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Unital designs with blocking sets
Discrete Applied Mathematics 163 (2014) 102–112
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Unital designs with blocking sets Abdullah Al-Azemi a , Anton Betten b,∗ , Dieter Betten c a
Department of Mathematics, Kuwait University, Kuwait
b
Department of Mathematics, Colorado State University, United States
c
Math. Seminar Univ. Kiel, Germany
article
info
Article history: Received 31 October 2010 Accepted 18 February 2013 Available online 22 March 2013 Keywords: Unital Design Finite geometry
abstract A unital 2-(28, 4, 1) design has 28 points, each block has size 4 and every pair of points is on exactly one block. A blocking set in a design is a subset of the point set with the property that every block intersects the blocking set nontrivially but no block is contained in the blocking set. In this work, we classify the unital 2-(28, 4, 1) designs with blocking sets. We find 68,806 unitals with a blocking set. Of these, 68,484 have a trivial automorphism group. © 2013 Elsevier B.V. All rights reserved.
1. Introduction and statement of results A finite incidence geometry is a pair (P , L). Here, P is a finite set whose elements are called points, and L is a set of subsets of P called lines, such that 1. each subset ℓ ∈ L has at least two points, and 2. each pair of points is on at most one block. A linear space is an incidence geometry (P , L) such that each pair of points is on exactly one line. We refer to [3] for more details on linear spaces. A 2-(v, k, λ) design is a pair (P , L) such that P is a set of size v, L is a set of k-subsets of P, and for each pair of points in P there are exactly λ elements in L that contain both points. Thus, a 2-(v, k, 1) design is a linear space with all lines of size k. Often, the elements of L in a design are called blocks (and the design is called a block system). A 2-(v, k, 1) design is also known as a Steiner 2-design. An incidence geometry (P , L) has a blocking set if there exists a subset B ⊂ P with the following properties: 1. B ∩ ℓ ̸= ∅ for all lines ℓ ∈ L and 2. ℓ ⊂ B for no line ℓ ∈ L. For n ≥ 2, a 2-(n3 + 1, n + 1, 1) design is called a unital design of order n. In particular, a 2-(28, 4, 1) unital design consists of 28 points and 63 lines of size 4, so that each pair of points is on exactly one block. A unital is a set S of n3 + 1 points in a projective plane π = (P , L) of order n such that |S ∩ ℓ| ∈ {1, n + 1} for all lines ℓ ∈ L. A classical example of a unital is the Hermitian unital which is the set of absolute points of a unitary polarity in the desarguesian projective plane of order q2 , PG(2, q2 ). Another example is the Ree unital which is a design on q3 + 1 points associated with the Ree group, see [10]. Unitals (i.e., unital designs that are embedded in projective planes) have received much attention. The unitals of order 3 were classified in [12]. The work [13] focuses on unitals in planes of order 16. The unitals of order 4 in the desarguesian projective plane of order 16 have been classified in [1]. The book [2] is dedicated to the study of unitals.
∗
Corresponding author. E-mail addresses: [email protected] (A. Al-Azemi), [email protected] (A. Betten), [email protected] (D. Betten).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.02.023
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Brouwer [6] was the first to show the difference between unitals and unital designs. A unital is a unital design but not conversely. The classification of all unital designs of order 3 seems to be beyond reach at the moment. Penttila and Royle in [12] put it this way: The problem of constructing all 2-(28, 4, 1) designs seems to be infeasible at this stage (though possibly not by much). The present work is a contribution to the classification of unital 2-(28, 4, 1) designs. Brouwer [6] found 138 unital designs of order 3. Seven more unital designs were found in [11]. In [9], unital designs with non-trivial automorphism group were classified. There are exactly 4466 such unital designs. Several unitals with trivial automorphism group were found in [5]. In the present article, we make the additional assumption that there exists a blocking set. Thus, we classify all unitals of order 3 admitting a blocking set. We find 68,806 such unitals, of which 68,484 have a trivial automorphism group. The paper is organized in the following way. In Section 2, we recall some basic definition of tactical decomposition. In Section 3, the notion of a blocking set is related to that of discrepancy. In Section 4, we consider all of the possible decomposition schemes of unitals admitting a blocking set. The classification procedure is described in Section 5. 2. Tactical decomposition In this section, we recall some basic definitions of tactical decompositions. For further reference, see [7]. In the following, we will consider an incidence geometry X = (P , L). For p ∈ P, we define
(p) = {ℓ ∈ L | p ∈ ℓ} the pencil of lines through p. A decomposition of an incidence geometry X = (P , L) is a pair (R, C ) of ordered partitions where R = (R1 , R2 , . . . , Rm ) is a partition of points and C = (C1 , C2 , . . . , Cn ) is a partition of blocks. For each i = 1, 2, . . . , m and each j = 1, 2, . . . , n, let ri,j = |{B ∈ Cj | p ∈ B}| with p ∈ Ri fixed. Let ci,j = |{p ∈ Ri | p ∈ B}|, for B ∈ Cj fixed. A decomposition (R, C ) of X is said to be point tactical if the number ri,j is independent of the choice p ∈ Ri for each i and for each j. It is said to be block tactical (or line tactical) if for each i and for each j, the number ci,j is independent of the choice B ∈ Cj . A point tactical and a block tactical decomposition is simply said to be tactical decomposition with respect to X. Given a point tactical decomposition (R, C ) of an incidence geometry, we define the decomposition scheme in the following way. These numbers are ai = |Ri | for 1 ≤ i ≤ m, bj = |Cj | for each 1 ≤ j ≤ n together with the integers ri,j and ci,j . In particular, if R = {R1 , . . . , Rm } and C = {C1 , . . . , Cn }, we use a form as in (1) to describe such a scheme.
→ a1
.. .
am
b1 r1,1
b2 r1,2
··· ···
rm,1
rm,2
···
.. .
.. .
bn r1,n
.. .
.
(1)
rm,n
Note that the horizontal arrow in the upper left corner is indicating that this scheme describes a point tactical decomposition. In a similar way, if (R, C ) is a block tactical decomposition, then the form in (2) describes the corresponding decomposition scheme. Notice that the horizontal arrow is replaced by a vertical one. This indicates that we have a block tactical decomposition.
↓ a1
.. .
am
b1 c1,1
b2 c1,2
··· ···
cm,1
cm,2
···
.. .
.. .
bn c1,n
.. .
.
(2)
cm,n
3. Discrepancy and blocking sets We recall the notion of discrepancy, here only applied to finite incidence geometries, see [4]. Let (P , L) be a finite incidence structure. Let Π = (A, B) be a non-trivial partition of the point set P with two classes (i.e., with P = A ∪ B, with A ∩ B = ∅ and with 1 < |A| < |P |). For each line ℓ ∈ L, one calculates the values |A ∩ ℓ| and |B ∩ ℓ| and takes the positive difference. The maximum of all such numbers is u(Π ) = max {||A ∩ ℓ| − |B ∩ ℓ||} . L
The discrepancy of (P , L) is the smallest value of u(Π ) when considering all non-trivial partitions of P with two classes:
∆ = min u(Π ). Π
In the present work, we are interested in 2-(28, 4, 1) designs. Since all blocks have length 4 in these designs, only the following three possibilities for the value of u exist: 4 − 0 = 4, 3 − 1 = 2 and 2 − 2 = 0. It follows that the discrepancy is one of the values 4, 2 or 0. The case ∆ = 0 does not arise:
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Proposition 1. The discrepancy ∆ of a 2-(28, 4, 1) design is either 2 or 4. Proof. To prove this, we have to exclude ∆ = 0. In such case, there would exist a partition Π = (A, B) such that the induced subgeometries on A and w on B only have 2-blocks. These subgeometries are also linear spaces. Therefore, the number of 2-lines must be of the form 2 , w ∈ Z. But the number of blocks is 63 and is not of this form. The notion of a blocking set (see Section 1) is closely related to the discrepancy: Proposition 2. A 2-(28, 4, 1) design has a blocking set if and only if the discrepancy is 2. Proof. Let B be a blocking set for the 2-(28, 4, 1) design. Then B and A = P \ B define a partition Π = (A, B) of P, where all differences are 3 − 1 = 2 or 2 − 2 = 0. Therefore, the maximum value of these differences is u(Π ) ∈ {0, 2}. But the case of u(Π ) = 0 is not possible as this would imply ∆ = 0, a contradiction. The minimum u(Π ) over all partitions Π is 2. So we get ∆ = 2. Conversely, if ∆ = 2, then there is a partition Π = (A, B) with u(Π ) = 2. Therefore we can take A (and also B) as blocking set. In this paper, we classify all 2-(28, 4, 1) unital designs with blocking sets. According to Proposition 2, this means we classify all 2-(28, 4, 1) unital designs with discrepancy equal to 2. 4. The possible decompositions We make now the assumption that the unitals have a blocking set, and we determine the possible tactical decompositions. Lemma 3. Let (P , L) be a 2-(28, 4, 1) unital design. The column tactical decomposition
↓ 12 16
21 3 1
3 2 2
39 1 3
does not arise. Proof. Let (Π , Ψ ) be the decomposition that gives rise to the decomposition scheme. That is, we have Π = (A, B) a partition of P with |A| = 12 and |B| = 16. Also Ψ = (L1 , L2 , L3 ) a partition of L with |L1 | = 21, |L2 | = 3 and |L3 | = 39. For p ∈ A, let xi = |(p) ∩ Li | for i = 1, 2, 3. Also, for p ∈ B, let yi = |(p) ∩ Li | for i = 1, 2, 3. Then 2x1 + x2 = 11, x1 + 2x2 + 3x3 = 16, x1 + x2 + x3 = 9, 3y1 + 2y2 + y3 = 12, y2 + 2y3 = 15, y1 + y2 + y3 = 9 which yields
(x1 , x2 , x3 ) ∈ {(5, 1, 3), (4, 3, 2)},
(y1 , y2 , y3 ) ∈ {(1, 1, 7), (0, 3, 6)}.
