Search for systems of linked symmetric 2 (36 , 15 , 6) designs - - PowerPoint PPT Presentation

Search for systems of linked symmetric 2 − (36, 15, 6) designs

Matan Ziv-Av

Ben-Gurion University of the Negev

AGT16, Plzeˇ n, October 2016.

1

Outline

1

Symmetric 2-designs

2

Systems of linked symmetric designs

3

Parameters (36, 15, 6)

4

Concluding remarks

2

Axioms

A symmetric 2-design with parameters (v, k, λ) (also called a square 2-design) is an incidence structure (B, P, I) such that

1

|P| = v;

2

|B| = v;

3

Each block is incident with exactly k points.

4

Each point is incident with exactly k blocks.

5

Each pair of blocks is incident with exactly λ points.

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Each pair of points is incident with exactly λ blocks.

Some necessary conditions on the parameters are:

λ(v − 1) = k(k − 1). Bruck-Ryser-Chowla: if v is even, then k − λ is a square; if v is odd then the equation x2 − (k − λ)y 2 − (−1)

v−1 2 λz2 = 0 has non zero integer solutions.

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Related designs

The dual of a symmetric 2-design is a symmetric 2-design with the same parameters. The complement of a symmetric 2-(v, k, λ) design (taking the complement of the incidence relation) is a symmetric 2-(v, v − k, v − 2k + λ) design.

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Projective planes

When λ = 1 we get the axioms of a projective plane. v = (k + 1)2 + (k + 1) + 1. Exists when k − 1 is a prime power. Does not exist for k − 1 = 6 and for k − 1 = 10.

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Biplanes

When λ = 2 we get a biplane. v = 1 + k(k−1)

2

. The order of a biplane is k − 2. For k = 2, 3, 4, 5, there exists a unique biplane. For k = 6 (v = 16) there are 3 biplanes. For k = 9 there are 4 biplanes. For k = 11, there are 5 biplanes. For k = 13, there are 2 known biplanes.

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Biplane of order 2

k = 4, v = 1 + k(k−1)

2

= 7. Parameters: (7, 4, 2). Complement design (7, 3, 1). Fano plane: B = {{1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {4, 5, 7}, {1, 5, 6}, {2, 6, 7}, {1, 3, 7}} Thus the unique (up to isomorphism) biplane of order 2 is B = {{3, 5, 6, 7}, {1, 4, 6, 7}, {1, 2, 5, 7}, {1, 2, 3, 6}, {2, 3, 4, 7}, {1, 3, 4, 5}, {2, 4, 5, 6}}

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Incidence matrix

With each symmetric 2-design we associate its incidence matrix. This is a v × v (0,1)-matrix with rows denoted by points and columns by blocks. The (i, j) entry is 1 if point i is incident to block j and 0 otherwise. The following holds for A, an incidence matrix of a symmetric 2-design: AAT = kI + λ(J − I)

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Polarities

A polarity of a symmetric design (P, B, I) is a bijection σ : B → P, such that xIσ(y) if and only if yIσ(x). It is easy to see that a polarity can be interpreted as a permutation of the columns of the incidence matrix such that the permuted matrix is symmetric. In this case we can rewrite the matrix equation: A2 = kI + λ(J − I) This is the matrix equation of a (v, k, λ, λ)-SRG.

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Biplane of order 2

B = {{3, 5, 6, 7}, {1, 4, 6, 7}, {1, 2, 5, 7}, {1, 2, 3, 6}, {2, 3, 4, 7}, {1, 3, 4, 5}, {2, 4, 5, 6}} A =   

0 1 1 1 0 1 0 0 0 1 1 1 0 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0

   σ = (2, 6, 5)(4, 7), A′ =   

0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 1 1 0 0 1 0

   Also: σ = (1, 2)(3, 5, 7, 4, 6), A′′ =   

1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0

   Those are not the matrices of a simple graph.

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Absolute points

If σ is a polarity of a design D = (P, B, I), then p ∈ P is an absolute point of σ if p ∈ σ(p). In the incidence matrix: if the i-th element of the diagonal is 1. Back to the biplane of order 2: A′′ =   

1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0

   This polarity has 4 absolute points.

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SRG (v, k, λ, λ)

If D = (P, B, I) is a design and σ its polarity with no absolute points, then its incidence matrix A is the adjacency matrix of a SRG with parameters (v, k, λ, λ). Conversely, if Γ = (V , E) is a (v, k, λ, λ), then (V , {Γ(v)|v ∈ V }, ∈) is a symmetric 2 − (v, k, λ) design. In matrices language: the adjacency matrix of Γ is an incidence matrix of a design. Isomorphism is not preserved when going from a design to a graph. Two non-isomorphic (v, k, λ, λ) SRGs may correspond to the same design. There are three (16, 6, 2) designs (biplanes of order 4), but only one of them admits a polarity with no absolute points. There are two (16, 6, 2, 2) SRGs (Shrikhande graph and L2(4)), so they both generate the same biplane.

