SLIDE 1 Large Sets of q-Analogs of Designs
Michael Braun, Michael Kiermaier, Axel Kohnert∗ , Reinhard Laue Universit¨ at Bayreuth, laue@uni-bayreuth.de
Abstract Joining small Large Sets of t-designs to form large Large Sets of t-designs allows to recursively construct infinite series of t-designs. This concept is generalized from ordinary designs over sets to designs
- ver finite vector spaces, i.e. designs over GF(q), using three types
- f joins. While there are only very few general constructions of such
q-designs known so far, from only one large set in the literature and two new ones in this paper this way many infinite series of Large Sets
- f q-designs with constant block sizes are derived.
Keywords: q-analog, t-design, Large Set, subspace design AMS classifications: Primary 51E20; Secondary 05B05, 05B25, 11Txx
∗† 11.12.2013
1
SLIDE 2 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 ⊆ V
k
V
k
each T ∈ V
t
each T ∈ V
t
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,¨ Osterg˚ ard,Wassermann,Vardy 2013 [7], Large Set: LSq[N](t, k, v) partition of V
k
- q into N disjoint t-(v, k, λ)q
with λ = [v−t
k−t]q
N .
LS2[3](2, 3, 8) Braun,Kohnert,¨ Osterg˚ ard,Wassermann 2013[8] Three disjoint 2-(8, 3, 21)2 that partition V
3
2
SLIDE 3 Table 1: Table of LS2[3](2, k, v)
v S t a r t : 3 ? 8
9
3 ? 5
?
3 ? 5 ? ? ? 9 ? 20
? ? ? ? ? ? 21
? ? ? ? ? ? 22
? ? ? ? ? 23
? ? ? ? ? 24
? ? ? ? 25 3 ? 5
? 11 ? ? 26
?
? ? ? 27
? ? ? 28
? ? 29
? ? 30
? 31 3 ? 5
? 11
? 32
?
?
33
3 ? 5 ? ? ? 9 ? 11 ? ? ? 15 ? 17
? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? 11 ? ? ? ? ? 17
3
SLIDE 4 Table 2: Table of LSq[2](2, k, v), q = 3, 5
v S t a r t ( N e w ) : 3 6
3 ? ? 10
? 11
? 12
13 3
3 ? ? ? 7 ? ? 18
? ? ? ? ? 19
? ? ? ? ? 20
? ? ? ? 21 3
? ? ? ? 22
? ? ? 23
? ? ? 24
? ? 25 3 ? ? ? 7
? ? 26
? ? ?
? 27
? ?
? 28
?
29 3
3 ? ? ? 7 ? ? ? ? ? ? ? 15 ? ? 34
? ? ? ? ? ? ? ? ? ? ? ? ? 35
? ? ? ? ? ? ? ? ? ? ? ? ? 36
? ? ? ? ? ? ? ? ? ? ? ? 37 3
? ? ? ? ? ? ? 15 ? ? ? ? 38
? ? ? ? ? ? ? ? ? ? ? 39
? ? ? ? ? ? ? ? ? ? ? 40
4
SLIDE 5 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
) Second partition Decompose each Bi into a join of two components:
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SLIDE 6
✭✭✭✭✭✭✭✭✭✭✭ ✭ ✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭
U + K K {0} U K1 k1 k2 V
r r r r r r
(U + K)/U = K2 K1 ∗U K2 = {K : U ∩ K = K1, (U + K)/U = K2} For K1 ⊆ U
k1
V/U
k2
K1 ∗U K2 = ∪{(K1 ∗U K2) : K1 ∈ K1, K2 ∈ K2} Partition by ordinary joins: q-Vandermonde: v1 + v2 k
=
k1=k
v1 k1
·
k − k1
q(v1−k1)(k−k1) Example 1.1. For v = 10, k = 3, v1 = 6, v2 = 4 the formula reads as 10 3
= q18 6
4 3
+ q10 6 1
4 2
+ q6 6 2
4 1
+ q0 6 3
4
.
6
SLIDE 7
- Avoiding join and Covering join
r r r
{0} K1 r U1
r
F U2
r
K ¯
F
KF
r
V
r
r ✭✭✭✭✭ ✭ ✭✭✭✭✭ ✭
✭
✭
W/U2 = K2 Avoiding join: K1∗ ¯
FK2 = {K ¯ F : K ¯ F∩U2 = K ¯ F∩U1 = K1, U2+K ¯ F = W}.
