactical refinement of FðuÞ if u is odd. 3. The classes in Fðu þ 1Þ that descend from a given class of CðuÞ are ordered in lexicographic fashion according to the values of rV (V 2 VCðuÞ ) if u is even, and according to the values of kB (B 2 BCðuÞ ) if u is odd. The paper Betten and Braun (1992) considers as an additional requirement that the terminal decomposition CðhÞ be tactical in both way. By repeating the refinement procedure sufficiently often, this can always be achieved. Since this condition leads to TDO-stacks that are potentially very deep, this requirement is somewhat impractical for our purposes, and hence we decide to drop it. In this case, we speak of a tactical decomposition stack. In any case, the decomposition stack depends only on the initial decomposition Cð0Þ and on the lexicographic ordering used in the refinement steps. If A is the automorphism group of the incidence structure ðP,LÞ, then the stabilizer of CðhÞ in A is the same as the stabilizer of Cð0Þ in A. A synthesis algorithm for tactical decomposition stacks of linear spaces has been described in Betten and Betten (2010). 3. The Buekenhout–Metz unital The construction of the unitals attributed to Buekenhout (1976) and Metz (1979) rely on a three-dimensional model of the Desarguesian projective plane. Consider the projective line PGð1,q2 Þ; a set of q2 þ 1 one-dimensional subspaces of a two-dimensional vector space over Fq2 . If we now change the scalars to the subfield Fq , we have instead a fourdimensional vector space over Fq and the points of PGð1,q2 Þ then become q2 þ 1 pairwise disjoint lines of PGð3,qÞ; otherwise known as the Desarguesian spread S of PGð3,qÞ. We then embed PGð3,qÞ as a hyperplane H in PGð4,qÞ and extract from this geometry the points of PGð4,qÞ not in H, and the planes of PGð4,qÞ not in H that are incident with a line of S. This new incidence structure of points and planes can now be viewed as the points and lines (respectively) of the Desarguesian affine plane AGð2,q2 Þ. To complete this affine plane to the Desarguesian projective plane of order q2, we need only institute the spread S itself and its members as the line at infinity, and the incidence relation for all of these objects is natural. Objects of PGð4,qÞ

Objects of PGð2,q2 Þ

Points of PGð4,qÞ not in H Planes of PGð4,qÞ not in H,meeting H in an element of S

Affine points Affine lines

Elements of S

Points at infinity

S

The line at infinity

´-Bruck/Bose construction whereby translation planes can all be This construction is an instance of the famous Andre constructed from projective space PGð2t1,qÞ endowed with a spread of (t1)-dimensional subspaces. See Andre´ (1954) and Bruck and Bose (1964) for more. ´ map. The affine Andre´ map is a Let us now look at the map underpinning this isomorphism, which we will call the Andre one-to-one correspondence between the points of AGð4,qÞ and the points of AGð2,q2 Þ using field reduction. In the following, we choose a basis ðo1 , o2 Þ for Fq2 over Fq and define the mapping

c : Fq2 -ðFq Þ2 , a/ða1 ,a2 Þ

ð1Þ

where a ¼ a1 o1 þ a2 o2 . Let a, b be coordinates for AGð2,q2 Þ and a0 ,a1 ,b0 ,b1 be coordinates for AGð4,qÞ. The affine Andre´ map is the following: AGð2,q2 Þ-AGð4,qÞ : ða, bÞ/ða0 ,a1 ,b0 ,b1 Þ

ð2Þ

where ða0 ,a1 Þ ¼ cðaÞ, and ðb0 ,b1 Þ ¼ cðbÞ. Observe that this map is one-to-one and hence can be inverted. Let a0 ,a1 ,b0 ,b1 ,z be coordinates for V 5 ¼ F5q and let a, b, g be coordinates for V 3 ¼ F3q2 . Let PGð4,qÞ ¼ PðV 5 Þ and PGð2,q2 Þ ¼ PðV 3 Þ. Let H1 be the hyperplane at infinity z ¼0 in PGð4,qÞ and let ‘1 be the line at infinity g ¼ 0 in PGð2,q2 Þ. We define a map PGð2,q2 Þ\‘1 -PGð4,qÞ\H1 : ða, b, gÞ-ða0 ,a1 ,b0 ,b1 ,1Þ where ða0 ,a1 Þ ¼ cða=gÞ and ðb0 ,b1 Þ ¼ cðb=gÞ. This map can be seen as a version of the affine Andre´ map when fixing the line at infinity ‘1 to be g ¼ 0. Since this map is not defined for the points on the line at infinity, we will in a moment define an extension of this map. Consider the hyperplane H1 in PGð4,qÞ given by z ¼0. The underlying vector space is isomorphic to an AGð4,qÞ. Therefore, we may employ the inverse map of (2). Projectively, this induces a map from the points on H1 to the q2 þ1 points on ‘1 in PGð2,q2 Þ. Observe that this map is qþ 1 to 1. Namely, a point in PGð2,q2 Þ with g ¼ 0 corresponds to q þ1 collinear points on H1 in PGð4,qÞ. That is, we have a one-to-one correspondence between the elements of a fixed spread of lines of H1 ¼ PGð3,qÞ and the points of ‘1 in PGð2,q2 Þ. At the level of points, we have a correspondence between PGð4,qÞ

