Michael Braun University of Applied Sciences Darmstadt ...alcoma15.uni-bayreuth.de/files/slides/invited/braun.pdf · A survey on designs over ﬁnite ﬁelds Michael Braun University

A survey on designs over finite fields

Michael BraunUniversity of Applied Sciences Darmstadt

ALCOMA15

dedicated to our friend Axel Kohnert (1962–2013)

goals of this talk ...


recall resultsof the past 30 years



focus onconstructions




computer resultsgroup actions





provide list ofknown construtions





provide list ofknown construtions

describe somenew results

first, let’s start with a typical slide on q-analogs ...


V set of cardinality n

P(V ) lattice of subsets of V

(Vk

)set of k-subsets of V

(nk

)binomial coefficient

Aut(P(V )) ≃ Sym(V )

Anti(P(V )) = · ◦ Aut(P(V ))


V set of cardinality n V n-dimensional vector space over Fq

P(V ) lattice of subsets of V L(V ) lattice of subspaces of V

(Vk


[Vk

]set of k-subspaces of V

(nk


[nk

]qq-binomial coefficent

Aut(P(V )) ≃ Sym(V ) Aut(L(V )) ≃ PΓL(V )

Anti(P(V )) = · ◦ Aut(P(V )) Anti(L(V )) = ·⊥ ◦ Aut(L(V ))


V set of cardinality n V n-dimensional vector space over Fq

P(V ) lattice of subsets of V L(V ) lattice of subspaces of V

(Vk


[Vk

]set of k-subspaces of V

(nk


[nk

]qq-binomial coefficent

Aut(P(V )) ≃ Sym(V ) Aut(L(V )) ≃ PΓL(V )

Anti(P(V )) = · ◦ Aut(P(V )) Anti(L(V )) = ·⊥ ◦ Aut(L(V ))

take any incidence structure on sets,replace sets by vector spaces,and cardinalities by dimensions

take a combinatorial design ...


(V,B) is a t-(n, k , λ) designiff V = {1, . . . , n}, B ⊆

(Vk

)blocks

∀T ∈(Vt

): |{B ∈ B | T ⊆ B}| = λ



(Vk

)blocks

∀T ∈(Vt

): |{B ∈ B | T ⊆ B}| = λ

... and consider its q-analog



(Vk

)blocks

∀T ∈(Vt

): |{B ∈ B | T ⊆ B}| = λ


(V,B) is a t-(n, k , λ; q) designiff V ≃ F

nq, B ⊆

[Vk

]blocks

∀T ∈[Vt

]: |{B ∈ B | T ⊆ B}| = λ



(Vk

)blocks

∀T ∈(Vt

): |{B ∈ B | T ⊆ B}| = λ



nq, B ⊆

[Vk

]blocks

∀T ∈[Vt

]: |{B ∈ B | T ⊆ B}| = λ

number of blocks |B| = λ[nt]q[kt ]q



(Vk

)blocks

∀T ∈(Vt

): |{B ∈ B | T ⊆ B}| = λ



nq, B ⊆

[Vk

]blocks

∀T ∈[Vt

]: |{B ∈ B | T ⊆ B}| = λ

number of blocks |B| = λ[nt]q[kt ]q

trivial design B =[Vk

]with λ =

[n−tk−t

]q

B., Kerber, Laue, SystematicConstruction of q-Analogs ofDesigns, 2005

Suzuki, 2-Designsover GF (q),1992

Thomas, Designsover FiniteFields, 1987

Suzuki, 2-Designsover GF (2m),1990

Ray-Chaudhuri, Schram, Designson Vectorspaces ConstructedUsing Quadratic Forms, 1992

Thomas, Designs and PartialGeometries over Finite Fields,1996

Abe, Yoshiara, On Suzuki’sConstruction of 2-Designsover GF (q), 1993

Itoh, A New Family of 2-Designsover GF (q) Admitting SLm(qℓ),1998

Tits, Sur les AnaloguesAlgebriques des GroupesSemi-Simples Complexes, 1957

Miyakawa, Munemasa, Yoshiara,On a Class of Small 2-Designsover GF (q), 1995

Kiermaier, Laue, Derivedand Residual SubspaceDesigns, 2014

B., Kiermaier, Nakic, On theAutomorphism Group of a Binaryq-Analog of the Fano Plane, 2015

Fazeli, Lovett, Vardy, Non-trivial t-Designs over FiniteFields Exist for All t, 2013

B., Etzion, Ostergard, Vardy, Wasser-mann, On the Existence of q-Analogsof Steiner Systems, 2013