Let w1 , w2 be the number of points of A of the first two types, and let w3 , w4 be the number of points of B of the latter two types. Then 5w1 + 12w2 + w3 ≤ 63, 3w1 + 6w2 + 7w3 + 18w4 ≤ 117,
w1 + w2 = 12, w3 + w4 = 16, w1 + w2 + w3 + w4 = 28. The first and third condition imply (w1 , w2 ) = (12, 0). Thus the first condition becomes w3 ≤ 3, which forces w4 ≥ 13. But then the second condition is violated. Lemma 4. Let (P , L) be a 2-(28, 4, 1) unital design with column tactical decomposition
↓ 13 15
24 3 1
6 2 2
33 1 . 3
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The decomposition can be refined uniquely to the row tactical decomposition
→ 7 6 9 6
24 6 5 2 1
6 0 2 0 2
33 3 2 . 7 6
Proof. We use notation similar to that of the proof of Lemma 3. Counting arguments show that
(x1 , x2 , x3 ) ∈ {(6, 0, 3), (5, 2, 2), (4, 4, 1), (3, 6, 0)} and
(y1 , y2 , y3 ) ∈ {(2, 0, 7), (1, 2, 6), (0, 4, 5)}. Let w1 , w2 , w3 , w4 be the number of points of A of the first 4 types. Let w5 , w6 , w7 be the number of points of B of the latter 3 types. Then
w1 + w2 + w3 + w4 + w5 + w6 + w7 = 28, w2 + 6w3 + 15w4 + w6 + 6w7 ≤ 15, 6w1 + 5w2 + 4w3 + 3w4 = 72, 2w2 + 4w3 + 6w4 = 12, 2w5 + w6 = 24. The only solution to this system is
(w1 , . . . , w7 ) = (7, 6, 0, 0, 9, 6, 0). Thus we get the unique refinement as claimed.
Theorem 5. Each 2-(28, 4, 1) unital design with a blocking set admits one of the following decomposition schemes
→
Type D: → 7 28 14 1 6 14 1 2
28 2 6
,
7 6 9 6
Type E: 24 6 6 0 5 2 2 0 1 2
33 3 2 7 6
.
(3)
The first decomposition (type D) is tactical while the second one (type E) is only point tactical. It is a refinement of the column tactical decomposition
↓ 13 15
24 3 1
6 2 2
33 1 . 3
(4)
Proof. Let Π = (A, B) be a partition of the point set P with ∆ = 2 = u(Π ). Let |A| = a and |B| = b. Then, the partition Π has the following block tactical decomposition.
↓ a b
x 3 1
y 2 2
z 1 3
.
Since a + b = 28, we can assume without loss of generality that a ≤ 14. Moreover, x + y + z = 63,
(5)
since there are 63 lines total. Counting flags (incident point–line pairs) inside A, we get 3x + 2y + z = 9a.
(6)
This is together with (5) implies 2x + y = 9a − 63.
(7)
Counting pairs of points (p, q) ∈ A × B, we get 3x + 4y + 3z = ab.
(8)
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Using (5), we find y = ab − 189.
(9)
Under the assumption a ≤ 14, this implies
(a, b, y) ∈ {(12, 16, 3), (13, 15, 6), (14, 14, 7)}.
(10)
Now, x can be computed by using (7) and z follows by using (5). In the three cases, we get
(x, z ) ∈ {(21, 39), (24, 33), (28, 28)}.
(11)
Therefore we arrive at the following three block tactical decompositions:
↓ 12 16
21 3 1
3 2 2
↓
39 1 , 3
24 3 1
13 15
6 2 2
↓
33 1 , 3
and
14 14
28 3 1
7 2 2
28 1 . 3
The first decomposition is ruled out by Lemma 3. The second decomposition refines to the decomposition of type E by Lemma 4. In the third case, it is easy to show that the decomposition is tactical (i.e., both point tactical and block tactical). This is the decomposition of type D. The decomposition of type D is suitable for computer classification. On the other hand, the decomposition of type E is still too coarse, so we wish to refine it. Refining the decomposition further would give too many cases. So, we concentrate on the subgeometry with the 6-block domain (i.e., the lines ℓ ∈ L with |A ∩ ℓ| = |B ∩ ℓ| = 2). In particular, we consider the partial decomposition
→ 6 6
6 2 2
which is tactical. The upper part
→ 6
6 2
generates either a 6-cycle or two 3-cycles. The same holds for the lower part. This gives four cases: 1. 2. 3. 4.
Two 3-cycles in the upper part and a 6-cycle in the lower part, 6 cycle in the upper part and two 3-cycles below, two 6-cycles, one in the upper part and one in the lower part, two 3-cycles in the upper part and two 3-cycles in the lower part.
The fourth case is easily seen to be impossible. The first three cases can be described by 1. → 3 3 2 0 3 6 1
3 0 2 1
,
3.
2. → 3 6 1 3 2 0 3
→
3 1 0 2
3 3 3 3
,
6 2 2 2 2
.
These schemes can be included in the decomposition scheme of type E as described in Theorem 6. Theorem 6. The unitals generated from the scheme of type E are exactly those which are generated from the schemes of type F, type G, and type H. These three types are presented in (12). Type F
→ 7 3 3 6 9
3 0 2 0 1 0
3 0 0 2 1 0
24 6 5 5 1 2
Type H
Type G 33 3 , 2 2 6 7
→ 7 6 3 3 9
3 0 1 2 0 0
3 0 1 0 2 0
24 6 5 1 1 2
33 3 , 2 6 6 7
→ and
7 3 3 3 3 9
6 0 2 2 2 2 0
24 6 5 5 1 1 2
33 3 2 . 2 6 6 7
5. Classification of the unital designs In this section, we describe the classification procedure for unital designs admitting a blocking set.