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

113

and PGð2,q2 Þ. This correspondence is one-to-one if restricted to AGð4,qÞ on one side and AGð2,q2 Þ on the other side. It is q þ1 to one between the points on H1 and the points on ‘1 . Another way of looking at this is as follows: The Andre´ map can be extended in the domain, namely, to the set of points of PGð2,q2 Þ, and in the range, namely, by including the elements of a fixed spread in H1 ¼ PGð3,qÞ as possible images of points on ‘1 . In this sense we have a one-to-one map between AGð4,qÞ [ S and PGð2,q2 Þ. Of course, upon identification of spread elements with the sets of points that they consist of, this map and the relation from above become the same object. Under the mapping described above, planes of AGð4,qÞ that are cosets of spread elements become lines of AGð2,q2 Þ and hence (after completion) lines of PGð2,q2 Þ. Namely, if d is a spread line, the parallel class of planes of the form x þ d (x 2 AGð4,qÞ) form the pencil of lines through the point at infinity corresponding to d. Projectively, this means that planes in PGð4,qÞ intersecting H1 in a line of the spread become lines of PGð2,q2 Þ. Let S be a Desarguesian spread of lines in H ¼ PGð3,qÞ. Embed H as a hyperplane in PGð4,qÞ. Then we have an isomorphism between AGLð2,q2 Þ and the stabilizer of S inside PGLð5,qÞ, which we denote as GS , for short. Both groups give rise to tactical decompositions. On the one hand, we have the affine group inducing the decomposition of the projective plane PGð2,q2 Þ with a distinguished line as

ð3Þ

On the other hand, we have the decomposition of the point/line incidence structure PGð4,qÞ given by the group GS , which is

ð4Þ

Buekenhout’s (1976) approach was to take the Andre preimage of a unital in PGð2,q2 Þ. Here, we consider only the case q of a unital U with one point at infinity, such as aq þ 1 þ b þ b ¼ 0 (whose point at infinity is ð0,1,0Þ). He observed the following: the set 1

U 0 :¼ Andre

ðUÞ

ð5Þ

is a cone over an elliptic quadric. It has size q3 þ qþ 1, since the unique point at infinity ð0,1,0Þ has a preimage of qþ 1 1 points under the Andre´ map. Namely, the line s :¼ Andre ð0,1,0Þ (here, we identify a line with the q þ 1 points on it) is a member of the Desarguesian spread in H1 which we call the special generator. There is a special point V on this line, called the vertex, and U 0 consists of q2 þ 1 lines through V, called generators. The other q2 lines are affine lines, we call them the ordinary generators. Modulo V, the set U 0 has the structure of an elliptic quadric. In terms of the unital design, it corresponds to the q2 blocks through ð0,1,0Þ (c.f. Fig. 1, where we have chosen a PGð3,qÞ containing an elliptic quadric, so that there is a second point on the line s, the point of tangency W). Let us now collect some facts about the Hermitian curve. In PGð2,q2 Þ, all secants intersect the Hermitian curve in Baer sublines. The Andre´ preimage of lines ‘ 2 PGð2,q2 Þ are planes in AGð4,qÞ whose projective completion is a spread line (namely, the spread line corresponding to ‘ \ ‘1 ). Consider any secant ‘ not through ð0,1,0Þ. Under the inverse Andre´ map, the Baer subline ‘ \ U corresponds to a conic in the plane in AGð4,qÞ that corresponds to ‘. Observe that this conic avoids the line at infinity. Thus, we may consider the conic as a conic in AGð2,qÞ.

Fig. 1. The configuration of Buekenhout.

114

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

Metz (1979) used the idea of Buekenhout to create new unitals. By forcing a secant line to intersect the unital in something other than a Baer subline, a unital can be constructed that is not a Hermitian curve. That this is almost always possible follows from the fact that there are more conics in AGð2,qÞ than there are Baer sublines of PGð1,q2 Þ provided q 42. This led to the following construction: Consider a plane p in the factor space PGð3,qÞ of lines through V in PGð4,qÞ. Let AGð2,qÞ be the affine part of p that is disjoint from the hyperplane H1 . For q4 2, there exists a conic in this AGð2,qÞ whose Andre´ image is not a Baer subline (because of the counting result mentioned before). We need more. Recall that s is the spread line through V, and write smodV for the point in the factor space PGð3,qÞ. We construct an elliptic quadric O in this PGð3,qÞ such that 1. O \ p is the conic that we have chosen, 2. sðmod VÞ 2 O, 3. the hyperplane at infinity H1 ðmod VÞ is the tangent plane to O at sðmod VÞ. Metz shows that this is always possible. Consider the preimage of this elliptic quadric under the factor map. This is a cone with base an elliptic quadric and vertex the point V. The Andre´ image of this cone is a unital. This unital possesses a secant (namely, ‘) that does not intersect in a Baer subline (because of the choice of the conic in p). Thus, this unital is not Hermitian. In Ebert (1992), a model for the Buekenhout–Metz unitals in even characteristic planes is presented (for odd characteristic, Baker and Ebert, 1992 does the job). In this model, one chooses two parameters m, n 2 Fq2 , n2 = Fq with ! trace