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SRG (v, k − 1, λ − 2, λ)

If a (v, k, λ) design has a polarity with v absolute points, then A − I is an adjacency matrix of a graph. Recall A2 = kI + λ(J − I), so (A − I)2 = (k − 1)I + (λ − 2)A + λ(J − I − A). That is, A − I is the adjacency matrix of a (v, k − 1, λ − 2, λ) SRG. Conversely, if Γ = (V , E) is a (v, k − 1, λ − 2, λ), then (V , {v ∪ Γ(v)|v ∈ V }, ∈) is a symmetric 2 − (v, k, λ) design. In matrices language: A(Γ) + I is an incidence matrix of a design. Again, two non-isomorphic graphs may correspond to the same design. The smallest example has parameters (35, 16, 6, 8).

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Symmetric 2-designs as association schemes

Let (P, B, I) be a symmetric (v, k, λ) 2-design. Let us take Ω = P ∪ B. R0 = ∆Ω. Let R1 be the relation on Ω containing pairs of distinct elements of the same type. R2 = I is the incidence relation. R3 is the remaining pairs. Then (R0, R1, R2, R3) is a symmetric association scheme. Let us denote the corresponding adjacency matrices by I, A1, A2, A3. A2

1 = (v − 1)I + (v − 2)A1,

A1A2 = (k − 1)A2 + kA3, A1A3 = (v − k)A2 + (v − k − 1)A3, A2

2 = kI + λA1,

A2A3 = (k − λ)A1, A2

3 = (v − k)I + (v − 2k − λ)A1.

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Symmetric 2-designs as association schemes

Conversely, an imprimitive symmetric association scheme of rank 4 and order 2v having

• ne equivalence relation with two equivalence classes of the same size corresponds to a

symmetric 2-design. Indeed if we take take P to be one equivalence class and B to be the other, I to be R2, then:

there are v points and v blocks. Each block is incident to p0

22 points, and vice versa;

Every two blocks are incident to exactly p1

22 common points, and vice versa.

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Generalization

We want to generalize symmetric designs. We look for rank 4 symmetric association schemes with R1 an equivalence relation with f equivalence classes of the same size. We need an extra requirement: For any two equivalence classes A, B, restriction of the association scheme to A ∪ B results in a rank 4 association scheme with the same parameters.

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Systems of linked symmetric designs

(Ω1, . . . , Ωf , I) is a homogeneous system of linked symmetric designs with parameters (v, k, λ) if:

Each pair (Ωi, Ωj, I) is a symmetric design with parameters (v, k, λ). For every distinct i, j, l and for any a ∈ Ωi, b ∈ Ωj, the number of c ∈ Ωl incident to both a and b depends only on whether a and b are incident.

For f = 2 we get a single symmetric design. The multiplication table becomes: A2

1 = (v − 1)I + (v − 2)A1,

A1A2 = (k − 1)A2 + kA3, A1A3 = (v − k)A2 + (v − k − 1)A3, A2

2 = (f − 1)kI + (f − 1)λA1 + (f − 2)x2A2 + (f − 2)x1A3,

A2A3 = (f − 1)(k − λ)A1 + (f − 2)(k − x2)A2 + (f − 2)(k − x1)A3, A2

3 = (f −1)(v −k)I +(f −1)(v −2k−λ)A1+(f −2)(v −2k+x2)A2+(f −2)(v −2k+x1)A3.

Here x1 is the number of c when a, b are incident and x2 is the number of c when they are not incident.

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History of systems of linked symmetric designs

Systems of linked symmetric designs were defined by Cameron (1972) for the investigation of groups with more than two pairwise non-equivalent representations as doubly transitive permutation groups. Cameron and Seidel (1973) constructed an infinite series of systems of linked designs with parameters (22t+2, 22t+1 − 2t, 22t − 2t) (f = 22t+1 − 1) Noda (1974) added a few constraints on the parameters of such systems. Among them: k − λ = u2, x1 = k(k+u)

v

, x2 = x1 − u. Mathon (1981) used the above results and using the method of admissible sets constructed all linked symmetric designs with parameters (16, 6, 2). Davis, Martin and Polhill (2013) described a new series of systems of linked designs with the same parameters. The authors suggest that there is evidence that the systems in this series are not isomorphic, but prove this only for the case t = 1.