(12)
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Table 1 Computing the list of starters for decomposition schemes of types D, F, G, and H. Type
Depth n
Number of generated cases
Number of starters
D F G H
14 13 13 13
787 59 13,365 13,369
787 32 191 191
By the results of Section 4, this is equivalent to classifying the geometries from the decomposition schemes of type D (as in (3)), and of types F, G, and H (as in (12)). We assume familiarity with the basics of classification algorithms of incidence geometries. A detailed introduction to such algorithms can be found in [8]. We begin by classifying geometries satisfying these decompositions partially. By this, we mean that we classify the partial incidence matrices of geometries admitting a decomposition. A partial incidence matrix is a matrix whose first n rows satisfy the conditions imposed by the decomposition as well as by the geometry, but whose remaining rows are all zero. The idea is to fill the remaining rows of the incidence matrix later. We say that n is the depth of the partial incidence matrix. The parameter n is chosen after some experimentation, and it varies from decomposition to decomposition. The goal is to first classify all partial incidence matrices at depth n under the group fixing the decomposition scheme and fixing the first n rows setwise. For type D we choose n = 14, and for all other decompositions schemes we choose n = 13. We represent each orbit of partial incidence matrices by its canonical representative. This is just the matrix that is least in the lexicographical ordering of all matrices in its isomorphism class. In the next step, we forget about the decomposition scheme. That is, we classify the partial incidence matrices under the group that fixes the first n rows but otherwise is allowed to move points across the boundaries of the decomposition. This eliminates copies of geometries that differ only by a rearrangement of the decomposition classes. For instance, in the case of type F we have two 3-cycles which can be permuted and hence result isomorphic copies. Therefore, we do an isomorphism test at line 14 (type D) and at line 13 (types F, G, and H) under the group that fixes the first 14 rows setwise. The resulting list of cases (for each decomposition scheme type) are called the list of starters (at depth n). The number of isomorphism types of partial incidence matrices at depth n, and the number of starters at depth n is presented in Table 1. For each starter at depth n, we run a computer search to generate the partial incidence matrices at some depth m with n < m. Here, m = 18 for type D and m = 19 for the remaining types. In order to find the full incidence matrices of the geometries that we are looking for, we need to complete the partial incidence matrices at depth m to depth 28. This can be formulated as a problem of finding all cliques in a certain graph: For each isomorphism type of starter S at depth m, we define a graph ΓS . The vertices of ΓS are the possibilities for the missing rows. These are the 0/1 vectors that could potentially be used to fill in additional rows of the partial incidence matrix. Of course, these 0/1 vectors have to satisfy the conditions that result from the decomposition (as well as from the fact that we create a linear space). Two vertices of ΓS are connected if the corresponding vectors can both be chosen simultaneously to extend the partial incidence matrix. This requires testing if the rows have exactly one position where both have an entry one. Also, it requires testing if the conditions on the column sums are satisfied. For each starter, we present the number of cliques in the second column of Table 2 (type D), Table 3 (type F), Table 4 (type G), and Table 5 (type H). Once the incidence matrices have been filled to depth 28 using the cliques of size 28 − m from the graph Γ , we perform an additional round of isomorphism testing. This produces the classification of all geometries for a given scheme with a given starter at depth n (in the third column). After that we consider the geometries that are obtained from all starters and perform another round of isomorph rejection, this time disregarding the decomposition schemes. The numbers that result from this are listed at the bottom as ‘‘merged’’. The result of this procedure is presented in Theorem 7. Theorem 7. The decomposition scheme of types D, F, G, and H generate the numbers of (pairwise non-isomorphic) unitals presented in Table 6. Considering the unitals from all decomposition schemes together, and performing another round of isomorphism testing leads to the following result: Theorem 8. The number of (pairwise non-isomorphic) unitals of order 3 with discrepancy 2 (i.e., having a blocking set) is 68,806. The distribution of automorphism group orders is
{1922 , 487 , 242 , 162 , 127 , 819 , 7, 63 , 487 , 3102 , 290 , 168484 }. 6. Note added in the proof In work that was done after this paper was first submitted, we checked our results against the list of unital designs with non-trivial automorphism group [9]. The number of unital designs with non-trivial automorphism group is 4466. We tested which of these had a blocking set and found that exactly 322 did. This agrees with our number of 68,806 − 68,484 = 322 unital designs with blocking set and non-trivial automorphism group from Theorem 8.
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Table 2 Unitals of type D, generated from the list of starters at line 14. Nr.
gen.
isot.
Nr.
gen.
isot.
Nr.
1 11 12 14 16 21 23 24 26 30 32 35 39 40 41 42 50 51 52 55 59 61 67 69 72 73 76 77 79 80 89 90 93 96 98 99 101 102 105 109 112 113 114 115 116 117 118 119 120 121 122 123 126 129 130 131 132 133 137 138 139 140 145 146 147 149 150 152
18 36 114 16 8 8 4 2 8 2 16 8 352 32 4 4 8 8 8 12 8 10 4 24 80 80 4 4 48 152 40 6 4 4 24 16 16 4 26 2 8 28 4 4 4 6 40 16 24 20 44 8 60 2 8 6 2 4 16 8 4 4 4 16 4 32 12 2
1 13 19 1 1 2 1 1 1 1 1 3 10 3 1 1 1 2 1 2 2 2 1 1 4 6 1 1 3 42 2 2 1 1 1 1 2 1 3 1 1 4 1 1 2 2 7 1 3 2 5 2 4 1 1 2 1 1 1 2 1 1 1 2 1 1 2 1
292 294 295 297 299 301 303 305 306 307 308 310 312 313 314 315 316 317 318 319 321 323 325 326 332 333 335 340 342 344 345 347 348 349 350 351 354 355 358 359 360 362 363 364 368 369 370 372 374 375 377 378 379 380 381 382 383 384 386 387 390 391 392 393 394 395 396 397
12 8 4 8 8 12 66 4 8 24 40 32 40 24 16 4 8 8 20 6 4 4 4 8 2 2 18 24 8 16 2 8 20 4 4 8 16 8 12 16 16 16 10 8 2 10 8 8 4 8 6 8 4 10 2 24 8 8 12 8 8 16 2 8 48 28 40 2
2 1 1 1 1 2 4 1 3 5 11 2 7 6 1 1 1 1 2 2 1 1 1 1 1 1 4 3 1 1 1 1 2 1 1 3 1 2 2 1 2 3 2 2 1 2 1 1 1 1 2 1 1 3 1 3 1 1 2 1 1 2 1 1 9 4 6 1
531 532 533 536 537 539 540 542 543 545 546 547 550 551 555 559 561 562 564 566 567 568 569 570 572 573 575 577 580 583 585 586 588 589 590 593 595 597 600 601 602 603 604 611 614 615 617 618 620 622 623 624 629 632 637 638 640 641 646 648 651 654 659 661 662 665 669 670
gen. 16 8 32 4 16 8 4 14 16 20 8 16 32 14 4 8 4 16 20 24 8 24 20 16 10 32 8 46 8 12 80 16 8 4 12 8 16 20 16 8 8 8 4 8 24 8 4 24 8 4 4 12 2 8 8 10 22 8 16 4 8 8 4 8 4 8 12 38
isot. 1 2 8 1 1 2 1 4 1 2 1 1 7 3 1 1 1 4 3 4 1 3 2 2 2 4 1 7 1 2 13 1 2 1 2 2 1 2 2 2 1 2 1 2 2 1 1 2 1 1 1 2 1 4 4 2 5 1 3 1 2 2 1 1 1 1 2 6
A. Al-Azemi et al. / Discrete Applied Mathematics 163 (2014) 102–112
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Table 2 (continued) Nr.
gen.
isot.
Nr.
gen.
isot.
Nr.
154 157 158 161 162 166 168 170 171 172 174 176 177 179 180 183 184 185 186 187 192 193 195 196 197 198 202 203 204 206 207 208 211 212 213 214 216 217 218 220 222 223 225 226 227 229 231 232 237 242 245 246 247 248 249 253 254 255 256 257 258 259 260 265 276 277 279 280 283 284
4 4 16 4 22 16 16 4 28 12 24 4 8 50 2 36 16 40 4 4 144 8 12 2 8 4 44 16 20 16 68 8 20 4 16 16 4 56 8 16 2 4 16 16 12 2 8 12 10 24 4 16 4 16 18 16 8 8 8 8 8 4 24 8 8 4 12 32 6 12
1 1 2 1 3 4 1 1 3 2 3 1 2 8 1 2 1 2 1 1 9 1 2 1 1 1 4 3 2 2 6 1 2 1 1 1 1 3 1 2 1 1 1 1 2 1 1 2 2 2 1 1 1 7 8 1 1 1 1 1 1 2 2 1 1 1 2 2 2 2
399 400 401 403 405 409 412 414 415 416 417 423 424 425 426 427 428 430 431 433 434 436 440 442 448 452 453 455 458 459 460 461 462 464 465 466 468 470 472 473 474 475 476 478 481 484 485 486 489 491 492 494 495 497 499 500 501 503 504 505 506 508 515 517 519 520 521 522 523 526
28 12 4 16 8 24 6 8 4 16 32 4 10 4 24 12 2 2 4 16 2 16 20 24 4 4 8 8 4 24 16 4 44 14 16 8 4 16 8 8 32 2 8 8 4 4 4 10 16 2 16 16 10 16 6 16 12 24 20 26 4 24 4 4 4 22 4 4 4 16
3 2 1 1 2 3 2 1 1 1 6 1 2 1 2 2 1 1 1 1 1 1 1 5 1 1 1 1 1 2 2 2 10 3 3 1 1 1 1 1 7 1 1 2 1 1 1 3 2 1 4 2 2 1 2 1 2 6 2 3 1 2 1 1 1 5 1 2 1 4
672 674 675 676 677 678 680 682 683 684 685 686 687 689 691 692 694 695 696 697 699 700 702 703 705 706 707 708 709 710 712 713 714 715 717 718 719 720 721 723 724 726 728 731 732 735 736 737 738 740 741 742 743 744 745 746 747 749 752 754 757 759 761 767 769 771 772 773 774 776
gen. 40 32 8 4 8 34 12 8 6 20 4 6 16 16 8 8 20 8 10 8 24 24 20 16 4 4 12 40 8 4 24 12 4 8 12 12 14 808 8 4 398 6 8 12 4 8 4 4 18 36 4 34 8 18 4 12 2 32 240 16 2 2 24 16 4 4 22 24 8 8
isot. 2 1 1 1 1 3 2 1 2 1 1 2 3 1 1 2 2 1 2 2 2 2 2 2 1 1 2 5 2 1 3 2 1 1 2 3 1 72 1 1 32 2 1 2 1 1 1 1 1 3 1 5 1 2 1 2 1 4 8 1 1 1 2 2 1 1 5 3 1 1
(continued on next page)
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A. Al-Azemi et al. / Discrete Applied Mathematics 163 (2014) 102–112
Table 2 (continued) Nr.
gen.
isot.
