Large Sets of q-Analogs of Designs

Michael Braun, Michael Kiermaier, Axel Kohnert∗, Reinhard LaueUniversitat Bayreuth,laue@uni-bayreuth.de

Abstract

Joining small Large Sets of t-designs to form large Large Sets oft-designs allows to recursively construct infinite series of t-designs.This concept is generalized from ordinary designs over sets to designsover finite vector spaces, i.e. designs over GF (q), using three typesof joins. While there are only very few general constructions of suchq-designs known so far, from only one large set in the literature andtwo new ones in this paper this way many infinite series of Large Setsof q-designs with constant block sizes are derived.

Keywords: q-analog, t-design, Large Set, subspace designAMS classifications: Primary 51E20; Secondary 05B05, 05B25, 11Txx

∗† 11.12.2013

1

1 Classic and Subspace t-designs

t-(v, k, λ) designD = (V,B) t-(v, k, λ)q designD = (V,B)

V point set size v V GF (q)-vector space dim v

B ⊆(Vk

)B ⊆

[Vk

]q

each T ∈(Vt

)in λ B ∈ B each T ∈

[Vt

]qin λ B ∈ B

Subspace designs Cameron 1974 [10]:

Infinite series for t = 2:

Thomas 1987 q = 2 [18] Suzuki q > 2 [14], [15], Itoh

1997 [12].

∀t∃ simple t-subspace design: Fazelli,Lovett,Vardy 2013

[11], q-analog to Teirlinck’s theorem [16]:

Computer search:

t = 2, 3 M.,S. Braun et al. 2005-2011 [4, 5, 9, 6],

2-(13, 3, 1)2 Braun,Etzion,Ostergard,Wassermann,Vardy

2013 [7],

Large Set:

LSq[N ](t, k, v) partition of[Vk

]q

intoN disjoint t-(v, k, λ)q

with λ =[v−tk−t]qN .

LS2[3](2, 3, 8) Braun,Kohnert,Ostergard,Wassermann 2013[8]

Three disjoint 2-(8, 3, 21)2 that partition[V3

]2

2

Table 1: Table of LS2[3](2, k, v)