mq þ 1 ¼0 ðnq þ nÞ2

For z 2 Fq2 and r 2 Fq , let 2

qþ1

P z,r ¼ ðz, mz þ nz

þr,1Þ

The unital U m, n is defined as fð0,1,0Þg [ fPz,r 9z 2 Fq2 , r 2 Fq g It is known that the unital U m, n is classical if and only if m ¼ 0. The elliptic quadric that gives rise to the unital U m, n is given by the equation f ða0 ,a1 Þ þ b1 z ¼ 0 where f ða0 ,a1 Þ ¼ ðm1 þn1 Þa20 þn1 a0 a1 þ ðm0 þ m1 þ ðm1 þ n1 ÞhÞa21 Here, with ða, b,1Þ affine coordinates for PGð2,q2 Þ\‘1 , the mapping (1) yields cðaÞ ¼ ða0 ,a1 Þ, cðbÞ ¼ ðb0 ,b1 Þ, cðmÞ ¼ ðm0 ,m1 Þ, cðnÞ ¼ ðn0 ,n1 Þ. In all of this, we have o 2 Fq2 \Fq with oq ¼ 1 þ o and h ¼ o þ oq with h þ h2 ¼ 1 but ha1. Ebert shows that this choice of o is always possible, and for q¼4 we may take o satisfying o4 ¼ o þ1 as above. The ‘gold plated’ copies of the two known unitals in PGð2,16Þ are the Hermitian curve H4 ¼ U 0, o

ð6Þ

and the second Buekenhout–Metz unital BM4 ¼ U 1, o

ð7Þ

Every block in a Hermitian unital is an Hð1,qÞ, which is the same as a Baer subline. We record some facts about the two known unitals in PGð2,16Þ. Lemma 2. The automorphism group of H4 ¼ U 0, o has order 249,600 and is transitive on points and on tangent lines. The automorphism group of BM 4 ¼ U 1, o has order 768. It has four orbits on points of PGð2,16Þ and four orbits on lines. The unital points, off unital points, secant lines, tangent lines, each fall into two orbits. The tactical decomposition by automorphisms is

ð8Þ

The vertex of the cone in the Andre´ model is the fixed point. The orbit on secant lines of length 16 corresponds to the 16 ordinary generators of the cone over the elliptic quadric. The 192 other secant lines form the second orbit. The unital H4 has all its blocks Baer sublines. The unital BM4 has 16 Baer blocks, corresponding to the orbit of length 16 on secant lines.

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

115

Remark 3. The unitals H4 and BM4 have the property that the q2 ¼ 16 secant lines through a point of the unital induce blocks that are Baer sublines. A result from Casse et al. (1996) shows that this property characterizes the class of Buekenhout–Metz unitals for q even. Since we are interested in excluding non-Buekenhout–Metz unitals in PGð2,16Þ, we could use this fact as an additional condition. However, we do not make use of this result. 1

Recall that the set U 0 ¼ Andre ðUÞ from (5) is a cone over an ellipic quadric. If q ¼ 4, the stabilizer of the cone U 0 has order 12,533,760. The elliptic quadric in PGð3,4Þ is stabilized by PGLð2,16Þ of order 16,320. This group is a section of the group that stabilizes the cone. The stabilizer of a PGð3,4Þ not containing the vertex is a subgroup of index 256. Inducing this subgroup on the factor space modulo the vertex gives the stabilizer of the elliptic cone, with a kernel of order 3. The stabilizer PGLð5,qÞS,U0 of the spread S and the unital preimage U 0 in PGLð5,qÞ is isomorphic to the stabilizer of a tangent line of the unital U. For q ¼ 4, there are several possibilities. If U ¼ H4 , the group PGLð5,qÞS,U0 has order 3840. For U ¼ BM4 , we have to distinguish cases, since the stabilizer of the unital BM4 has two orbits on tangent lines if q¼ 4. If the fixed line is at infinity, the stabilizer PGLð5,qÞS,U 0 has order 768. If a tangent line at infinity belongs to the orbit of length 64, the group PGLð5,qÞS,U 0 has order 12.

