Construction of new large sets of designs over the binary field - - PowerPoint PPT Presentation
Construction of new large sets of designs over the binary field Alfred Wassermann Department of Mathematics, Universitt Bayreuth, Germany joint work with Michael Kiermaier and Reinhard Laue DARNEC 15 Istanbul Outline Designs over
Alfred Wassermann
Department of Mathematics, Universität Bayreuth, Germany
joint work with Michael Kiermaier and Reinhard Laue DARNEC ’15 Istanbul
◮ Designs over finite fields ◮ Computer construction ◮ Infinite series of large sets
◮ vector space V = F q ◮ Grassmannian: Gq(, k) := {U ≤ F q : dim U = k}
2
G2(4, 0) G2(4, 1) G2(4, 2) G2(4, 3) G2(4, 4)
◮ Gaussian coefficient:
k
= (q − 1)(q−1 − 1) · · · (q−k+1 − 1) (qk − 1)(qk−1 − 1) · · · (q − 1)
◮ |Gq(, k)| =
k
◮ Cameron (1974), Delsarte (1976) ◮ B ⊆ Gq(, k): set of k-subspaces (blocks) ◮ (F q, B): t-(, k, λ; q) design over Fq
each t-subspace of F
q is contained in
exactly λ blocks of B
◮ B set: simple design ◮ B multiset: non-simple design
◮ B = Gq(, k) is a t-(, k,
−t
k−t
design trivial 1-(4, 2, 7; 2) design
◮ B = Gq(, k) is a t-(, k,
−t
k−t
design trivial 1-(4, 2, 7; 2) design 1-(4, 2, 1; 2) design
◮ |B| = λ[
t]q
[
k t]q
◮ Necessary conditions:
λ = λ −
t−
k−
t−
∈ Z for = 0, . . . , t
t-(, k, λ; q) design →
◮ dual design: t-(, − k, λ; q) ◮ derived design: (t − 1)-( − 1, k − 1, λ; q) ◮ residual design: (t − 1)-( − 1, k, μ; q), where
μ = λ · −k
1
k−t+1
1
◮ Gq(, k) is a t-(, k,
−t
k−t
◮ Large set LSq[N](t, k, ):
partition of Gq(, k) into N disjoint t-(, k, λ; q) designs LS2[7](1, 2, 4)
◮ Necessary: N · λ =
−t
k−t
Designs over finite fields:
◮ GL(, q) = {M ∈ F× q
: M invertible}
◮ σ ∈ GL(, q) automorphism: Bσ = B
◮ Singer cycle:
◮ take ∈ F
q as an element of Fq
◮ (Fq \ {0}, ·) is a cyclic group G of order q − 1, i.e. ◮ G = 〈σ〉 ◮ G ≤ GL(, q) is called Singer cycle
◮ Frobenius automorphism:
◮ ϕ : Fq → Fq, U → Uq ◮ |〈ϕ〉| =
◮ |〈σ, ϕ〉| = · (q − 1)
◮ incidence matrix between t-subset and k-subsets:
Mt,k = (m,j), where m,j =
if T ⊂ Kj else
◮ solve
Mt,k · = λ λ . . . λ for 0/1-vector
Construction of designs with prescribed automorphism group
◮ choose group G acting on V, i.e. G ≤ S ◮ search for t-designs D = (V, B) having G as a group
i.e. for all g ∈ G and K ∈ B =⇒ Kg ∈ B.
◮ construct D = (V, B) as
union of orbits of G on k-subsets.
Definition
◮ K ⊂ V and |K| = k: KG := {Kg | g ∈ G} ◮ T ⊂ V and |T| = t: TG := {Tg | g ∈ G} ◮ Let
KG
1 ∪ KG 2 ∪ . . . ∪ KG n ⊆
V k
TG
1 ∪ TG 2 ∪ . . . ∪ TG m =
V t
MG
t,k = (m,j) where m,j := |{K ∈ KG j | T ⊂ K}|
Theorem (Kramer and Mesner, 1976)
The union of orbits corresponding to the 1s in a {0, 1} vector which solves MG
t,k · =
λ λ . . . λ is a t-(, k, λ) design having G as an automorphism group.
◮ LS2[3](2, 3, 8): Braun, Kohnert, Östergård, W.
(2014)
◮ Three disjoint 2-(8, 3, 21; 2) designs ◮ Group: 〈σ〉 in GL(8, 2) of order 255
◮ LS3[2](2, 3, 6): Braun (2005)
◮ T
wo disjoint 2-(6, 3, 20; 3) designs
◮ LS5[2](2, 3, 6): Braun, Kiermaier, Kohnert, Laue
(2014)
◮ T
wo disjoint 2-(6, 3, 78; 5) designs
◮ LS2[3](2, 4, 8)
◮ Three disjoint 2-(8, 4, 217; 2) designs ◮ Group: 〈σ5, ϕ2〉 in GL(8, 2) of order 204
Theorem (Kiermaier, Laue 2015)
◮ derived large set:
LSq[N](t, k, ) → LSq[N](t − 1, k − 1, − 1)
◮ q-analog of Van Trung, Van Leyenhorst, Driessen:
LSq[N](t, k − 1, − 1) and LSq[N](t, k, − 1) → LSq[N](t, k, )
LSq[3](2, 3, 8) LSq[3](2, 4, 8) LSq[3](2, 5, 8) LSq[3](2, 4, 9) LSq[3](2, 5, 9) LSq[3](2, 5, 10)
v
3 4 8
9
10
3 4 5
5
3 4 5 ? ? ? 9 10 20
5 ? ? ? ? 10 21
? ? ? ? ? 11 22
? ? ? ? ? 23
? ? ? ? ? 24
? ? ? ? 25 3 4 5
10 11 ? ? 26
5
? 27
? ? 28
? ? 29
? ? 30
? 31 3 4 5
10 11 -
32
5
33
34
3 4 5 ? ? ? 9 10 11 ? ? ? 15 16 17 -
5 ? ? ? ? 10 11 ? ? ? ? 16 17 -
? ? ? ? ? 11 ? ? ? ? ? 17 -