vS t a r t : 3 ? 8

- ? 9- - ? 10

- - - 11- - - - 12

- - - - 133 ? 5 - 14

- ? ? - - 15- - 5 - - - 16

- - - - - - 17- - - - - - - 18

- - - - - - - 193 ? 5 ? ? ? 9 ? 20

- ? ? ? ? ? ? ? 21- - ? ? ? ? ? ? ? 22

- - - ? ? ? ? ? ? 23- - - - ? ? ? ? ? ? 24

- - - - - ? ? ? ? ? 253 ? 5 - - - ? ? 11 ? ? 26

- ? ? - - - - ? ? ? ? 27- - 5 - - - - - 11 ? ? ? 28

- - - - - - - - - ? ? ? 29- - - - - - - - - - ? ? ? 30

- - - - - - - - - - - ? ? 313 ? 5 - - - ? ? 11 - - - 15 ? 32

- ? ? - - - - ? ? - - - - ? 33- - ? - - - - - ? - - - - - ? 34

- - - - - - - - - - - - - - - 35- - - - - - - - - - - - - - - 36

- - - - - - - - - - - - - - - - 373 ? 5 ? ? ? 9 ? 11 ? ? ? 15 ? 17 - - 38

- ? ? ? ? ? ? ? ? ? ? ? ? ? ? - - 39- - 5 ? ? ? ? ? 11 ? ? ? ? ? 17 - - - 40

3

Table 2: Table of LSq[2](2, k, v), q = 3, 5

vS t a r t ( N e w ) : 3 6

- 7- - 8

- - 93 ? ? 10

- ? ? 11- - ? ? 12

- - - ? 133 - - - 7 14

- - - - - 15- - - - - - 16

- - - - - - 173 ? ? ? 7 ? ? 18

- ? ? ? ? ? ? 19- - ? ? ? ? ? ? 20

- - - ? ? ? ? ? 213 - - - 7 ? ? ? ? 22

- - - - - ? ? ? ? 23- - - - - - ? ? ? ? 24

- - - - - - - ? ? ? 253 ? ? ? 7 - - - ? ? ? 26

- ? ? ? ? - - - - ? ? 27- - ? ? ? - - - - - ? ? 28

- - - ? ? - - - - - - ? 293 - - - 7 - - - ? - - - 15 30

- - - - - - - - - - - - - 31- - - - - - - - - - - - - - 32

- - - - - - - - - - - - - - 333 ? ? ? 7 ? ? ? ? ? ? ? 15 ? ? 34

- ? ? ? ? ? ? ? ? ? ? ? ? ? ? 35- - ? ? ? ? ? ? ? ? ? ? ? ? ? ? 36

- - - ? ? ? ? ? ? ? ? ? ? ? ? ? 373 - - - 7 ? ? ? ? ? ? ? 15 ? ? ? ? 38

- - - - - ? ? ? ? ? ? ? ? ? ? ? ? 39- - - - - ? ? ? ? ? ? ? ? ? ? ? ? 40

4

Recursion: (classical strategy, Khosrovshahi, Ajoodani-

Namini [1, 2, 3] Teirlinck [17])

Partition the problem, insert small Large Sets

into parts to obtain large Large Sets.

Notation: Part(S) set of partitions of set S,

First partition

{B1, . . . ,Bm} ∈ Part([V

k

]q

)

Second partition

Decompose each Bi into a join of two components:

5

• Ordinary join at U ≤ V :

((((((((

(((((((

(((((((

((((((((U + K

K

{0}

U

K1

k1

k2 V

r

rr r

rr

(U + K)/U = K2

K1 ∗U K2 = {K : U ∩K = K1, (U +K)/U = K2}

For K1 ⊆[Uk1

]q, K2 ⊆

[V/Uk2

]q

K1 ∗U K2 = ∪{(K1 ∗U K2) : K1 ∈ K1, K2 ∈ K2}

Partition by ordinary joins: q-Vandermonde:[v1 + v2

k

]q

=

k1=k∑k1=0

[v1

k1

]q

·[

v2

k − k1

]q

q(v1−k1)(k−k1)

Example 1.1. For v = 10, k = 3, v1 = 6, v2 = 4

the formula reads as[10

3

]q

= q18

[6

0

]q

[4

3

]q

+ q10

[6

1

]q

[4

2

]q

+ q6

[6

2

]q

[4

1

]q

+ q0

[6

3

]q

[4

0

]q

.

6

• Avoiding join and Covering join

r

r r

{0}K1 r

U1 rFU2 r

KF

KFr

Vr

���

���

Wr

((((((

((((((

���

((((((

���(((

(((((((

(((((

W/U2 = K2

Avoiding join:

K1∗FK2 = {KF : KF∩U2 = KF∩U1 = K1, U2+KF = W}.

K1 ⊆[Uk1

]q, K2 ⊆

[V/Uk2

]q

K1 ∗F K2 = ∪{(K1 ∗F K2) : K1 ∈ K1, K2 ∈ K2}

Covering join:

K1∗FK2 = {KF : KF∩U1 = K1, U1+KF = U2+KF = W}.

K1 ⊆[Uk1

]q, K2 ⊆

[V/Uk2

]q

K1 ∗F K2 = ∪{(K1 ∗F K2) : K1 ∈ K1, K2 ∈ K2}

7

V = V0 > V1 > . . . > Vv = {0} each Fi = Vi−1/Viof dimension 1. v ≥ k + s.

Partition by avoiding joins:[v

k

]q

=

k∑i=0

q(v−k−s)i[v − s− i

i

]q

[s + i− 1

k − i

]q

.

Example 1.2. For v = 10, k = 3, s = 3 the

formula reads as[10

3

]q

= q0

[3

0

]q

[6

3

]q

+ q4

[4

1

]q

[5

2

]q

+ q8

[5

2

]q

[4

1

]q

+ q12

[6

3

]q

[3

0

]q

.