4. Starter configurations In the following, we will consider putative parameters of unitals that refine the initial decomposition scheme. The parameters we consider are tactical decomposition stacks as defined in Section 2. Starting from the decomposition with two row-classes and two column classes, we make additional choices to arrive at starter configurations. Any unital must admit at least one of these decomposition stacks. Our goal is to create decomposition stacks that are sufficiently fine, such that the classification problem of unitals admitting any of these stacks is manageable. We use the synthesis algorithm from Betten and Betten (2010) to create refined decompositions. Later, we classify the possibilities for these starter configurations up to the group of the plane. Let C ¼ ðC0 , C1 , C2 , C3 Þ be the partition stack associated with the tactical decomposition of a putative unital U as in (1). C0 is the set of points of U, C1 is the set of secant lines, C2 is the set of points not in U and C3 is the set of tangent lines. In the following schemes, the subscripts refer to the classes of C. Thus, 2081 represents C1 , which is a class of size 208.

We decide to select two points from the unital and distinguish them from the other unital points. In the partition, this is realized by splitting off the 63 unital points that are not selected. This means that there is now a new class C4 that contains these 63 points. The old class C0 only contains the two selected points, which we call P1 and P2. Thus, we have the following row-tactical decomposition

Here, a horizontal line is introduced to separate points on the unital and points off the unital. Likewise, a vertical line is introduced to distinguish between secant lines and tangent lines. This decomposition is row-tactical but not column-tactical, so we can refine again. This yields the following columntactical decomposition.

Here, the class C1 is split as C1 -C1 [ C5 [ C6 , with C1 the line ‘0 through P1 and P2, C5 the secants through either P1 or P2, and C6 the remaining secants. Similarly, the tangent line-set C3 splits as C3 -C3 [ C7 , with C3 the two tangents ‘1 on P1 and ‘2 on P2 and C7 the rest.

116

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

Fig. 2. Four refinements after three steps.

After another refinement step, we find the following row-tactical decomposition:

Here, C4 splits as C4 -C4 [ C8 with C4 ¼ fP 3 ,P4 ,P 5 g. Thus, we have exhibited the block B0 ¼ fP 1 , . . . ,P5 g ¼ U \ ‘0

ð9Þ

which is the intersection of the unital and the line ‘0 , Also C2 -C2 [ C9 [ C10 [ C11 . Here, C2 are the 12 points on ‘0 that are off the unital, C11 is the point Q off the unital where the tangent lines ‘1 and ‘2 meet and C10 are the remaining 30 points on ‘1 and ‘2 . The next refinement is no longer unique. Let T be the set of tangent lines on Q and put T 1 ¼ f‘ 2 T 9‘ \ ‘0 2 B0 g,

T 2 ¼ f‘ 2 T 9‘ \ ‘02 = B0 g

ð10Þ

so that T ¼ T 1 [ T 2 . We let T i ¼ f‘ \ U9‘ 2 T i g

i ¼ 1,2

ð11Þ

and put T ¼ T 1 [ T 2 . Observe that T 2 \ ‘0 ¼ ptyset. Put t i ¼ 9T i 9 ¼ 9T i 9

i ¼ 1,2

ð12Þ

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

117

so 2 r t 1 r 5. These cases correspond to four possible column-tactical refinements, as shown in Fig. 2. (The column classes that make up T i are indicated using bold face.) In each of the four cases, we can perform yet another refinement step. The number of refinements is 23 for case t 1 ¼ 2, five for case t 1 ¼ 3, and one refinement for each of case t 1 ¼ 4 and case t 1 ¼ 5. Since we use these decompositions only implicitly, there is no need to display these 30 cases. So far, we have distinguished two points in the unital. This led to a number of tactical decompositions. These tactical decompositions are used for the purposes of classification. Each decomposition is considered in turn, and choices are made in all possible ways to satisfy a given decomposition. A choice corresponds to a set-partition C that is compatible with the given decomposition. Two choices are equivalent if one can be obtained from the other under a collineation of the plane. For this reason, we consider the orbits of the collineation group on all possible choices C that are compatible with one tactical decomposition at a time. For the sake of efficiency, we decide to split the search into three parts. In the first part, the set-partition C is determined only partially, but we consider the orbits of the collineation group G on the set of possible partitions. In the second part, we determine the set-partition C completely, but we no longer do isomorphism testing. In Part III, the setpartitions C that were found in Part II are checked for possible isomorphisms.