Nr.
gen.
288 290 291
12 32 4
2 2 1
527 528 530
4 16 24
isot.
Nr.
1 1 2
778 780 783
gen.
isot.
4 6 8
Total:
1 2 1 1015
Merged:
436
Table 3 Unitals of type F, generated from the list of starters at line 13. Nr.
gen.
isot.
Nr.
gen.
isot.
Nr.
gen.
1 2 3 4 5 6 7 8 9 10 11
4106 4526 4340 4268 4494 4110 4190 4196 4162 4460 4616
534 557 525 554 571 533 546 563 525 591 596
12 13 14 15 16 17 18 19 20 21 22
2362 4600 4312 1354 2030 1958 4170 4418 4502 4350 2220
206 571 533 106 277 261 276 277 276 540 265
23 24 25 26 27 28 29 30 31 32
2000 4550 4470 4514 2232 2378 4286 4292 4678 4456
isot. 252 277 561 566 279 276 283 549 561 554
Total:
13,841
Merged:
13,819
Table 4 Unitals of type G, generated from the list of starters at line 13. Nr.
gen.
isot.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
499 489 537 480 657 454 561 382 627 561 468 524 567 562 568 429 439 546 425 408 431 500 614 648 529 616 476 516 493 433 477 502 382 547 537
86 95 90 94 107 91 94 79 95 90 84 92 92 98 89 78 88 91 71 77 84 80 93 116 96 96 92 93 91 87 84 93 81 87 95
Nr. 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
gen.
isot.
Nr.
gen.
463 216 253 475 662 578 488 238 542 556 503 545 431 506 229 422 286 298 403 533 488 567 586 499 401 514 606 573 476 437 403 396 479 478 574
95 38 41 89 103 98 71 36 48 94 48 90 39 89 21 39 50 45 79 90 83 90 96 98 81 106 100 94 89 82 76 81 85 94 92
129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163
442 467 599 674 524 449 396 509 525 538 579 466 448 523 385 477 414 412 427 414 480 422 555 688 399 512 354 548 481 517 522 542 447 528 523
isot. 71 83 106 106 79 85 81 45 81 76 87 76 84 41 71 81 63 72 65 67 38 54 86 98 67 77 69 39 77 79 79 81 30 71 85
A. Al-Azemi et al. / Discrete Applied Mathematics 163 (2014) 102–112
111
Table 4 (continued) Nr.
gen.
isot.
Nr.
gen.
isot.
Nr.
gen.
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
389 571 516 633 642 517 490 521 454 478 445 622 418 565 483 392 502 512 513 510 498 587 390 214 278 516 471 564 553
74 101 81 97 106 103 81 94 78 91 85 93 82 90 79 77 91 91 97 92 95 103 586 39 44 93 88 97 115
100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
755 255 700 628 193 570 644 435 530 501 276 381 495 288 268 590 665 538 552 487 564 482 468 545 504 505 591 466 451
60 37 119 57 38 99 104 81 90 90 57 65 103 46 52 89 106 84 115 94 98 89 80 84 81 86 46 86 88
164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191
404 483 399 474 570 556 442 494 430 299 491 611 666 477 466 534 96 514 531 354 395 585 486 256 324 374 390 226
isot. 70 37 72 75 89 85 38 67 68 55 69 86 89 67 72 76 21 70 80 59 65 90 69 20 23 47 52 26
Total:
15,114
Merged:
14,901
Table 5 Unitals of type H, generated from the list of starters at line 13. Nr.
gen.
isot.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
3660 3266 1316 3284 954 1041 3475 3204 3607 4051 3654 3839 3678 3418 4153 3656 3327 3158 4209 3400 3738 3453 3779 3579 3068 3660 3118 3295 3197 3591 3618 3152
258 239 82 237 78 72 249 229 253 296 250 264 268 248 293 258 237 229 277 250 258 250 258 246 217 253 223 240 239 248 269 225
Nr. 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
gen.
isot.
Nr.
gen.
1059 1354 3602 3551 3027 3747 3490 3737 1533 3700 3467 1807 1555 3740 3311 3656 1071 965 3495 3330 3437 3040 3351 3388 3378 2887 3491 3697 3143 3720 2967 3259
76 89 261 262 217 276 247 260 114 266 246 128 107 269 119 127 73 39 252 256 234 229 235 236 248 216 257 254 219 249 215 240
129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
3519 3568 3135 3303 3430 3222 2946 3792 3208 3808 3994 3053 2991 3170 3155 3333 3168 1694 1433 2700 1663 1722 1038 1564 2498 3164 3199 3219 2880 2875 3200 3225
isot. 245 253 223 243 133 126 113 131 123 133 275 205 204 211 214 204 220 113 92 187 107 117 70 102 175 211 207 221 196 193 205 218
(continued on next page)
112
A. Al-Azemi et al. / Discrete Applied Mathematics 163 (2014) 102–112 Table 5 (continued) Nr.
gen.
isot.
Nr.
gen.
isot.
Nr.
gen.
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
3336 3221 3057 3726 3676 3426 3476 3317 3457 2995 3607 3037 3310 2993 3488 3446 3493 3511 3564 3450 3268 3537 3140 3306 3710 3517 3510 3856 3673 4183 3274 1306
251 223 233 265 260 250 246 243 247 223 252 226 245 229 238 238 243 246 261 244 243 256 213 231 262 260 251 273 257 294 235 94
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
2941 3517 3061 3151 3308 3304 3056 3496 3334 3584 3227 3358 3456 3718 3388 3584 3370 3892 3400 3198 3134 3543 3439 3316 3879 3567 3091 2880 3224 3325 3711 3498
207 125 230 124 118 121 211 244 237 244 235 236 236 259 244 248 236 258 244 220 234 245 245 226 280 248 218 212 250 227 255 249
161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191
3484 3248 2813 2913 3036 2779 2992 3273 3211 2860 2852 2995 2795 3423 3053 2897 2226 2696 2452 2626 2663 2918 2716 3125 1119 2561 974 1632 2268 888 816
isot. 242 224 207 203 207 181 207 211 213 200 194 209 182 226 215 196 163 175 171 171 173 204 192 203 76 186 63 106 140 63 51
Total:
40,031
Merged:
39,886
Table 6 The generated unitals from the decomposition of types D, F, G, and H. The corresponding distributions of automorphism group orders (ago’s) are presented in the third column. Type
Solutions
Distribution of ago’s
D F G H
436 13,819 14,901 39,886
{1225 , 278 , 33 , 487 , 63 , 7, 819 , 127 , 162 , 242 , 487 , 1922 } {113,770 , 24 , 345 } {114,846 , 21 , 354 } {139,876 , 27 , 33 }
Total:
69,042
References [1] John Bamberg, Anton Betten, Cheryl Praeger, Alfred Wassermann, Unitals in the desarguesian projective plane of order sixteen, J. Statist. Plann. Inference (2012) http://dx.doi.org/10.1016/j.jspi.2012.10.006. [2] Susan Barwick, Gary Ebert, Unitals in Projective Planes, in: Springer Monographs in Mathematics, Springer, New York, 2008. [3] Lynn Margaret Batten, Albrecht Beutelspacher, The Theory of Finite Linear Spaces, Cambridge University Press, Cambridge, 1993, Combinatorics of points and lines. [4] József Beck, Vera T. Sós, Discrepancy theory, in: Handbook of Combinatorics, Vols. 1, 2, Elsevier, Amsterdam, 1995, pp. 1405–1446. [5] Anton Betten, Dieter Betten, Vladimir D. Tonchev, Unitals and codes, Discrete Math. 267 (1–3) (2003) 23–33. Combinatorics 2000 (Gaeta). [6] A.E. Brouwer, Some unitals on 28 points and their embeddings in projective planes of order 9, in: Geometries and Groups (Berlin, 1981), in: Lecture Notes in Math., vol. 893, Springer, Berlin, 1981, pp. 183–188. [7] Peter Dembowski, Finite Geometries, in: Classics in Mathematics, Springer-Verlag, Berlin, 1997, Reprint of the 1968 original. [8] P. Kaski, P. Östergård, Classification Algorithms for Codes and Designs, in: Algorithms and Computation in Mathematics, vol. 15, Springer-Verlag, Berlin, 2006. [9] Vedran Krčadinac, Steiner 2-designs S (2, 4, 28) with nontrivial automorphisms, Glas. Mat. Ser. III 37(57) (2) (2002) 259–268. [10] Heinz Lüneburg, Some remarks concerning the Ree groups of type (G2 ), J. Algebra 3 (1966) 256–259. [11] R. Mathon, A. Rosa, 2-(v, k, λ) designs of small order, in: C.J. Colbourn, J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, New York, 1996, pp. 3–41. [12] Tim Penttila, Gordon F. Royle, Sets of type (m, n) in the affine and projective planes of order nine, Des. Codes Cryptogr. 6 (3) (1995) 229–245. [13] Stoicho D. Stoichev, Vladimir D. Tonchev, Unital designs in planes of order 16, Discrete Appl. Math. 102 (1–2) (2000) 151–158. Coding, cryptography and computer security (Lethbridge, AB, 1998).