K1 ⊆ U
k1
V/U
k2
K1 ∗ ¯
F K2 = ∪{(K1 ∗ ¯ F K2) : K1 ∈ K1, K2 ∈ K2}
Covering join: K1∗FK2 = {KF : KF∩U1 = K1, U1+KF = U2+KF = W}. K1 ⊆ U
k1
V/U
k2
K1 ∗F K2 = ∪{(K1 ∗F K2) : K1 ∈ K1, K2 ∈ K2}
7
SLIDE 8 V = V0 > V1 > . . . > Vv = {0} each Fi = Vi−1/Vi
- f dimension 1. v ≥ k + s.
Partition by avoiding joins: v k
=
k
q(v−k−s)i v − s − i i
s + i − 1 k − i
. Example 1.2. For v = 10, k = 3, s = 3 the formula reads as 10 3
= q0 3
6 3
+ q4 4 1
5 2
+ q8 5 2
4 1
+ q12 6 3
3
. Partition by covering joins: k = k1 + k2 + 1 v k
=
v−k2−1
q(i−k1)(k2+1) i k1
v − i − 1 k2
. Example 1.3. For v = 10, k1 = 1, k2 = 1 the formula reads as 10 3
= 8 1
+ q2 2 1
7 1
+ q4 3 1
6 1
+ q6 4 1
5 1
+ q8 5 1
4 1
+ q10 6 1
3 1
+ q12 7 1
2 1
.
8
SLIDE 9 {B1, . . . , Bm} ∈ Part( V
k
- q) is (N, t) partitionable:
∀i Bi = B(i)
1 ˙
∪ . . . ˙ ∪B(i)
N
∀T ∈ V t
, ∀B(i)
j
|{B ∈ B(i)
j
: T ⊆ B}| = λ(T, i), independent of j. If {B1, . . . , Bm} ∈ Part( V
k
able then the designs Dj = ∪m
i=1B(i) j
for j = 1, . . . , N form an LSq[N](t, k, v).
9
SLIDE 10 Theorem 1.1. Joining Partitions ∗ one of the 3 joins (either U1 = U2 or dim(U1/U2) = 1): {K1
1, . . . , K1 N} ∈ Part(
U2 k1
) (N, t) partition,M ⊆ V/U1 k2
= ⇒ {K1
1 ∗ M, . . . , K1 N ∗ M)} is an (N, t) partition
M ⊆ U2 k1
, {K2
1, . . . , K2 N} ∈ Part(
V/U1 k2
) (N, t) partition = ⇒ {M ∗ K2
1, . . . , M ∗ K2 N)} is an (N, t) partition
{K1
1, . . . , K1 N} ∈ Part(
U2 k1
) (N, t1) partition, {K2
1, . . . , K2 N} ∈ Part(
V/U1 k2
) (N, t2) partition, Lat an N × N Latin Square. = ⇒ {∪Lat(r,s)=aK1
r ∗K2 s : a = 1, . . . , N} is an (N, t1+t2+
1) partition. Proof analog to the classical case.
10
SLIDE 11 Doubling the point set by Ordinary join: LS2[3](2, 3, 8) ↔dual LS2[3](2, 5, 8) derived LS2[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
LS2[3](1, 4, 8) ∗U LS2[3](0, 1, 8), LS2[3](2, 3, 8) ∗U V/U
2
U
2
LS2[3](0, 1, 8) ∗U LS2[3](1, 4, 8) U
———————————————– LS2[3](2, 5, 16) dual to LS2[3](2, 11, 16)
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SLIDE 12 Theorem 1.2. Let {0} ≤ Vn < Vn−1 < . . . < V0 ≤ V be a chain of subspaces such that each Fi = Vi−1/Vi has 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
= ˙ ∪a−1
i=0
V0 k − i
∗V0 V/V0 i
˙ ∪ ˙ ∪b−1
i=0
k − a − i
∗ ¯
Fi+1
V/Vi a + i
˙ ∪ ˙ ∪c
i=0
k − a − b − i
∗Vn V/Vn a + b + i
Theorem 1.3. ∃ LSq[N](t, t + 1, u), LSq[N](t, k, u), LSq[N](t, k, v), ∀k−t
i=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
SLIDE 13
- Proof. Let {0} ≤ Vt < Vt−1 < . . . < V0 ≤ V , Fi =
Vi−1/Vi dim(Fi) = 1, dim(V0) = v, dim(V/Vt) = u. Bi (N, t)− partition V0
k
V/V0
LSq[N](t, k, v) ∗V0 V/V0
V1
k−1
F1
V/V0
1
LSq[N](t-1, k-1, v-1) ∗ ¯
F1 LSq[N](0, 1, u-t)
V2
k−2
F2
V/V1
2
LSq[N](t-2, k-2, v-2) ∗ ¯
F1 LSq[N](1, 2, u-t+1)
· · · · Vt
k−t
Ft
V/Vt
t
LSq[N](0, k-t, v-t) ∗ ¯
Ft LSq[N](t-1, t, u-1)
k−t−1
V/Vt
t+1
k−t−1
- q ∗Vt LSq[N](t, t+1, u)
- Vt
k−t−2
V/Vt
t+2
1 , k-t-2, v-t) ∗Vt LSq[N](t(2) 2 , t+2, u)
· · · ·
k−t−i
V/Vt
t+i
LSq[N](t(i)
1 , k-t-i, v-t) ∗Vt LSq[N](t(i) 2 , t+i, u)
· · · · Vt
V/Vt
k
Vt
13
SLIDE 14
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
SLIDE 15
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
SLIDE 16 1.1 Covering Join Constructions