5. The search: Part I Part I of the search amounts to making all choices for C0 , C4 , C2 , C10 , C11 in Fig. 2. The two large classes C8 and C9 will be decided upon in Part II. Let G ¼ PGLð3,q2 Þ be the full collineation group of PGð2,q2 Þ. We proceed in 6 steps: Algorithm. The classification algorithm in Part I. Step 1: Up to projective equivalence, choose a line ‘0 . This line defines the class C1 . Let Gð1Þ ¼ G‘0 be the stabilizer of ‘0 in G. Step 2: Up to equivalence under Gð1Þ , choose a set B0 of q þ1 points on ‘0 . This set defines the union C2 [ C3 . Let ð1Þ Gð2Þ ¼ Gð1Þ B0 ¼ G‘0 ,B0 be the stabilizer of B0 in G . Step 3: Up to equivalence under Gð2Þ , choose 2 points P1 ,P 2 2 B0 . This set defines the class C2 . Let Gð3Þ ¼ Gð2Þ fP 1 ,P 2 g be the stabilizer of fP1 ,P 2 g in Gð2Þ . Step 4: Up to equivalence under Gð3Þ , choose a point Q2 = ‘0 . This point defines the class C11 . Let Gð4Þ ¼ Gð3Þ Q be the stabilizer of Q in Gð3Þ . Step 5: Up to equivalence under Gð4Þ , select a set T of q þ 1 lines from the pencil ðQ Þ containing the lines P 1 Q and P 2 Q . ð4Þ These lines in T serve as the q þ 1 tangent lines through Q. Let Gð5Þ ¼ Gð4Þ T be the stabilizer of T in G . Step 6: Up to equivalence under Gð5Þ , choose a set T2 of t2 points, different from Q and not on ‘0 , one on each line of T 2 , to ð5Þ serve as points of tangency for the lines in T 2 . Let Gð6Þ ¼ Gð5Þ T 2 be the stabilizer of T2 in G .

We will now give some detailed comments on the way this algorithm proceeds. The group G ¼ PGLð3,q2 Þ is transitive on lines, so we decide to choose ‘0 to be the line g ¼ 0. The stabilizer is Gð1Þ ¼ AGLð2,q2 Þ. For q¼4, this group has order 62,668,800. Under this group, there are 3 possibilities for subsets B0 of ‘0 of size 5, see Table 1 (where ‘s.o.’ is the stabilizer order, ‘o.l.’ stands for orbit length, and ‘c.t.’ stands for circle type, see below). As Sherk points out in Sherk (1983), these three types can be distinguished by their circle-type. The circle-type is an isomorphism invariant. To define it, regard PGð1,q2 Þ as the inversive plane IðqÞ. This is a 3-design whose blocks are the Baer sublines (for details, we refer to Dembowski, 1997). For a set S D PGð1,q2 Þ, let ci be the number of circles that intersect S in exactly i points. We use exponential notation ici to indicate that there are ci circles that each intersect the set in i points. The set R2ð0Þ has 10 circles, each intersecting the set in three points. The set R2ð1Þ has one circle with four points and six circles with three points each. The set R2ð2Þ is a Baer subline itself. The possibilities for pairs of points in each of these three cases are listed in Table 2. In all seven cases, the stabilizer Gð3Þ is transitive on the 256 points off ‘0 . Thus, we may take Q ¼ ð0,0,1Þ in all cases. This is the representative R4 at level 4, see Table 3. At depth 5, we find 1778 cases, with an average stabilizer order of 16.15. At depth 6, we find 199,496 cases total, with an average stabilizer order of 1.01. Table 1 The possibilities for the first block. C2

R2ð ¼ B0 Þ

s.o.

o.l.

c.t.

0/3 1/3 2/3

fð1,0,0Þ,ð0,1,0Þ,ð1,1,0Þ,ðo,1,0Þ,ðo4 ,1,0Þg fð1,0,0Þ,ð0,1,0Þ,ð1,1,0Þ,ðo,1,0Þ,ðo5 ,1,0Þg fð1,0,0Þ,ð0,1,0Þ,ð1,1,0Þ,ðo5 ,1,0Þ,ðo10 ,1,0Þg

30,720 15,360 921,600

2040 4080 68

(310) (36,41) (51)

118

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

Table 2 The possibilities for pairs of points in B0. C2

C3

G3

R3ð ¼ fP 1 ,P 2 gÞ

s.o.

o.l.

0/3 0/3 0/3

0/3 1/3 2/3

0/7 1/7 2/7

fð1,0,0Þ,ð0,1,0Þg fð0,1,0Þ,ð1,1,0Þg fð0,1,0Þ,ðo,1,0Þg

7680 15,360 7680

4 2 4

1/3 1/3 1/3

0/3 1/3 2/3

3/7 4/7 5/7

fð1,0,0Þ,ð0,1,0Þg fð1,0,0Þ,ð1,1,0Þg fð1,0,0Þ,ðo,1,0Þg

7680 3840 3840

2 4 4

2/3

0/1

6/7

fð1,0,0Þ,ð0,1,0Þg

92160

10

Table 3 The possibilities for the point Q. G4

R4 ð ¼ fQ gÞ

s.o.

o.l.

