Tight relative 2-designs on 2 shells in Johnson scheme Yan Zhu, - - PowerPoint PPT Presentation

Tight relative 2-designs on 2 shells in Johnson scheme

Yan Zhu, Eiichi Bannai and Etsuko Bannai zhuyan870311@sina.com

Shanghai Jiao Tong University

March 15, 2014

Outline

• 1. Introduction
• 2. Main results
• 3. Construction of some examples
• 4. Future work

2 / 27

Association scheme

X = a finite set, {R0, R1, . . . , Rd} = the set of relations on X (i.e. Ri ⊆ X × X ),

R0 = {(x, x)|x ∈ X }. R0

R1 . . . Rd = X × X , and Ri Rj = ∅ if i = j .

tRi = Rj for some j ∈ {0, 1, . . . , d} ,where tRi = {(y, x)|(x, y) ∈ Ri}. |{z ∈ X |(x, z) ∈ Ri, (z, y) ∈ Rj }| = pk

i,j is a constant whenever

(x, y) ∈ Rk. Then X = (X , {Ri}0≤i≤d) is an association scheme. Moreover, it is symmetric if tRi = Ri.

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Adjacency matrix

The i-th adjacency matrix Ai of X is defined by (Ai)xy =

• 1,

if (x, y) ∈ Ri 0,

• therwise

A0 = I . A0 + A1 + . . . + Ad = J. tAi = Aj AiAj = d

i=0 pk i,j Ak = Aj Ai.

{A0, A1, . . . , Ad} form an associative commutative algebra which is called the Bose-Mesner algebra of the association scheme.

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Matrix version

Symmetric association scheme: X = (X , {Ri}i=0,...,d). Adjacency matrices: A0, . . . , Ad. Primitive idempotents: E0, . . . , Ed. Bose-Mesner algebra: C[A0, . . . , Ad] = C[E0, . . . , Ed]

AiAj =

d

• i=0

pk

i,j Ak

and Ei ◦ Ej = 1 |X |

d

• i=0

qk

i,j Ek.

Eigenmatrices:

(A0, . . . , Ad) = (E0, . . . , Ed)P (1) (E0, . . . , Ed) = 1 |X |(A0, . . . , Ad)Q (2) i.e., Ai =

d

• j =0

Pi(j )Ej and Ei = 1 |X |

d

• j =0

Qi(j )Aj

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X = (X , {Ri}0≤i≤d) is called a P-polynomial scheme with respect to the

• rdering A0, A1, . . . , Ad, if there exist some polynomials vi(x) of degree i

such that Ai = vi(A1). X = (X , {Ri}0≤i≤d) is called a Q-polynomial scheme with respect to the

• rdering E0, E1, . . . , Ed, if there exist some polynomials v ∗

i (x) of degree i

such that Ei = v ∗

i (E1).

Definition 1.1 Let V be a set of cardinality v and let d be a positive integer with d ≤ v

2.

Let X be the set of d-element subsets of V . Define Ri by (x, y) ∈ Ri if |x ∩ y| = d − i. Then X = (X , {Ri}0≤i≤d) is a symmetric association scheme of class d and is called Johnson scheme J(v, d).

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Some notations

X = (X , {Ri}0≤i≤d): a symmetric association scheme. u0 ∈ X : a fixed point arbitrarily. Xi = {x ∈ X |(u0, x) ∈ Ri}, then X0, X1, . . . , Xd are called shells of X. F(X ): the vector space consists of all the real valued functions on X . Lj (X ): the subspace of F(X ) spanned by all the columns of Ej . F(X ) = L0(X )⊥L1(X )⊥ . . . ⊥Ld(X ). Denote mj = dim(Lj (X )) = rank(Ej ), ki = |{y ∈ X |(u0, y) ∈ Ri}|.

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Definition 1.2 [1] Let (Y , w) be a weighted subset of X with positive function w on Y . (Y , w) is called a relative t-design with respect to u0 if the following condition holds.

p

• i=1
• x∈Xri

Wri |Xri |f (x) =

• y∈Y

w(y)f (y) (3) for any function f ∈ L0(X )⊥L1(X )⊥ . . . ⊥Lt(X ), where Wri =

y∈Yri w(y), i = 1, 2, . . . , p.

Let {r1, r2, . . . , rp} = {r|Xr Y = ∅} and S = Xr1 Xr2 . . . Xrp, we say Y is supported by p shells. Denote Yri = Y Xri , i = 1, 2, . . . , p.