LS2[3](2, 11, 20): For i = 5, . . . , 14 let Fi = Vi+1/Vi and Si = Vi k1
∗Fi V/Vi+1 k2
. 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 V11
5
LS2[3](1, 5, 9) ∗F9 LS2[3](1, 5, 10) LS2[3](1, 5, 10) ∗F10 LS2[3](1, 5, 9) V11
5
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
SLIDE 17 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 t2 6r + 5 ∗ 6m + 2 2 6r + 6 0 6m + 1 1 6r + 7 1 6m 6r + 8 2 6m − 1 ∗ 6r + 9 1 6m − 2 1 6r + 10 1 6m − 3 1
17
SLIDE 18 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
V7
3
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
SLIDE 19
Larger n: The number of rows grows by 4 of the pat- tern: v1 t1 v2 t2 4r − 1 ∗ 4m + 2 2 4r 0 4m + 1 1 4r + 1 1 4m 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
SLIDE 20
THANK YOU for YOUR PATIENCE
20
SLIDE 21 References
[1] Khosrovshahi, Gholamreza B. and Ajoodani-Namini, Shahin, Combining t-designs. J. Comb. Theory, Ser. A 58 (1991), 26-34. [2] Ajoodani-Namini, S. and Khosrovashahi, G.B., More on halving the complete designs, Discrete Mathematics 135 (1994), 29–37. [3] Ajoodani-Namini, Shahin, Extending large sets of t-designs, J. Com- binatorial Theory, Ser. A 76 (1996), 139-144. [4] Braun, Michael, Kerber, Adalbert and Laue, Reinhard, Sys- tematic construction of q-analogs of designs, Designs, Codes, Cryptog- raphy 34, (2005), 55-70. [5] Braun, Michael, Some new designs over finite fields, Bayreuther
- Math. Schr. 74 (2005), 58–68.
[6] Braun, Michael, Designs over finite fields - revisited Fq10, Ghent (2011). [7] Braun, Michael, Etzion, Tuvi, ¨ Osterg˚ ard, Patric R., Vardy, Alexander and Wassermann, Alfred, Existence of q-analogs of Steiner systems, ArXiv: 1304.1462v2. [8] Braun, Michael, Kohnert, Axel, ¨ Osterg˚ ard, Patric R. and Wassermann, Alfred, Large sets of t-designs over finite fields J. Combinatorial Theory, Ser. A 12(C) (2014), 195-202. [9] Braun, Stefanie, Construction of q-analogs of combinatorial designs, Alcoma’10 Thurnau, 2010. [10] Cameron, P. J., ”Generalization of Fisher’s inequality to fields with more than one element”. In: Combinatorics. Proceedings of the British Combinatorial Conference 1973, London Mathematical Society Lecture Note Series 13 (1974), 9–13, Cambridge. [11] Fazeli, Arman, Lovett, Shachar and Vardy, Alexander, Non- trivial t-designs over finite fields exist for all t, Electronic Colloquium
- n Computational Complexity (ECCC) 20: 126 (2013)
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SLIDE 22
[12] Itoh, Toyoharu, A new family of 2-designs over GF(q) admitting SLm(ql), Geom. Dedicata 69 (1998), 261–286. [13] Kiermaier, Michael, Laue, Reinhard, Derived and residual sub- space designs, arXiv:1405.5432 [14] Suzuki, H., 2-designs over GF(2m), Graphs Combin. 6 (1990), 293–296. [15] Suzuki, H., 2-designs over GF(q), Graphs Combin. 8 (1992), 381–389. [16] Tierlinck, Luc, Non-trivial t-designs without repeated blocks exist for all t, Discrete Math. 65 (1987), 301–311. [17] Tierlinck, Luc, Locally trivial t-designs and t-designs without repeated blocks, Discrete Math. 77 (1989), 345-356. [18] Thomas, Simon, Designs over finite fields, Geom. Dedicata 24 (1987), 237–242. 22