0/7 1/7 2/7

{(0,0,1)} {(0,0,1)} {(0,0,1)}

30 60 30

256 256 256

3/7 4/7 5/7

{(0,0,1)} {(0,0,1)} {(0,0,1)}

30 15 15

256 256 256

6/7

{(0,0,1)}

360

256

Table 4 The eight equations. P

xP ¼ q3 þ 19A9

P2C

P

y‘ ¼ q3 þ 19B9

‘2C

P

xP ¼ q þ 19‘ \ A9

ð8‘ 2 AÞ

P2‘\C

P

xP ¼ 19‘ \ A9

P2‘\C

qy‘ þ P

P

ð8‘ 2 BÞ

xP ¼ q þ 19‘ \ A9

ð8‘ 2 CÞ

P2‘\C

y‘ ¼ 19ðPÞ \ B9

ð8P 2 AÞ

‘2ðPÞ\C

P

y‘ ¼ q þ 19ðPÞ \ B9

‘2ðPÞ\C

P

qxP þ

ð8P 2 BÞ

y‘ ¼ q þ 19ðPÞ \ B9

ð8P 2 CÞ

‘2ðPÞ\C

6. The search: Part II We will now determine the set of unitals associated to a starter configuration from Part I. In terms of the decomposition schemes in Fig. 2, we decide upon the sets C8 and C9 . This means that we have to choose a subset C8 of size 60 from a set of size 225. We do not use the group any more, mostly because isomorphism testing is expensive and also because almost all starter configurations have trivial automorphism group, so that the gain no longer justifies the means. Instead, we decide to transform the problem into one of solving a system of linear Diophantine equations, something for which standard software is available. In detail, we proceed as follows: Let A be the known unital points at this time (i.e., the points of B0 , and the t2 points from Step 6). Let B be the known not-unital points at this time. That is, the point Q , and the points on the tangent lines through Q other than the points of tangency. Specifically, B ¼ ½T \ðfQ g [ B0 Þ Let C be the set of undecided points (the rest). Let A be the set of known secants at this time: 1. The line ‘,

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

119

Table 5 The cases of configurations with solutions. Case

C1

C2

C3

C4

C5

C6

70,341 141,518 156,661 197,382

0/1 0/1 0/1 0/1

1/3 1/3 1/3 2/3

0/3 1/3 2/3 0/1

0/1 0/1 0/1 0/1

104/245 427/455 156/455 11/26

90/120 123/225 60/225 0/1

Table 6 The Diophantine systems with solutions. Case

Size of system

# 1’s

# solutions

70,341 141,518 156,661 197,382

536  366 536  366 536  366 536  360

117 117 117 120

1 1 1 78

Table 7 The identification of solutions.

2. 3. 4. 5.

Case

Arises in

70,341 141,518 156,661 197,382

BM4 BM4 BM4 BM4 , H4

the q2 lines on either P1 or P2 that do not pass through Q , the q2 q lines ðQ Þ\T . For each point P 2 B0 \fP 1 ,P2 g such that PQ 2 T , the q2 lines on P that are not in T . Lastly, for each of the t2 points P 2 T 2 , the q2 lines on P that are not in T .

Let B be the set of known tangents at this time, i.e., B ¼ T , let C be the set of undecided lines (i.e., the rest), and let U be the unital that we wish to construct. We assume that A D U. The equations that we establish have Boolean decision variables of two types. The first set of variables represents the choice of points from C that – together with the points in A – form the unital U. Specifically, for P 2 C, the variable xP is 1 if P is chosen to be part of the unital (otherwise, xP ¼0). The second set of variables represents the choice of lines from C that – together with lines in B – form the tangent lines. For ‘ 2 C, the variable y‘ is one if ‘ is a tangent line to U (otherwise, y‘ ¼ 0). The equations are of eight types, listed in Table 4. Using the above mentioned LLL-based solver for linear Diophantine systems of equations (Wassermann, 1998), together with the commercial solver Gurobi, we determine that only four configurations give rise to unitals, see Table 5. The configurations are the orbits under G ¼ PGLð3,16Þ from Step I. The size of the coefficient matrix of the respective systems and the number of decision variables that have to equal one are listed in Table 6. The column ‘# solutions’ is the number of unitals (not isomorphism types) that contain a particular instance of this configuration. These numbers were determined with the LLL-based solver, which is able to determine the complete solution set of the respective systems. 7. The search: Part III—classification In this final step, we will identify the unitals that we found in Part II. We will prove that the solutions in the four cases belong to the unitals H4 and BM4 as indicated in Table 7. The first three configurations have t 1 ¼ 2, the last one has t 1 ¼ 5. Observe that the last configuration appears in both unitals. In fact, it appears in BM4 in two different ways. More information about these cases is given in Table 8. In that table, GC n is the stabilizer of the configuration C n . This group has been called Gð6Þ in Table 5. In case 197,382, B0 is a Baer subline and t 1 ¼ 5. That is, the tangent lines on Q are tangent to the unital in the 5 points of B0 . Any line in the Hermitian unital has this form, with Q ¼ ‘0? . For each unital, we count the number of secants that induce Baer sublines as blocks. We summarize this information in Table 9. The three solutions with 208 Baer sublines in case 197,382 are Hermitian unitals. The corresponding three