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Unital designs with blocking sets Abdullah Al-Azemi a , Anton Betten b,∗ , Dieter Betten c a
Department of Mathematics, Kuwait University, Kuwait
b
Department of Mathematics, Colorado State University, United States
c
Math. Seminar Univ. Kiel, Germany
article
info
Article history: Received 31 October 2010 Accepted 18 February 2013 Available online 22 March 2013 Keywords: Unital Design Finite geometry
abstract A unital 2-(28, 4, 1) design has 28 points, each block has size 4 and every pair of points is on exactly one block. A blocking set in a design is a subset of the point set with the property that every block intersects the blocking set nontrivially but no block is contained in the blocking set. In this work, we classify the unital 2-(28, 4, 1) designs with blocking sets. We find 68,806 unitals with a blocking set. Of these, 68,484 have a trivial automorphism group. © 2013 Elsevier B.V. All rights reserved.
1. Introduction and statement of results A finite incidence geometry is a pair (P , L). Here, P is a finite set whose elements are called points, and L is a set of subsets of P called lines, such that 1. each subset ℓ ∈ L has at least two points, and 2. each pair of points is on at most one block. A linear space is an incidence geometry (P , L) such that each pair of points is on exactly one line. We refer to [3] for more details on linear spaces. A 2-(v, k, λ) design is a pair (P , L) such that P is a set of size v, L is a set of k-subsets of P, and for each pair of points in P there are exactly λ elements in L that contain both points. Thus, a 2-(v, k, 1) design is a linear space with all lines of size k. Often, the elements of L in a design are called blocks (and the design is called a block system). A 2-(v, k, 1) design is also known as a Steiner 2-design. An incidence geometry (P , L) has a blocking set if there exists a subset B ⊂ P with the following properties: 1. B ∩ ℓ ̸= ∅ for all lines ℓ ∈ L and 2. ℓ ⊂ B for no line ℓ ∈ L. For n ≥ 2, a 2-(n3 + 1, n + 1, 1) design is called a unital design of order n. In particular, a 2-(28, 4, 1) unital design consists of 28 points and 63 lines of size 4, so that each pair of points is on exactly one block. A unital is a set S of n3 + 1 points in a projective plane π = (P , L) of order n such that |S ∩ ℓ| ∈ {1, n + 1} for all lines ℓ ∈ L. A classical example of a unital is the Hermitian unital which is the set of absolute points of a unitary polarity in the desarguesian projective plane of order q2 , PG(2, q2 ). Another example is the Ree unital which is a design on q3 + 1 points associated with the Ree group, see [10]. Unitals (i.e., unital designs that are embedded in projective planes) have received much attention. The unitals of order 3 were classified in [12]. The work [13] focuses on unitals in planes of order 16. The unitals of order 4 in the desarguesian projective plane of order 16 have been classified in [1]. The book [2] is dedicated to the study of unitals.
∗
Corresponding author. E-mail addresses: [email protected] (A. Al-Azemi), [email protected] (A. Betten), [email protected] (D. Betten).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.02.023
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103
Brouwer [6] was the first to show the difference between unitals and unital designs. A unital is a unital design but not conversely. The classification of all unital designs of order 3 seems to be beyond reach at the moment. Penttila and Royle in [12] put it this way: The problem of constructing all 2-(28, 4, 1) designs seems to be infeasible at this stage (though possibly not by much). The present work is a contribution to the classification of unital 2-(28, 4, 1) designs. Brouwer [6] found 138 unital designs of order 3. Seven more unital designs were found in [11]. In [9], unital designs with non-trivial automorphism group were classified. There are exactly 4466 such unital designs. Several unitals with trivial automorphism group were found in [5]. In the present article, we make the additional assumption that there exists a blocking set. Thus, we classify all unitals of order 3 admitting a blocking set. We find 68,806 such unitals, of which 68,484 have a trivial automorphism group. The paper is organized in the following way. In Section 2, we recall some basic definition of tactical decomposition. In Section 3, the notion of a blocking set is related to that of discrepancy. In Section 4, we consider all of the possible decomposition schemes of unitals admitting a blocking set. The classification procedure is described in Section 5. 2. Tactical decomposition In this section, we recall some basic definitions of tactical decompositions. For further reference, see [7]. In the following, we will consider an incidence geometry X = (P , L). For p ∈ P, we define
(p) = {ℓ ∈ L | p ∈ ℓ} the pencil of lines through p. A decomposition of an incidence geometry X = (P , L) is a pair (R, C ) of ordered partitions where R = (R1 , R2 , . . . , Rm ) is a partition of points and C = (C1 , C2 , . . . , Cn ) is a partition of blocks. For each i = 1, 2, . . . , m and each j = 1, 2, . . . , n, let ri,j = |{B ∈ Cj | p ∈ B}| with p ∈ Ri fixed. Let ci,j = |{p ∈ Ri | p ∈ B}|, for B ∈ Cj fixed. A decomposition (R, C ) of X is said to be point tactical if the number ri,j is independent of the choice p ∈ Ri for each i and for each j. It is said to be block tactical (or line tactical) if for each i and for each j, the number ci,j is independent of the choice B ∈ Cj . A point tactical and a block tactical decomposition is simply said to be tactical decomposition with respect to X. Given a point tactical decomposition (R, C ) of an incidence geometry, we define the decomposition scheme in the following way. These numbers are ai = |Ri | for 1 ≤ i ≤ m, bj = |Cj | for each 1 ≤ j ≤ n together with the integers ri,j and ci,j . In particular, if R = {R1 , . . . , Rm } and C = {C1 , . . . , Cn }, we use a form as in (1) to describe such a scheme.
→ a1
.. .
am
b1 r1,1
b2 r1,2
··· ···
rm,1
rm,2
···
.. .
.. .
bn r1,n
.. .
.
(1)
rm,n
Note that the horizontal arrow in the upper left corner is indicating that this scheme describes a point tactical decomposition. In a similar way, if (R, C ) is a block tactical decomposition, then the form in (2) describes the corresponding decomposition scheme. Notice that the horizontal arrow is replaced by a vertical one. This indicates that we have a block tactical decomposition.
↓ a1
.. .
am
b1 c1,1
b2 c1,2
··· ···
cm,1
cm,2
···
.. .
.. .
bn c1,n
.. .
.