Partition by covering joins: k = k1 + k2 + 1[v

k

]q

=

v−k2−1∑i=k1

q(i−k1)(k2+1)

[i

k1

]q

[v − i− 1

k2

]q

.

Example 1.3. For v = 10, k1 = 1, k2 = 1 the

formula reads as[10

3

]q

=

[8

1

]q

+ q2

[2

1

]q

[7

1

]q

+ q4

[3

1

]q

[6

1

]q

+

q6

[4

1

]q

[5

1

]q

+ q8

[5

1

]q

[4

1

]q

+ q10

[6

1

]q

[3

1

]q

+ q12

[7

1

]q

[2

1

]q

.

8

{B1, . . . ,Bm} ∈ Part([Vk

]q) is (N, t) partitionable:

∀i Bi = B(i)1 ∪ . . . ∪B

(i)N

∀T ∈[V

t

]q

,∀B(i)j |{B ∈ B

(i)j : T ⊆ B}| = λ(T, i),

independent of j.

If {B1, . . . ,Bm} ∈ Part([Vk

]q) is (N, t) partition-

able then the designs

Dj = ∪mi=1B(i)j for j = 1, . . . , N

form an LSq[N ](t, k, v).

9

Theorem 1.1. Joining Partitions ∗ one of the 3

joins (either U1 = U2 or dim(U1/U2) = 1):

{K11, . . . ,K1

N} ∈ Part([U2

k1

]q

) (N, t) partition,M ⊆[V/U1

k2

]q

=⇒ {K11 ∗M, . . . ,K1

N ∗M)} is an (N, t) partition

M ⊆[U2

k1

]q

, {K21, . . . ,K2

N} ∈ Part([V/U1

k2

]q

) (N, t) partition

=⇒ {M ∗ K21, . . . ,M ∗ K2

N)} is an (N, t) partition

{K11, . . . ,K1

N} ∈ Part([U2

k1

]q

) (N, t1) partition,

{K21, . . . ,K2

N} ∈ Part([V/U1

k2

]q

) (N, t2) partition,

Lat an N ×N Latin Square.

=⇒

{∪Lat(r,s)=aK1r ∗K2

s : a = 1, . . . , N} is an (N, t1 +t2 +

1) partition.

Proof analog to the classical case.

10

Doubling the point set by Ordinary join:

LS2[3](2, 3, 8)↔dual LS2[3](2, 5, 8)

derivedLS2[3](2, 5, 8) = LS2[3](1, 4, 7)

residual LS2[3](2, 3, 8) = LS2[3](1, 3, 7)

——————————————–

Point extension: LS2[3](1, 4, 8)

Ordinary joins at U < V, dim(U) = 8, dim(V ) = 16:

{LS2[3](2, 5, 8) ∗U[V/U

0

]2,

LS2[3](1, 4, 8) ∗U LS2[3](0, 1, 8),

LS2[3](2, 3, 8) ∗U[V/U

2

]2,[

U2

]2∗U LS2[3](2, 3, 8),

LS2[3](0, 1, 8) ∗U LS2[3](1, 4, 8)[U0

]2∗U LS2[3](2, 5, 8)}.

———————————————–

LS2[3](2,5,16) dual to LS2[3](2,11,16)

11

Theorem 1.2. Let {0} ≤ Vn < Vn−1 < . . . < V0 ≤ V

be a chain of subspaces such that each Fi = Vi−1/Vihas dimension 1. Then for k with 1 ≤ k ≤ dim(V )

and k = a + b + c, a + b = n, a, b, c non-negative

integers, there is the following partition.

[V

k

]q

= ∪a−1i=0

[V0

k − i

]q

∗V0

[V/V0

i

]q

∪ ∪b−1i=0

[Vi+1

k − a− i

]q

∗ ¯Fi+1

[V/Via + i

]q

∪ ∪ci=0

[Vn

k − a− b− i

]q

∗Vn[V/Vn

a + b + i

]q

Theorem 1.3.