120

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

Table 8 More information about the soluble cases. Case n

t1

t2

Q

T2

9GC n 9

70,341 141,518 156,661 197,382

2 2 2 5

3 3 3 0

(0,0,1) (0,0,1) (0,0,1) (0,0,1)

fðo14 ,1,1Þ,ðo6 , o8 ,1Þ,ðo4 , o10 ,1Þg fðo3 ,1,1Þ,ðo14 , o8 ,1Þ,ðo5 , o11 ,1Þg fðo4 ,1,1Þ,ðo3 , o4 ,1Þ,ðo3 , o10 ,1Þg |

2 1 1 360

Table 9 Number of Baer sublines. Case

# of solutions

# Baer sublines

70,341 141,518 156,661 197,382 197,382

1 1 1 3 75

16 16 16 208 16

Table 10 Orbits on pairs of points on a line of type II in the BM4-unital. i

Orbit representative

Q

s.o.

1 2 3

fP 0,0 ,P 1,0 g fP 0,0 ,P o13 ,g g fP 0,0 ,P o2 ,1 g

ðo,0,1Þ ðo9 ,0,1Þ ðo14 ,0,1Þ

2 1 1

Hermitian forms are

abq þ aq b þ o5i gq þ 1 , i ¼ 0,1,2 Observe that these Hermitian forms have the line ‘1 as secant. Let us now consider the question of which starter configurations appear in the two known unitals. Recall from Lemma 2 that the automorphism group of H4 acts transitively on the blocks of the unital, which are all Baer sublines. The tangent lines at the five points of the block are concurrent. The stabilizer of a block, of order 1200, acts transitively on the pairs of points from that block. Hence the stabilizer of a pair has order 120. This configuration is isomorphic to starter case 197,382. By Lemma 2, the automorphism group of BM4 has two orbits on blocks. One consists of the 16 secants through the point at infinity ð0,1,0Þ, which are Baer sublines, such as fP0,r 9r 2 Fq g ¼ fð0,1,0Þ,ð0,0,1Þ,ð0,1,1Þ,ð0, o5 ,1Þ,ð0, o10 ,1Þg The second orbit is represented by fP0,0 ,P 1,0 ,P o13 ,g ,P o2 ,1 ,P a9 ,g 2 g ¼ fð0,0,1Þ,ð1, o4 ,1Þ,ðo13 , o2 ,1Þ,ðo2 , o6 ,1Þ,ðo9 , o13 ,1Þg and has length 192. Each block is isomorphic to R2ð1Þ, with circle type ð36 ,41 Þ (i.e., not a Baer subline). The stabilizer of a block has order 4. The stabilizer of the Baer subline has two orbits on pairs of points from the Baer subline. The first orbit is the set of pairs containing the vertex, represented by fð0,1,0Þ,ð0,0,1Þg. The stabilizer of this pair has order 12. The second orbit is represented by fð0,0,1Þ,ð0,1,1Þg, with a stabilizer of order 8. In both cases, the intersection point of the tangent lines is Q ¼ ð1,0,0Þ. In both cases, t 1 ¼ 5, that is, the five tangent lines through Q are the lines that connect the points of the Baer subline to Q . Both cases are isomorphic to configuration 197,382. That is, configuration 197,382 is embedded in BM4 in two different ways. Considering the second block orbit, we find three orbits of the block stabilizer on pairs of points from that block. Table 10 gives more information about these cases. In all cases, t 1 ¼ 2, that is, there are three tangent lines on Q whose point of tangency is not on the line that we stabilized. Table 11 indicates the points of tangency for the three cases, as well as the stabilizer of the configuration inside the automorphism group of BM4. Finally, it indicates the isomorphism type of starter sets corresponding to the configuration. Lemma 4. Let G be the full collineation group of the projective plane p. Let U be a unital in p with automorphism group A. Let C be a starter configuration. Let D1 , . . . ,Dk be representatives for the A-orbits of configurations in U that are isomorphic to C. Then

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

121

Table 11 Refining the information of Table 10. i

Pts. of tangency

s.o.

Case

1 2 3

fP o9 ,0 ,P o13 ,1 ,P o2 ,0 g fP o9 ,g2 ,P o2 ,g ,P o11 ,g2 g fP o3 ,1 ,Po10 ,g ,Po,g g

2 1 1

70,341 141,518 156,661

Table 12 Number of unitals containing a fixed starter configuration. Case

# of H4

# of BM 4

# unitals total

2 2 ¼1 1 1 ¼1 1 1 ¼1 360 360 12 þ 8

1

70,341

0

141,518

0

156,661

0

197,382

360 120

¼3

1 1 ¼ 75

78

the number of unitals containing C and isomorphic to U is k X 9GC 9 9A Di 9 i¼1

Proof. We use the notation G(x) for the orbit of x under the group G. Consider the containment relation between the elements of the set G(C) and the elements of the set GðUÞ. Let a be the number of configurations isomorphic to C that are contained in U. Let b be the number of unitals in G(U) that contain C. Simple double counting yields a9GðUÞ9 ¼ b9GðCÞ9, and P we have a ¼ ki ¼ 1 9AðDi Þ9. Therefore, b¼

k k k X 9GðUÞ9 X 9GC 9 X 9A9 9GC 9 ¼ 9AðDi Þ9 ¼ 9GðCÞ9 i ¼ 1 9A9 i ¼ 1 9ADi 9 9A Di 9 i¼1