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Non power of 4 - smallest feasible parameters

All known systems of linked designs have order power of 4: The infinite series by Cameron, Seidel and by Davis, Martin, Polhill. The 19 systems with f = 3, 4, 5, 6, 7 and parameters (16, 6, 2) discovered by Mathon. The smallest feasible sets of parameters for which the existence of a system of linked designs is unknown is:

(36, 15, 6) (45, 12, 3).

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Admissible sets

Mathon’s method: Let P = [1, v], B be a collection of subsets of P of size k. If a design D = (P, B) is a part of a system of linked designs, then there exists another design (P, B′) such that each block C ∈ B′ intersect each block of B in either x1 or x2 points. Define a k−subset of P as admissible for design D if it intersects each block of B in either x1 or x2 points. The algorithm is then: enumerate all admissible sets, and construct all possible B′ from them. In the case (v, k, λ) = (16, 6, 2), then k − λ = 4 = u2, so u = ±2. x1 = k(k+u)

v

, so u = 2, x1 = 3, x2 = x1 − u = 1.

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Spence - SRGs (36, 15, 6, 6)

• E. Spence constructed 32584 SRGs with parameters (36, 15, 6, 6).

Later Spence and McKay proved that this list is exhaustive. A computer readable version of the list is available from Spence’s web page. By using a computer, we found that those SRGs generate 31284 pairwise non-isomorphic symmetric 2 − (36, 15, 6) designs.

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Spence - SRGs (36, 14, 4, 6)

• E. Spence constructed 180 SRGs with parameters (36, 14, 4, 6).

Later Spence and McKay proved that this list is exhaustive. A computer readable version of the list is available from Spence’s web page. Again, using a computer we found that those SRGs generate 180 non-isomorphic designs. 13 of those also have polarities with no absolute points, i.e. they appeared in the previous list, so we get 167 new designs.

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Spence - others

The number of absolute points of a polarity in a (36, 15, 6) design is divisible by 6. The previous two lists include all designs with polarities with 0 or 36 absolute points. Spence provides (non-exhaustive) lists of designs with 6, 12, 18, 24 and 30 absolute points. The lists include 120, 6, 381, 71, 39 designs respectively. Of those 617 designs, there are 534 pairwise non-isomorphic designs. 519 of them did not appear previously. Altogether, from all data provided by Spence, we generate a list of 31970 designs.

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Crnkovi´ c

• D. Crnkovi´

c (1999) constructed and classified (36, 15, 6) designs with some constrains on the automorphism groups. There exist 38 designs with an automorphism of order 5. Of them, 17 appear in Spence’s lists. There are 4 designs with an automorphism of order 7. Three of them appear in Spence lists.

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Consolidation

Altogether 31970 + 22 = 31992 designs are known. Crnkovi´ c also found some designs with automorphisms of order 3 and 4, but does not provide enough information to reconstruct them. I did not find in literature other information about known designs, though it is possible I missed some relevant papers.

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Results using Mathon’s methods

For each of the 31992 designs I calculated all admissible sets. That is, sets that intersect each of the 36 blocks of the design in either 5 or 8 points. I did not test that it intersects exactly 15 sets in 5 points. Need to test 36

15

• ≈ 232 sets.

This can be easily reduced to 15

5

21

10

• +

15

8

21

7

• ≈ 230.

Less than 300 designs have enough admissible sets. For the designs that do have enough admissible sets, no system of linked designs can be constructed.

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Search for further symmetric (36, 15, 6) 2-designs

One way to settle the question of existence of a system of linked designs with parameters (36, 15, 6) is to construct all symmetric designs with this parameters. The first step in the process is to determine the six sets that contain the points 1 and 2. There are about 1500 options up to equivalence. For each of those cases, all options of selecting the remaining 30 sets need to be determined. Unfortunately, the estimated runtime for each case is about 1000 CPU years. Partial results:

No designs where the maximum size of intersections of three blocks is 3. A few designs that do not appear in any of the above lists.

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Other potential solutions

There may be other ways to solve the problem: Theory of association schemes. It may be possible to show that an association scheme of rank 4 and order 108 with the requires properties does not exist. Or maybe construct such a scheme directly and not by using smaller schemes (=designs) as building blocks. Theory of systems of linked symmetric designs: perhaps all systems of linked symmetric designs have 4n points for some n.

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Some references

Cameron, Peter J. On groups with several doubly-transitive permutation

• representations. Math. Z. 128 (1972), 114.

Mathon, Rudolf The systems of linked 2 − (16, 6, 2) designs. Ars Combin. 11 (1981), 131148. Crnkovi´ c, Dean On symmetric (36,15,6) designs. Glas. Mat. Ser. III 34(54) (1999), no. 2, 105108. B McKay and E Spence, Classification of regular two-graphs on 36 and 38 vertices. Australasian Journal of Combinatorics, vol.24, p.293 (2001)

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Thank you for your attention