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Theorem 1.3 [1] Let (Y , w) be a relative 2e-design of a Q-polynomial scheme. Then the following inequality holds. |Y | ≥ dim(L0(S) + L1(S) + . . . + Le(S)), (4) where Lj (S) = {f |S, f ∈ Lj (X )}, j = 0, 1, . . . , e. Definition 1.4 If equality holds in (4), then (Y , w) is called a tight relative 2e-design with respect to u0. Theorem 1.5 [2] Let X = (X , {Ri}0≤i≤d) be a Q-polynomial scheme. Let G be the automorphism group of X. Let (Y , w) be a tight relative 2e-design with respect to u0. Assume that the stabilizer Gu0 of u0 acts transitively on every shell Xr, 1 ≤ r ≤ d. Then the weighted function w of any tight relative 2e-design (Y , w) is constant on each Yri (1 ≤ i ≤ p).

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Formula for some parameters

Pi(j ) =

i

• t=0

(−1)t j t d − j i − t v − d − j i − t

• .

(5) Qj (i) = Pi(j )mj ki . (6) mj = v j

• −

v j − 1

• and

ki = d i v − d i

• .

(7) We consider 2-designs on 2 shells, i.e., e = 1 and p = 2. E0 = 1 |X |J, E1 = 1 |X |

d

• j =0

Q1(j )Aj .

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Theorem 2.1 Take a sequence elements from X as u0 = {1, 2, . . . , d}, ui = {1, 2, . . . , d − 1, d + i + 1}, (1 ≤ i ≤ v − d − 1) ui = {1, 2, . . . , d, d + 1} \ {i − (v − d) + 1}, (v − d ≤ i ≤ v − 1) i.e., u1 = {1, 2, . . . , d − 1, d + 2} u2 = {1, 2, . . . , d − 1, d + 3} . . . uv−d−1 = {1, 2, . . . , d − 1, v} uv−d = {2, 3, . . . , d − 1, d, d + 1} uv−d+1 = {1, 3, . . . , d − 1, d, d + 1} . . . uv−1 = {1, 2, . . . , d − 1, d + 1} Then {φ0|S, φ1|S, . . . , φv−1|S} is a basis of L0(S) + L1(S), where S = Xr1 Xr2.

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Some notations

φ0(x) = φ(0)

u0 (x) = |X |E0(x, u0) ≡ 1

(8) φi(x) = φ(1)

ui (x) = |X |E1(x, ui)

(9)

Inner product is defined by

< f , g >=

2

• i=1

Wri |Xri |

• x∈Xri

f (x)g(x). (10) d0 =< φ0, φ0 >, c0 =< φi, φi >, for 1 ≤ i ≤ v − 1 c1,5 =< φ0, φi >, for 1 ≤ i ≤ v − 1 c1,1 =< φi, φj >, for 1 ≤ i = j ≤ v − d − 1 c1,2 =< φi, φj >, for v − d ≤ i = j ≤ v − 2 c1,3 =< φi, φv−1 >, for 1 ≤ i ≤ v − d − 1 c1,4 =< φi, φv−1 >, for v − d ≤ i ≤ v − 2 c2 =< φi, φj >, for 1 ≤ i ≤ v − d − 1, v − d ≤ j ≤ v − 2 c1,1 = c1,3 and c1,2 = c1,4

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idea of the proof: Denote U ∗ = U \ {u0} and S = Xr1 Xr2. Let M be a submatrix of E1. M is indexed by S × U ∗ whose (x, ui)-entry is defined by (M)x,ui = |X |E1(x, ui) for any (x, ui) ∈ S × U ∗. (tMM)(uj , uk) =

2

• i=1

Wri |Xri |

• x∈Xri

φj (x)φk(x).

tMM =

                  c0 c1,1 · · · c1,1 c2 c2 · · · c2 c1,1 c1,1 c0 · · · c1,1 c2 c2 · · · c2 c1,1 . . . ... . . . . . . . . . . . . . . . . . . c1,1 c1,1 · · · c0 c2 c2 · · · c2 c1,1 c2 c2 · · · c2 c0 c1,2 · · · c1,2 c1,2 c2 c2 · · · c2 c1,2 c0 · · · c1,2 c1,2 . . . ... . . . . . . . . . . . . . . . . . . c2 c2 · · · c2 c1,2 c1,2 · · · c0 c1,2 c1,1 c1,1 · · · c1,1 c1,2 c1,2 · · · c1,2 c0                   (11)