Ray-Chaudhuri, Singhi,q-Analogues of t-Designsand Their Existence, 1989

Etzion, Vardy, On q-Analogsof Steiner Systems andCovering Designs, 2011

Nakic, Pavcevic, TacticalDecompositions of Designsover Finite Fields, 2013

Cameron, Generalisation ofFisher’s Inequality to Fields withmore than One Element, 1974

B., New 3-Designsover the BinaryField, 2013

Schwartz, Etzion, Codesand Anticodes in theGrassman Graph, 2002

Metsch, Bose-Burton TypeTheorems for Finite Projective,Affine and Polar Spaces, 1999

B., Kohnert, Ostergard, Wasser-mann, Large Sets of t-Designsover Finite Fields, 2014

B., A Note on the Existence ofNon-Simple Designs over FiniteFields, 2012

Cameron, LocallySymmetric Designs,1974

B., Some NewDesigns overFinite Fields, 2005

Heden, Sissokho, On theExistence of a (2, 3)-Spreadin V (7, 2), 2011

Suzuki, On theInequalities of t-Designsover a Finite Field, 1990

Kiermaier, Pavcevic,Intersection Numbers forSubspace Designs, 2014

B., q-Analogs of t-WiseBalanced Designs overFinite Fields, 2014

B., Kiermaier, Kohnert,Laue, Large Sets ofSubspace Designs, 2014

Delsarte, Association Schemesand t-Designs in RegularSemilattices, 1976






















































































































































































































































































































































a first example ...

a first example ... 1-(4, 2, 3; 2)

a first example ... 1-(4, 2, 3; 2)

1 11 01 00 0

1 11 00 01 0

1 10 01 01 0

1 01 11 00 0

1 01 10 01 0

0 01 11 01 0

1 01 01 10 0

1 00 01 11 0

0 01 01 11 0

1 01 00 01 1

1 00 01 01 1

0 01 01 01 1

1 11 10 10 1

1 10 11 10 1

0 11 11 10 1

a first example ... 1-(4, 2, 3; 2)

1 11 01 00 0

1 11 00 01 0

1 10 01 01 0

1 01 11 00 0

1 01 10 01 0

0 01 11 01 0

1 01 01 10 0

1 00 01 11 0

0 01 01 11 0

1 01 00 01 1

1 00 01 01 1

0 01 01 01 1

1 11 10 10 1

1 10 11 10 1

0 11 11 10 1

verification is quite uncomfortable ...

... but the incidence matrix makes it easy


0 0 0 0 0 0 0 0 0 1 1 1 1 1 10 0 0 0 0 0 1 1 1 0 0 0 1 1 10 0 0 1 1 1 0 0 0 0 0 0 1 1 11 1 1 0 0 0 0 0 0 0 0 0 1 1 1, , , , , , , , , , , , , , ,

0 1 1 0 1 1 0 1 1 1 1 1 0 0 01 0 1 1 0 1 1 1 1 0 1 1 0 1 11 1 0 1 1 1 1 0 1 1 0 1 1 0 11 1 1 1 1 0 1 1 0 1 1 0 1 1 0


0 0 0 0 0 0 0 0 0 1 1 1 1 1 10 0 0 0 0 0 1 1 1 0 0 0 1 1 10 0 0 1 1 1 0 0 0 0 0 0 1 1 11 1 1 0 0 0 0 0 0 0 0 0 1 1 1, , , , , , , , , , , , , , ,

0 1 1 0 1 1 0 1 1 1 1 1 0 0 01 0 1 1 0 1 1 1 1 0 1 1 0 1 11 1 0 1 1 1 1 0 1 1 0 1 1 0 11 1 1 1 1 0 1 1 0 1 1 0 1 1 0

1000

0100

0010

0001

1100

1010

1001

0110

0101

0011

1110

1101

1011

0111

1111


0 0 0 0 0 0 0 0 0 1 1 1 1 1 10 0 0 0 0 0 1 1 1 0 0 0 1 1 10 0 0 1 1 1 0 0 0 0 0 0 1 1 11 1 1 0 0 0 0 0 0 0 0 0 1 1 1, , , , , , , , , , , , , , ,

0 1 1 0 1 1 0 1 1 1 1 1 0 0 01 0 1 1 0 1 1 1 1 0 1 1 0 1 11 1 0 1 1 1 1 0 1 1 0 1 1 0 11 1 1 1 1 0 1 1 0 1 1 0 1 1 0