&

The number of unitals containing a fixed starter configuration can now be computed. See Table 12. 8. Some additional comments on the algorithms Since the proof that we presented is based on computer work, it seems appropriate to add some additional comments on the performance of the algorithms. Our goal is to give more insight into the details and thereby raise the confidence level in the correctness of our proof. Concerning the LLL-based solver for Diophantine equations, we report that almost all cases could be handled in around 30 s each. Some 10,000 cases caused trouble, of which 2,000 cases had to be handed over to Gurobi, which seemed to perform well on those. The LLL-solver has two stages. In the beginning, an LLL-reduced basis is computed. In the second part, which we call exhaustive enumeration, the solutions to the system are enumerated by tracing through the space spanned by the reduced basis. Here, enumerating means that all solutions are listed, once each, and that it is shown that there are no other solutions. As it turns out, the time for the enumeration depends very much on the ordering on input vectors for the LLL-reduction, and we do not have a good idea about what makes some bases work better than others. The four cases listed in Table 6 could all be enumerated from the LLL-basis. The first three cases took around 5 min to compute, the fourth case took around 35 min. For classifying orbits of group on sets, we use the algorithm Leiterspiel from Schmalz (1992). The size of the subset is restricted, and one can require that the subsets require a certain G-invariant condition. Most importantly, the algorithm builds up a data structure that holds information about the orbits in question. This data structure is very useful to preform isomorphism testing during the run of the algorithm, and at a later stage. This enables one to compute isomorphisms between configurations that appear inside the ‘gold plated’ unitals H4 and BM4 as defined in (6) and (7) and the classification of starter configurations. References ¨ Andre´, Johannes, 1954. Uber nicht-desarguessche ebenen mit transitiver translationsgruppe. Mathematische Zeitschrift 60, 156–186. Baker, R.D., Ebert, G.L., 1992. On Buekenhout-Metz unitals of odd order. Journal of Combinatorial Theory, Series A 60 (1), 67–84. Barwick, Susan, Ebert, Gary, 2008. Unitals in Projective Planes, Springer Monographs in Mathematics. Springer, New York. Betten, Anton, Betten, Dieter, 2010. There is no Drake/Larson linear space on 30 points. Journal of Combinatorial Designs 18 (1), 48–70.

122

J. Bamberg et al. / Journal of Statistical Planning and Inference 144 (2014) 110–122

Betten, Dieter, Braun, Mathias, 1992. A tactical decomposition for incidence structures. In: Combinatorics ’90 (Gaeta, 1990), Ann. Discrete Math., vol. 52. North-Holland, Amsterdam, pp. 37–43. Bruck, R.H., Bose, R.C., 1964. The construction of translation planes from projective spaces. Journal of Algebra 1, 85–102. Buekenhout, F., 1976. Existence of unitals in finite translation planes of order q2 with a kernel of order q. Geometriae Dedicata 5 (2), 189–194. Casse, L.R.A., O’Keefe, Christine M., Penttila, Tim, 1996. Characterizations of Buekenhout–Metz unitals. Geometriae Dedicata 59 (1), 29–42. Dembowski, Peter, 1997. Finite Geometries, Classics in Mathematics. Springer-Verlag, Berlin Reprint of the 1968 original. Ebert, G.L., 1992. On Buekenhout–Metz unitals of even order. European Journal of Combinatorics 13 (2), 109–117. Gurobi Optimizer Version 4.0, 2011. Gurobi Optimization Inc. /http://www.gurobi.com/S. Metz, Rudolf, 1979. On a class of unitals. Geometriae Dedicata 8 (1), 125–126. Moore, Eliakim Hastings, 1896a. Tactical Memoranda I–III. The American Journal of Mathematics 18 (3), 264–290. Moore, Eliakim Hastings, 1896b. Tactical Memoranda I–III. The American Journal of Mathematics 18 (4), 291–303. Penttila, Tim, Royle, Gordon F., 1995. Sets of type (m,n) in the affine and projective planes of order nine. Designs, Codes and Cryptography 6 (3), 229–245. ¨ Schmalz, Bernd, 1992. t-Designs zu vorgegebener Automorphismengruppe. Bayreuther Mathematische Schriften 41, 164. Dissertation, Universitat Bayreuth, Bayreuth, 1992. Sherk, F.A., 1983. Translation planes of order 16. In: Finite Geometries (Pullman, Wash., 1981), Lecture Notes in Pure and Appl. Math., vol. 82. Dekker, New York, 1983, pp. 401–412. Wassermann, A., 1998. Finding simple t-designs with enumeration techniques. Journal of Combinatorial Designs 6, 79–90.