(2)
cm,n
3. Discrepancy and blocking sets We recall the notion of discrepancy, here only applied to finite incidence geometries, see [4]. Let (P , L) be a finite incidence structure. Let Π = (A, B) be a non-trivial partition of the point set P with two classes (i.e., with P = A ∪ B, with A ∩ B = ∅ and with 1 < |A| < |P |). For each line ℓ ∈ L, one calculates the values |A ∩ ℓ| and |B ∩ ℓ| and takes the positive difference. The maximum of all such numbers is u(Π ) = max {||A ∩ ℓ| − |B ∩ ℓ||} . L
The discrepancy of (P , L) is the smallest value of u(Π ) when considering all non-trivial partitions of P with two classes:
∆ = min u(Π ). Π
In the present work, we are interested in 2-(28, 4, 1) designs. Since all blocks have length 4 in these designs, only the following three possibilities for the value of u exist: 4 − 0 = 4, 3 − 1 = 2 and 2 − 2 = 0. It follows that the discrepancy is one of the values 4, 2 or 0. The case ∆ = 0 does not arise:
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Proposition 1. The discrepancy ∆ of a 2-(28, 4, 1) design is either 2 or 4. Proof. To prove this, we have to exclude ∆ = 0. In such case, there would exist a partition Π = (A, B) such that the induced subgeometries on A and w on B only have 2-blocks. These subgeometries are also linear spaces. Therefore, the number of 2-lines must be of the form 2 , w ∈ Z. But the number of blocks is 63 and is not of this form. The notion of a blocking set (see Section 1) is closely related to the discrepancy: Proposition 2. A 2-(28, 4, 1) design has a blocking set if and only if the discrepancy is 2. Proof. Let B be a blocking set for the 2-(28, 4, 1) design. Then B and A = P \ B define a partition Π = (A, B) of P, where all differences are 3 − 1 = 2 or 2 − 2 = 0. Therefore, the maximum value of these differences is u(Π ) ∈ {0, 2}. But the case of u(Π ) = 0 is not possible as this would imply ∆ = 0, a contradiction. The minimum u(Π ) over all partitions Π is 2. So we get ∆ = 2. Conversely, if ∆ = 2, then there is a partition Π = (A, B) with u(Π ) = 2. Therefore we can take A (and also B) as blocking set. In this paper, we classify all 2-(28, 4, 1) unital designs with blocking sets. According to Proposition 2, this means we classify all 2-(28, 4, 1) unital designs with discrepancy equal to 2. 4. The possible decompositions We make now the assumption that the unitals have a blocking set, and we determine the possible tactical decompositions. Lemma 3. Let (P , L) be a 2-(28, 4, 1) unital design. The column tactical decomposition
↓ 12 16
21 3 1
3 2 2
39 1 3
does not arise. Proof. Let (Π , Ψ ) be the decomposition that gives rise to the decomposition scheme. That is, we have Π = (A, B) a partition of P with |A| = 12 and |B| = 16. Also Ψ = (L1 , L2 , L3 ) a partition of L with |L1 | = 21, |L2 | = 3 and |L3 | = 39. For p ∈ A, let xi = |(p) ∩ Li | for i = 1, 2, 3. Also, for p ∈ B, let yi = |(p) ∩ Li | for i = 1, 2, 3. Then 2x1 + x2 = 11, x1 + 2x2 + 3x3 = 16, x1 + x2 + x3 = 9, 3y1 + 2y2 + y3 = 12, y2 + 2y3 = 15, y1 + y2 + y3 = 9 which yields
(x1 , x2 , x3 ) ∈ {(5, 1, 3), (4, 3, 2)},
(y1 , y2 , y3 ) ∈ {(1, 1, 7), (0, 3, 6)}.
Let w1 , w2 be the number of points of A of the first two types, and let w3 , w4 be the number of points of B of the latter two types. Then 5w1 + 12w2 + w3 ≤ 63, 3w1 + 6w2 + 7w3 + 18w4 ≤ 117,
w1 + w2 = 12, w3 + w4 = 16, w1 + w2 + w3 + w4 = 28. The first and third condition imply (w1 , w2 ) = (12, 0). Thus the first condition becomes w3 ≤ 3, which forces w4 ≥ 13. But then the second condition is violated. Lemma 4. Let (P , L) be a 2-(28, 4, 1) unital design with column tactical decomposition
↓ 13 15
24 3 1
6 2 2
33 1 . 3
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105
The decomposition can be refined uniquely to the row tactical decomposition
→ 7 6 9 6
24 6 5 2 1
6 0 2 0 2
33 3 2 . 7 6
Proof. We use notation similar to that of the proof of Lemma 3. Counting arguments show that
(x1 , x2 , x3 ) ∈ {(6, 0, 3), (5, 2, 2), (4, 4, 1), (3, 6, 0)} and
(y1 , y2 , y3 ) ∈ {(2, 0, 7), (1, 2, 6), (0, 4, 5)}. Let w1 , w2 , w3 , w4 be the number of points of A of the first 4 types. Let w5 , w6 , w7 be the number of points of B of the latter 3 types. Then
w1 + w2 + w3 + w4 + w5 + w6 + w7 = 28, w2 + 6w3 + 15w4 + w6 + 6w7 ≤ 15, 6w1 + 5w2 + 4w3 + 3w4 = 72, 2w2 + 4w3 + 6w4 = 12, 2w5 + w6 = 24. The only solution to this system is
(w1 , . . . , w7 ) = (7, 6, 0, 0, 9, 6, 0). Thus we get the unique refinement as claimed.
Theorem 5. Each 2-(28, 4, 1) unital design with a blocking set admits one of the following decomposition schemes
→
Type D: → 7 28 14 1 6 14 1 2
28 2 6
,
7 6 9 6
Type E: 24 6 6 0 5 2 2 0 1 2
33 3 2 7 6
.
(3)
The first decomposition (type D) is tactical while the second one (type E) is only point tactical. It is a refinement of the column tactical decomposition
↓ 13 15
24 3 1
6 2 2
33 1 . 3
(4)
Proof. Let Π = (A, B) be a partition of the point set P with ∆ = 2 = u(Π ). Let |A| = a and |B| = b. Then, the partition Π has the following block tactical decomposition.
↓ a b
x 3 1
y 2 2
z 1 3
.
Since a + b = 28, we can assume without loss of generality that a ≤ 14. Moreover, x + y + z = 63,
(5)
since there are 63 lines total. Counting flags (incident point–line pairs) inside A, we get 3x + 2y + z = 9a.
(6)
This is together with (5) implies 2x + y = 9a − 63.
(7)
Counting pairs of points (p, q) ∈ A × B, we get 3x + 4y + 3z = ab.
(8)
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A. Al-Azemi et al. / Discrete Applied Mathematics 163 (2014) 102–112
Using (5), we find y = ab − 189.
(9)
Under the assumption a ≤ 14, this implies
(a, b, y) ∈ {(12, 16, 3), (13, 15, 6), (14, 14, 7)}.
(10)
Now, x can be computed by using (7) and z follows by using (5). In the three cases, we get
(x, z ) ∈ {(21, 39), (24, 33), (28, 28)}.
(11)
Therefore we arrive at the following three block tactical decompositions:
↓ 12 16
21 3 1
3 2 2
↓
39 1 , 3
24 3 1
13 15
6 2 2
↓
33 1 , 3
and
14 14
28 3 1
7 2 2
28 1 . 3
The first decomposition is ruled out by Lemma 3. The second decomposition refines to the decomposition of type E by Lemma 4. In the third case, it is easy to show that the decomposition is tactical (i.e., both point tactical and block tactical). This is the decomposition of type D. The decomposition of type D is suitable for computer classification. On the other hand, the decomposition of type E is still too coarse, so we wish to refine it. Refining the decomposition further would give too many cases. So, we concentrate on the subgeometry with the 6-block domain (i.e., the lines ℓ ∈ L with |A ∩ ℓ| = |B ∩ ℓ| = 2). In particular, we consider the partial decomposition
→ 6 6
6 2 2
which is tactical. The upper part
→ 6
6 2
generates either a 6-cycle or two 3-cycles. The same holds for the lower part. This gives four cases: 1. 2. 3. 4.
Two 3-cycles in the upper part and a 6-cycle in the lower part, 6 cycle in the upper part and two 3-cycles below, two 6-cycles, one in the upper part and one in the lower part, two 3-cycles in the upper part and two 3-cycles in the lower part.
The fourth case is easily seen to be impossible. The first three cases can be described by 1. → 3 3 2 0 3 6 1
3 0 2 1
,
3.
2. → 3 6 1 3 2 0 3
→
3 1 0 2
3 3 3 3
,
6 2 2 2 2
.
These schemes can be included in the decomposition scheme of type E as described in Theorem 6. Theorem 6. The unitals generated from the scheme of type E are exactly those which are generated from the schemes of type F, type G, and type H. These three types are presented in (12). Type F
→ 7 3 3 6 9
3 0 2 0 1 0
3 0 0 2 1 0
24 6 5 5 1 2
Type H
Type G 33 3 , 2 2 6 7
→ 7 6 3 3 9
3 0 1 2 0 0
3 0 1 0 2 0
24 6 5 1 1 2
33 3 , 2 6 6 7
→ and
7 3 3 3 3 9
6 0 2 2 2 2 0
24 6 5 5 1 1 2
33 3 2 . 2 6 6 7
5. Classification of the unital designs In this section, we describe the classification procedure for unital designs admitting a blocking set.