∃LSq[N ](t, t + 1, u), LSq[N ](t, k, u), LSq[N ](t, k, v),

∀k−ti=2 ∃LSq[N ](t(i)1 , k − t− i, v − t),

LSq[N ](t(i)2 , t + i, u) : t

(i)1 + t

(i)2 + 1 ≥ t

=⇒∃LSq[N ](t, k, u + v − t).

12

Proof. Let {0} ≤ Vt < Vt−1 < . . . < V0 ≤ V , Fi =

Vi−1/Vidim(Fi) = 1, dim(V0) = v, dim(V/Vt) = u.

Bi (N, t)− partition[V0k

]q∗V0

[V/V0

0

]q

LSq[N ](t, k, v) ∗V0

[V/V0

0

]q[

V1k−1

]q∗F1

[V/V0

1

]q

LSq[N ](t-1, k-1, v-1) ∗F1LSq[N ](0, 1, u-t)[

V2k−2

]q∗F2

[V/V1

2

]q

LSq[N ](t-2, k-2, v-2) ∗F1LSq[N ](1, 2, u-t+1)

· ·· ·[

Vtk−t]q∗Ft[V/Vtt

]q

LSq[N ](0, k-t, v-t) ∗Ft LSq[N ](t-1, t, u-1)[Vt

k−t−1

]q∗Vt[V/Vtt+1

]q

[Vt

k−t−1

]q∗Vt LSq[N ](t, t+1, u)[

Vtk−t−2

]q∗Vt[V/Vtt+2

]q

LSq[N ](t(2)1 , k-t-2, v-t) ∗Vt LSq[N ](t

(2)2 , t+2, u)

· ·· ·[

Vtk−t−i

]q∗Vt[V/Vtt+i

]q

LSq[N ](t(i)1 , k-t-i, v-t) ∗Vt LSq[N ](t

(i)2 , t+i, u)

· ·· ·[

Vt0

]q∗Vt[V/Vtk

]q

[Vt0

]q∗Vt ∗Vt LSq[N ](t, k, u)

13

Corollary 1.4. Analog to classical result

∃LSq[N ](t, k, u), LSq[N ](t, i, v) for t + 1 ≤ i ≤ k

=⇒ ∀n∃LSq[N ](t, k, u + n(v − t))

Corollary 1.5. ∃LSq[N ](2, k, v), LSq[N ](2, k + 2, v)

=⇒ ∃LSq[N ](2, k + 2, 2v − 2)

Corollary 1.6.

∀n∃LS2[3](2, 3, 6n + 2), ∀n∃LS2[3](2, 5, 6n + 2)

Corollary 1.7. If ∃LSq[N ](t, k, v1),LSq[N ](0, i, v1−t)for 1 ≤ i < k − t− 1, and

∃LSq[N ](t, t+1, v2),LSq[N ](t, k, v2),LSq[N ](t−1, i, v2)

for t + 2 ≤ i < k,

=⇒ ∀n∃LSq[N ](t, k, v2 + n(v1 − t))Theorem 1.8. If p = 2 · 3a + 1 is a prime then all

LS2[3](1, k, p) exist.

Proof: Prescribe Singer cycle as a group of automor-

phisms.

Theorem 1.9. v = 2 · 3a + 2, p = 2 · 3a + 1 prime,

∃LS(2, k, v)

=⇒ ∀n∃LSq[N ](2, k, v + n(v − 2)).

=⇒ ∀n∃LSq[N ](2, v − k, v + n(v − 2)).

14

Since many p = 2 · 3a + 1 are primes, the duals of

a large set of series starting at smaller dimensions give

many new infinite series.

Are there infinitely many primes p of this form?

a = 0, 1, 2, 4, 5, 6, 9, 16, 17, 30, 54, 57, 60, 65, 132,

180, 320, 696, 782, 822, 897, 1252, 1454, 4217, 5480, 6225,

7842, 12096, 13782, 17720, 43956, 64822, 82780, 105106,

152529, 165896, 191814, 529680, 1074726, 1086112, 1175232

are known: N.J.A. Sloane, at al. The On-Line Ency-

clopedia of Integer Sequences.