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Orthonormal basis

Gram-Schmidt’s method: {φ1, . . . , φv−1, φ0} − → {ϕ1, ϕ2, . . . , ϕv}. ϕ1 = φ1 c0 , ϕi = 1 √Di−1Di

• < φ1, φ1 >

< φ2, φ1 > . . . < φi, φ1 > < φ1, φ2 > < φ2, φ2 > . . . < φi, φ2 > . . . · · · · · · . . . < φ1, φi−1 > < φ2, φi−1 > . . . < φi, φi−1 > φ1 φ2 . . . φi

• (12)

The Gram determinant Di is given by Di =

• < φ1, φ1 >

< φ2, φ1 > . . . < φi, φ1 > < φ1, φ2 > < φ2, φ2 > . . . < φi, φ2 > . . . · · · · · · . . . < φ1, φi−1 > < φ2, φi−1 > . . . < φi, φi−1 > < φ1, φi > < φ2, φi > . . . < φi, φi >

• 14 / 27

Property of orthonormal basis

Let H be a matrix whose rows are indexed by Y with v columns whose (y, i)-entry is defined by

• w(y)ϕi(y).

Then (tHH)i,j = δi,j and (H tH)x,y = δx,y imply       

• y∈Y

w(y)ϕi(y)ϕj (y) = δi,j

v

• i=1

w(y)ϕi(x)ϕi(y) = δx,y

x ∈ Xr,

x = {1, 2, . . . , d − r, d + 1, d + 2, . . . , d + r} 1 wr =

• x∈Yr

ϕ2

i (x).

(13)

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x, y ∈ Xr and (x, y) ∈ Rα.

y = {1, 2, . . . , a, d − r + 1, . . . , 2d − 2r − a, d + 1, . . . , 2d − α − a, d + r + 1, . . . , 2r + α + a}, d − 2r ≤ a ≤ d − r

v

• i=1

ϕi(x)ϕi(y) = f (Wr, v, d, r, α, a) g(Wr, v, d, r, α, a). (14)

x ∈ Xr1, y ∈ Xr2, (x, y) ∈ Rγ, r1 < r2.

i). d − r1 ≤ 2d − 2r2 − a y = {1, 2, . . . , a, d − r2 + 1, . . . , 2d − 2r2 − a, d + 1, . . . , 2d − γ − a + r1 − r2, d + r1 + 1, . . . , 2r2 + γ + a} ii). d − r1 > 2d − 2r2 − a y = {1, 2, . . . , a, d − r2 + 1, . . . , 2d − 2r2 − a, d + 1, . . . , d + r2 − γ, d + r1 + 1, . . . , d + r1 + γ}

v

• i=1

ϕi(x)ϕi(y) = f ′(Wr1, Wr2, v, d, r1, r2, a, γ) g′(Wr1, Wr2, v, d, r1, r2, a, γ). (15)

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Determine parameter set

{v, d, r1, r2, α1, α2, γ, Nr1, Nr2,

wr1 wr2 }.

Step 1:

Given r1, r2, solve the equations

v

• i=1

ϕi(x)ϕi(y) = 0 for x, y ∈ Xrj (j = 1, 2). If the numerator of these two expressions have a common factor k1Wr1 − k2Wr2 with k1

k2 ∈ Q>0, then keep the parameters

v, d, r1, r2, α1, a1, α2, a2 such that

Wr1 Wr2 is positive and rational.

Step 2:

Substitute the parameters above into the equation

v

• i=1

ϕi(x)ϕi(y) = 0 for x ∈ Xr1, y ∈ Xr2 and solve γ from it. List possible parameters such that γ is an integer.

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Determine parameter set

Step 3:

Assume wr1 = 1, then Wr1 = Nr1, Wr2 = (v − Nr1)wr2. Substitute these into 1 wr1 =

v

• i=1

ϕ2

i (x)

x ∈ Xr1 and obtain wr2(v, Nr1).

Step 4:

Substitute the parameters into

v

• i=1

ϕi(x)ϕi(y) = 0 for x, y ∈ Xr1. and solve Nr1. Similarly, we can obtain Nr2 and keep the integral solutions.