(12)
A. Al-Azemi et al. / Discrete Applied Mathematics 163 (2014) 102–112
107
Table 1 Computing the list of starters for decomposition schemes of types D, F, G, and H. Type
Depth n
Number of generated cases
Number of starters
D F G H
14 13 13 13
787 59 13,365 13,369
787 32 191 191
By the results of Section 4, this is equivalent to classifying the geometries from the decomposition schemes of type D (as in (3)), and of types F, G, and H (as in (12)). We assume familiarity with the basics of classification algorithms of incidence geometries. A detailed introduction to such algorithms can be found in [8]. We begin by classifying geometries satisfying these decompositions partially. By this, we mean that we classify the partial incidence matrices of geometries admitting a decomposition. A partial incidence matrix is a matrix whose first n rows satisfy the conditions imposed by the decomposition as well as by the geometry, but whose remaining rows are all zero. The idea is to fill the remaining rows of the incidence matrix later. We say that n is the depth of the partial incidence matrix. The parameter n is chosen after some experimentation, and it varies from decomposition to decomposition. The goal is to first classify all partial incidence matrices at depth n under the group fixing the decomposition scheme and fixing the first n rows setwise. For type D we choose n = 14, and for all other decompositions schemes we choose n = 13. We represent each orbit of partial incidence matrices by its canonical representative. This is just the matrix that is least in the lexicographical ordering of all matrices in its isomorphism class. In the next step, we forget about the decomposition scheme. That is, we classify the partial incidence matrices under the group that fixes the first n rows but otherwise is allowed to move points across the boundaries of the decomposition. This eliminates copies of geometries that differ only by a rearrangement of the decomposition classes. For instance, in the case of type F we have two 3-cycles which can be permuted and hence result isomorphic copies. Therefore, we do an isomorphism test at line 14 (type D) and at line 13 (types F, G, and H) under the group that fixes the first 14 rows setwise. The resulting list of cases (for each decomposition scheme type) are called the list of starters (at depth n). The number of isomorphism types of partial incidence matrices at depth n, and the number of starters at depth n is presented in Table 1. For each starter at depth n, we run a computer search to generate the partial incidence matrices at some depth m with n < m. Here, m = 18 for type D and m = 19 for the remaining types. In order to find the full incidence matrices of the geometries that we are looking for, we need to complete the partial incidence matrices at depth m to depth 28. This can be formulated as a problem of finding all cliques in a certain graph: For each isomorphism type of starter S at depth m, we define a graph ΓS . The vertices of ΓS are the possibilities for the missing rows. These are the 0/1 vectors that could potentially be used to fill in additional rows of the partial incidence matrix. Of course, these 0/1 vectors have to satisfy the conditions that result from the decomposition (as well as from the fact that we create a linear space). Two vertices of ΓS are connected if the corresponding vectors can both be chosen simultaneously to extend the partial incidence matrix. This requires testing if the rows have exactly one position where both have an entry one. Also, it requires testing if the conditions on the column sums are satisfied. For each starter, we present the number of cliques in the second column of Table 2 (type D), Table 3 (type F), Table 4 (type G), and Table 5 (type H). Once the incidence matrices have been filled to depth 28 using the cliques of size 28 − m from the graph Γ , we perform an additional round of isomorphism testing. This produces the classification of all geometries for a given scheme with a given starter at depth n (in the third column). After that we consider the geometries that are obtained from all starters and perform another round of isomorph rejection, this time disregarding the decomposition schemes. The numbers that result from this are listed at the bottom as ‘‘merged’’. The result of this procedure is presented in Theorem 7. Theorem 7. The decomposition scheme of types D, F, G, and H generate the numbers of (pairwise non-isomorphic) unitals presented in Table 6. Considering the unitals from all decomposition schemes together, and performing another round of isomorphism testing leads to the following result: Theorem 8. The number of (pairwise non-isomorphic) unitals of order 3 with discrepancy 2 (i.e., having a blocking set) is 68,806. The distribution of automorphism group orders is
{1922 , 487 , 242 , 162 , 127 , 819 , 7, 63 , 487 , 3102 , 290 , 168484 }. 6. Note added in the proof In work that was done after this paper was first submitted, we checked our results against the list of unital designs with non-trivial automorphism group [9]. The number of unital designs with non-trivial automorphism group is 4466. We tested which of these had a blocking set and found that exactly 322 did. This agrees with our number of 68,806 − 68,484 = 322 unital designs with blocking set and non-trivial automorphism group from Theorem 8.
108
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Table 2 Unitals of type D, generated from the list of starters at line 14. Nr.
gen.
isot.
Nr.
gen.
isot.
Nr.
1 11 12 14 16 21 23 24 26 30 32 35 39 40 41 42 50 51 52 55 59 61 67 69 72 73 76 77 79 80 89 90 93 96 98 99 101 102 105 109 112 113 114 115 116 117 118 119 120 121 122 123 126 129 130 131 132 133 137 138 139 140 145 146 147 149 150 152
18 36 114 16 8 8 4 2 8 2 16 8 352 32 4 4 8 8 8 12 8 10 4 24 80 80 4 4 48 152 40 6 4 4 24 16 16 4 26 2 8 28 4 4 4 6 40 16 24 20 44 8 60 2 8 6 2 4 16 8 4 4 4 16 4 32 12 2
1 13 19 1 1 2 1 1 1 1 1 3 10 3 1 1 1 2 1 2 2 2 1 1 4 6 1 1 3 42 2 2 1 1 1 1 2 1 3 1 1 4 1 1 2 2 7 1 3 2 5 2 4 1 1 2 1 1 1 2 1 1 1 2 1 1 2 1
292 294 295 297 299 301 303 305 306 307 308 310 312 313 314 315 316 317 318 319 321 323 325 326 332 333 335 340 342 344 345 347 348 349 350 351 354 355 358 359 360 362 363 364 368 369 370 372 374 375 377 378 379 380 381 382 383 384 386 387 390 391 392 393 394 395 396 397
12 8 4 8 8 12 66 4 8 24 40 32 40 24 16 4 8 8 20 6 4 4 4 8 2 2 18 24 8 16 2 8 20 4 4 8 16 8 12 16 16 16 10 8 2 10 8 8 4 8 6 8 4 10 2 24 8 8 12 8 8 16 2 8 48 28 40 2
2 1 1 1 1 2 4 1 3 5 11 2 7 6 1 1 1 1 2 2 1 1 1 1 1 1 4 3 1 1 1 1 2 1 1 3 1 2 2 1 2 3 2 2 1 2 1 1 1 1 2 1 1 3 1 3 1 1 2 1 1 2 1 1 9 4 6 1
531 532 533 536 537 539 540 542 543 545 546 547 550 551 555 559 561 562 564 566 567 568 569 570 572 573 575 577 580 583 585 586 588 589 590 593 595 597 600 601 602 603 604 611 614 615 617 618 620 622 623 624 629 632 637 638 640 641 646 648 651 654 659 661 662 665 669 670
gen. 16 8 32 4 16 8 4 14 16 20 8 16 32 14 4 8 4 16 20 24 8 24 20 16 10 32 8 46 8 12 80 16 8 4 12 8 16 20 16 8 8 8 4 8 24 8 4 24 8 4 4 12 2 8 8 10 22 8 16 4 8 8 4 8 4 8 12 38
isot. 1 2 8 1 1 2 1 4 1 2 1 1 7 3 1 1 1 4 3 4 1 3 2 2 2 4 1 7 1 2 13 1 2 1 2 2 1 2 2 2 1 2 1 2 2 1 1 2 1 1 1 2 1 4 4 2 5 1 3 1 2 2 1 1 1 1 2 6
A. Al-Azemi et al. / Discrete Applied Mathematics 163 (2014) 102–112
109
Table 2 (continued) Nr.
gen.
isot.
Nr.
gen.
isot.
Nr.