15

1.1 Covering Join Constructions

LS2[3](2, 11, 20):

For i = 5, . . . , 14 let Fi = Vi+1/Vi and

Si =

[Vik1

]2

∗Fi[V/Vi+1

k2

]2

.

11 = 5 + 1 + 5, 20 = v1 + 1 + v2.

{V5} ∗F5 LS2[3](2, 5, 14)

LS2[3](0, 5, 6) ∗F6 LS2[3](1, 5, 13)

LS2[3](1, 5, 7) ∗F7 LS2[3](0, 5, 12)

LS2[3](2, 5, 8) ∗F8

[V115

]2

LS2[3](1, 5, 9) ∗F9 LS2[3](1, 5, 10)

LS2[3](1, 5, 10) ∗F10 LS2[3](1, 5, 9)[V115

]2∗F11 LS2[3](2, 5, 8)

LS2[3](1, 5, 12) ∗F12 LS2[3](1, 5, 7)

LS2[3](1, 5, 13) ∗F13 LS2[3](0, 5, 6)

LS2[3](2, 5, 14) ∗F14 {V5}

All the corresponding Large Sets exist, using residual

Large Sets and those for t = 1.

16

Theorem 1.10. ∀(n ≥ 2)∃LS2[3](2, 11, 6n + 2).

Proof. Use LS2[3](2, 5, 6n + 2) with 6n + 2 = 14, 20, . . .

and their residuals. The additional 1-design Large Sets

also exist by recursion 1.4.

The number of rows grows by 6 of the pattern:

v1 t1 v2 t26r + 5 ∗ 6m + 2 2

6r + 6 0 6m + 1 1

6r + 7 1 6m 0

6r + 8 2 6m− 1 ∗6r + 9 1 6m− 2 1

6r + 10 1 6m− 3 1

17

Two new Large Sets: Halvings LSq[2](2, 3, 6), q = 3, 5

Theorem 1.11.

∃LSq[N ](2, 3, 6), 4n− 1 > 2s − 1

=⇒∃LSq[N ](2, 2s − 1, 4n + 2)

Proof. s = 2: Use recursion Theorem 1.4

∃LSq[N ](2, 3, 6) =⇒ ∀n∃LSq[N ](2, 3, 4n + 2).

s = 3

∃LSq[N ](2, 3, 6) =⇒ ∀n∃LSq[N ](2, 7, 4n + 2).

Start with n = 2, use the covering join, 7 = 3 + 3 + 1:

{V3} ∗F3 LSq[N ](2, 3, 10)

LSq[N ](0, 3, 4) ∗F4 LSq[N ](1, 3, 9)

LSq[N ](1, 3, 5) ∗F5 LSq[N ](0, 3, 8)

LSq[N ](2, 3, 6) ∗F6

[V/V7

3

]q[

V73

]q∗F7 LSq[N ](2, 3, 6)

LSq[N ](0, 3, 8) ∗F12 LSq[N ](1, 3, 5)

LSq[N ](1, 3, 9) ∗F13 LSq[N ](0, 3, 4)

LSq[N ](2, 3, 10) ∗F14 {V/V8}

Each

LSq[N ](t1, k, v1) ∗F LSq[N ](t2, k, v2)

for t1, t2 ∈ {0, 1} joins residual Large Sets.

18

Larger n: The number of rows grows by 4 of the pat-

tern:

v1 t1 v2 t24r − 1 ∗ 4m + 2 2

4r 0 4m + 1 1

4r + 1 1 4m 0

4r + 2 2 4m− 1 ∗Induction on s only iterates the pattern with the pre-

vious value of s, using

2k + 1 = 2(2s− 1) + 1 = 2s+1− 2 + 1 = 2s+1− 1.

Corollary 1.12. For q = 3, 5 there exist infinite se-

ries of halvings for all k = 2s − 1.

19

THANK YOU for YOUR PATIENCE

20

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