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List of all possible parameters

4 ≤ v ≤ 30 v d r1 r2 α1 a1 α2 a2 γ Nr1 Nr2

wr1 wr2

16 6 3 5 4 0,1,2 4 0,1 4 10 6 1

• 19

9 6 8 5 2 3 4 13 6

8 13

× 22 10 6 10 8 4 2 5 16 6

1 4

× 28 12 4 12 6 8 4 9 24 4

2 3

× 28 12 8 12 8 4 4 6 24 4

1 2

× 30 14 8 14 11 6 2 7 22 8

2 11

×

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List of all possible parameters

31 ≤ v ≤ 50 v d r1 r2 α1 a1 α2 a2 γ Nr1 Nr2

wr1 wr2

32 12 10 12 9 2 6 6 27 5

2 3

× 36 15 7 10 9 2,..,6 9 0,..,4 9 15 21 1

• 36

15 7 11 9 1 12 2 9 33 3

10 7

× 36 16 10 16 10 5 4 8 31 5

12 31

× 40 15 10 15 9 7 9 33 7

7 11

× 45 12 8 11 9 0,1,2,3 9 0,1 9 33 12 1

• 45

15 10 15 10 9 10 39 6

54 65

× 45 18 12 18 11 2 7 10 37 8

24 37

× 50 20 15 20 12 2 8 10 44 6

32 55

× 50 18 16 18 14 2 9 9 42 8

9 14

×

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Computer search

tight relative 2-designs from SRGs

SRGs relative 2-designs (16, 6, 2, 2) 2 2 (36, 15, 6, 6) 32, 548 31 (45, 12, 3, 3) 78 29

tight relative 2-designs from Symmetric 2-designs with non-null

polarity Symmetric 2-designs relative 2-designs (16, 6, 2) 3 2 (36, 15, 6) 617∗ 339 (45, 12, 3) 6∗

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Some examples from SRGs

{v, d, r1, r2, Nr1, Nr2} = {16, 6, 3, 5, 10, 6} − →2-(16,6,2) designs

G = Z4 × Z4

Base block D and B = {gD|g ∈ G}. D u0 1 2 3

• ×

× × 1 ×

• 2

×

• 3

×

• =

⇒ 1 2 3 ×

• ×

1 ×

• ×
• 2

× ×

• 3
• Xr1 = {gD|g ∈ G′},

where G′ = {(0, 0), (0, 1), (0, 2), (1, 1), (1, 3), (2, 2), (2, 3), (1, 0), (2, 0), (3, 0)}. Xr2 = B \ Xr1.

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Some examples from SRGs

{v, d, r1, r2, Nr1, Nr2} = {36, 15, 7, 10, 15, 21} − → 2-(36,15,6) designs

G = Z6 × Z6

Base block D and B = {gD|g ∈ G}. D u0 1 2 3 4 5

• ×

× × × × 1 × ×

• 2

×

• ×
• 3

×

• ×
• 4

×

• ×
• 5

×

• ×

= ⇒ 1 2 3 4 5 × × ×

• ×

× 1 ×

• ×

× × 2 ×

• ×
• ×

3

• ×

×

• 4

×

• 5
• 23 / 27

Some examples from SRGs

{v, d, r1, r2, Nr1, Nr2} = {36, 15, 7, 10, 15, 21} − → 2-(36,15,6) designs

G = S3 × S3

Base block D and B = {gD|g ∈ G} D u0 1 g1 g2 g3 g4 g5 1

• ×

× × × × g1 × ×

• g2

×

• ×
• g3

×

• ×
• g4

×

• ×
• g5

×

• ×

= ⇒ 1 g1 g2 g3 g4 g5 1 × × × × × × g1

• ×

× ×

• g2
• ×

× ×

• g3
• ×

× ×

• g4
• g5
• 1 = Id,

g1 = (12), g2 = (13), g3 = (23), g4 = (123), g5 = (132)

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Future work

Does every tight relative 2-design on 2 shells in J(v, d) have the

structure of coherent configuration?

Are there any more examples constructed from other symmetric

2-designs?

25 / 27

Reference

• Ei. Bannai, Et. Bannai, Remarks on the concepts of t−designs, J.

Appl Math Comput. 40 no.1-2,(2012),195-207.

• Ei. Bannai, Et. Bannai, Hi. Bannai, On the existence of tight

relative 2-designs on binary Hamming association schemes, arXiv:1304.5760

• Ei. Bannai, Et. Bannai, S. Suda, H. Tanaka, On relative

t-designs in polynomial association schemes, arXiv:1303.7163

• Ei. Bannai, Ta. Ito, Algebraic combinatorics I: Association schemes,

Benjamin/Cummings, Menlo Park, CA, 1984.

• Th. Beth, D.Jungnickel, H. Lenz, Design theory,

Bibliographisches Instistu, 1985.

• B. McKay , E. Spence, Classification of regular two-graphs on 36

and 38 vertices, Australasian Journal of Combinatorics, vol.24, 2001

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Thank you!