154 157 158 161 162 166 168 170 171 172 174 176 177 179 180 183 184 185 186 187 192 193 195 196 197 198 202 203 204 206 207 208 211 212 213 214 216 217 218 220 222 223 225 226 227 229 231 232 237 242 245 246 247 248 249 253 254 255 256 257 258 259 260 265 276 277 279 280 283 284
4 4 16 4 22 16 16 4 28 12 24 4 8 50 2 36 16 40 4 4 144 8 12 2 8 4 44 16 20 16 68 8 20 4 16 16 4 56 8 16 2 4 16 16 12 2 8 12 10 24 4 16 4 16 18 16 8 8 8 8 8 4 24 8 8 4 12 32 6 12
1 1 2 1 3 4 1 1 3 2 3 1 2 8 1 2 1 2 1 1 9 1 2 1 1 1 4 3 2 2 6 1 2 1 1 1 1 3 1 2 1 1 1 1 2 1 1 2 2 2 1 1 1 7 8 1 1 1 1 1 1 2 2 1 1 1 2 2 2 2
399 400 401 403 405 409 412 414 415 416 417 423 424 425 426 427 428 430 431 433 434 436 440 442 448 452 453 455 458 459 460 461 462 464 465 466 468 470 472 473 474 475 476 478 481 484 485 486 489 491 492 494 495 497 499 500 501 503 504 505 506 508 515 517 519 520 521 522 523 526
28 12 4 16 8 24 6 8 4 16 32 4 10 4 24 12 2 2 4 16 2 16 20 24 4 4 8 8 4 24 16 4 44 14 16 8 4 16 8 8 32 2 8 8 4 4 4 10 16 2 16 16 10 16 6 16 12 24 20 26 4 24 4 4 4 22 4 4 4 16
3 2 1 1 2 3 2 1 1 1 6 1 2 1 2 2 1 1 1 1 1 1 1 5 1 1 1 1 1 2 2 2 10 3 3 1 1 1 1 1 7 1 1 2 1 1 1 3 2 1 4 2 2 1 2 1 2 6 2 3 1 2 1 1 1 5 1 2 1 4
672 674 675 676 677 678 680 682 683 684 685 686 687 689 691 692 694 695 696 697 699 700 702 703 705 706 707 708 709 710 712 713 714 715 717 718 719 720 721 723 724 726 728 731 732 735 736 737 738 740 741 742 743 744 745 746 747 749 752 754 757 759 761 767 769 771 772 773 774 776
gen. 40 32 8 4 8 34 12 8 6 20 4 6 16 16 8 8 20 8 10 8 24 24 20 16 4 4 12 40 8 4 24 12 4 8 12 12 14 808 8 4 398 6 8 12 4 8 4 4 18 36 4 34 8 18 4 12 2 32 240 16 2 2 24 16 4 4 22 24 8 8
isot. 2 1 1 1 1 3 2 1 2 1 1 2 3 1 1 2 2 1 2 2 2 2 2 2 1 1 2 5 2 1 3 2 1 1 2 3 1 72 1 1 32 2 1 2 1 1 1 1 1 3 1 5 1 2 1 2 1 4 8 1 1 1 2 2 1 1 5 3 1 1
(continued on next page)
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A. Al-Azemi et al. / Discrete Applied Mathematics 163 (2014) 102–112
Table 2 (continued) Nr.
gen.
isot.
Nr.
gen.
288 290 291
12 32 4
2 2 1
527 528 530
4 16 24
isot.
Nr.
1 1 2
778 780 783
gen.
isot.
4 6 8
Total:
1 2 1 1015
Merged:
436
Table 3 Unitals of type F, generated from the list of starters at line 13. Nr.
gen.
isot.
Nr.
gen.
isot.
Nr.
gen.
1 2 3 4 5 6 7 8 9 10 11
4106 4526 4340 4268 4494 4110 4190 4196 4162 4460 4616
534 557 525 554 571 533 546 563 525 591 596
12 13 14 15 16 17 18 19 20 21 22
2362 4600 4312 1354 2030 1958 4170 4418 4502 4350 2220
206 571 533 106 277 261 276 277 276 540 265
23 24 25 26 27 28 29 30 31 32
2000 4550 4470 4514 2232 2378 4286 4292 4678 4456
isot. 252 277 561 566 279 276 283 549 561 554
Total:
13,841
Merged:
13,819
Table 4 Unitals of type G, generated from the list of starters at line 13. Nr.
gen.
isot.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
499 489 537 480 657 454 561 382 627 561 468 524 567 562 568 429 439 546 425 408 431 500 614 648 529 616 476 516 493 433 477 502 382 547 537
86 95 90 94 107 91 94 79 95 90 84 92 92 98 89 78 88 91 71 77 84 80 93 116 96 96 92 93 91 87 84 93 81 87 95
Nr. 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
gen.
isot.
Nr.
gen.
463 216 253 475 662 578 488 238 542 556 503 545 431 506 229 422 286 298 403 533 488 567 586 499 401 514 606 573 476 437 403 396 479 478 574
95 38 41 89 103 98 71 36 48 94 48 90 39 89 21 39 50 45 79 90 83 90 96 98 81 106 100 94 89 82 76 81 85 94 92
129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163
442 467 599 674 524 449 396 509 525 538 579 466 448 523 385 477 414 412 427 414 480 422 555 688 399 512 354 548 481 517 522 542 447 528 523
isot. 71 83 106 106 79 85 81 45 81 76 87 76 84 41 71 81 63 72 65 67 38 54 86 98 67 77 69 39 77 79 79 81 30 71 85
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111
Table 4 (continued) Nr.
gen.
isot.
Nr.
gen.
isot.
Nr.
gen.
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
389 571 516 633 642 517 490 521 454 478 445 622 418 565 483 392 502 512 513 510 498 587 390 214 278 516 471 564 553
74 101 81 97 106 103 81 94 78 91 85 93 82 90 79 77 91 91 97 92 95 103 586 39 44 93 88 97 115
100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
755 255 700 628 193 570 644 435 530 501 276 381 495 288 268 590 665 538 552 487 564 482 468 545 504 505 591 466 451
60 37 119 57 38 99 104 81 90 90 57 65 103 46 52 89 106 84 115 94 98 89 80 84 81 86 46 86 88
164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191
404 483 399 474 570 556 442 494 430 299 491 611 666 477 466 534 96 514 531 354 395 585 486 256 324 374 390 226
isot. 70 37 72 75 89 85 38 67 68 55 69 86 89 67 72 76 21 70 80 59 65 90 69 20 23 47 52 26
Total:
15,114
Merged:
14,901
Table 5 Unitals of type H, generated from the list of starters at line 13. Nr.
gen.
isot.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
3660 3266 1316 3284 954 1041 3475 3204 3607 4051 3654 3839 3678 3418 4153 3656 3327 3158 4209 3400 3738 3453 3779 3579 3068 3660 3118 3295 3197 3591 3618 3152
258 239 82 237 78 72 249 229 253 296 250 264 268 248 293 258 237 229 277 250 258 250 258 246 217 253 223 240 239 248 269 225
Nr. 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
gen.
isot.
Nr.
gen.
1059 1354 3602 3551 3027 3747 3490 3737 1533 3700 3467 1807 1555 3740 3311 3656 1071 965 3495 3330 3437 3040 3351 3388 3378 2887 3491 3697 3143 3720 2967 3259
76 89 261 262 217 276 247 260 114 266 246 128 107 269 119 127 73 39 252 256 234 229 235 236 248 216 257 254 219 249 215 240
129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
3519 3568 3135 3303 3430 3222 2946 3792 3208 3808 3994 3053 2991 3170 3155 3333 3168 1694 1433 2700 1663 1722 1038 1564 2498 3164 3199 3219 2880 2875 3200 3225
isot. 245 253 223 243 133 126 113 131 123 133 275 205 204 211 214 204 220 113 92 187 107 117 70 102 175 211 207 221 196 193 205 218
(continued on next page)
112
A. Al-Azemi et al. / Discrete Applied Mathematics 163 (2014) 102–112 Table 5 (continued) Nr.
gen.
isot.
Nr.
gen.
isot.
Nr.
gen.
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
3336 3221 3057 3726 3676 3426 3476 3317 3457 2995 3607 3037 3310 2993 3488 3446 3493 3511 3564 3450 3268 3537 3140 3306 3710 3517 3510 3856 3673 4183 3274 1306
251 223 233 265 260 250 246 243 247 223 252 226 245 229 238 238 243 246 261 244 243 256 213 231 262 260 251 273 257 294 235 94
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
2941 3517 3061 3151 3308 3304 3056 3496 3334 3584 3227 3358 3456 3718 3388 3584 3370 3892 3400 3198 3134 3543 3439 3316 3879 3567 3091 2880 3224 3325 3711 3498
207 125 230 124 118 121 211 244 237 244 235 236 236 259 244 248 236 258 244 220 234 245 245 226 280 248 218 212 250 227 255 249
161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191
3484 3248 2813 2913 3036 2779 2992 3273 3211 2860 2852 2995 2795 3423 3053 2897 2226 2696 2452 2626 2663 2918 2716 3125 1119 2561 974 1632 2268 888 816
isot. 242 224 207 203 207 181 207 211 213 200 194 209 182 226 215 196 163 175 171 171 173 204 192 203 76 186 63 106 140 63 51
Total:
40,031
Merged:
39,886
Table 6 The generated unitals from the decomposition of types D, F, G, and H. The corresponding distributions of automorphism group orders (ago’s) are presented in the third column. Type
Solutions
Distribution of ago’s
D F G H
436 13,819 14,901 39,886
{1225 , 278 , 33 , 487 , 63 , 7, 819 , 127 , 162 , 242 , 487 , 1922 } {113,770 , 24 , 345 } {114,846 , 21 , 354 } {139,876 , 27 , 33 }
Total:
69,042
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