1000 1 1 1

0100 1 1 1

0010 1 1 1

0001 1 1 1

1100 1 1 1

1010 1 1 1

1001 1 1 1

0110 1 1 1

0101 1 1 1

0011 1 1 1

1110 1 1 1

1101 1 1 1

1011 1 1 1

0111 1 1 1

1111 1 1 1


0 0 0 0 0 0 0 0 0 1 1 1 1 1 10 0 0 0 0 0 1 1 1 0 0 0 1 1 10 0 0 1 1 1 0 0 0 0 0 0 1 1 11 1 1 0 0 0 0 0 0 0 0 0 1 1 1, , , , , , , , , , , , , , ,

0 1 1 0 1 1 0 1 1 1 1 1 0 0 01 0 1 1 0 1 1 1 1 0 1 1 0 1 11 1 0 1 1 1 1 0 1 1 0 1 1 0 11 1 1 1 1 0 1 1 0 1 1 0 1 1 0

1000 1 1 1

0100 1 1 1

0010 1 1 1

0001 1 1 1

1100 1 1 1

1010 1 1 1

1001 1 1 1

0110 1 1 1

0101 1 1 1

0011 1 1 1

1110 1 1 1

1101 1 1 1

1011 1 1 1

0111 1 1 1

1111 1 1 1

row sum is λ = 3 for all rows

now, group actions come into play ...

now, group actions come into play ... extend action of A=Aut(L(V ))to subsets of L(V ) by

gB := {gB | B ∈ B}


gB := {gB | B ∈ B}✘✘✘✘✾isomorphism class of B= orbitA(B) := {gB | g ∈ A}



✄✄✄✄✎

automorphism group of B= stabilizerAB := {g ∈ A | gB = B}



✄✄✄✄✎


✄✄✄✎

✘✘✘✾Orbit-Stabilizer-TheoremA(B) → A/AB

AgB = gABg−1

G=AB=AgB⇒g ∈NA(G )



✄✄✄✄✎


✄✄✄✎


AgB = gABg−1


⇒ candidates for possibleautomorphism groupscan be reduced todifferent conjugacy classes



✄✄✄✄✎


✄✄✄✎


AgB = gABg−1


⇒ candidates for possibleautomorphism groupscan be reduced todifferent conjugacy classes

⇒ the normalizer ofautomorphims groups giveshints for classification issues(several papers by Laue)

our 1-(4, 2, 3; 2) design admitsG ≃ Sym(4) as group of automorphisms


0 0 0 0 0 0 0 0 0 1 1 1 1 1 10 0 0 0 0 0 1 1 1 0 0 0 1 1 10 0 0 1 1 1 0 0 0 0 0 0 1 1 11 1 1 0 0 0 0 0 0 0 0 0 1 1 1, , , , , , , , , , , , , , ,

0 1 1 0 1 1 0 1 1 1 1 1 0 0 01 0 1 1 0 1 1 1 1 0 1 1 0 1 11 1 0 1 1 1 1 0 1 1 0 1 1 0 11 1 1 1 1 0 1 1 0 1 1 0 1 1 0

1000 1 1 1

0100 1 1 1

0010 1 1 1

0001 1 1 1

1100 1 1 1

1010 1 1 1

1001 1 1 1

0110 1 1 1

0101 1 1 1

0011 1 1 1

1110 1 1 1

1101 1 1 1

1011 1 1 1

0111 1 1 1

1111 1 1 1


0 0 0 0 0 0 0 0 0 1 1 1 1 1 10 0 0 0 0 0 1 1 1 0 0 0 1 1 10 0 0 1 1 1 0 0 0 0 0 0 1 1 11 1 1 0 0 0 0 0 0 0 0 0 1 1 1, , , , , , , , , , , , , , ,

0 1 1 0 1 1 0 1 1 1 1 1 0 0 01 0 1 1 0 1 1 1 1 0 1 1 0 1 11 1 0 1 1 1 1 0 1 1 0 1 1 0 11 1 1 1 1 0 1 1 0 1 1 0 1 1 0

1000 1 1 1

0100 1 1 1

0010 1 1 1

0001 1 1 1

1100 1 1 1

1010 1 1 1

1001 1 1 1

0110 1 1 1

0101 1 1 1

0011 1 1 1

1110 1 1 1

1101 1 1 1

1011 1 1 1

0111 1 1 1

1111 1 1 1


0 0 0 0 0 0 0 0 0 1 1 1 1 1 10 0 0 0 0 0 1 1 1 0 0 0 1 1 10 0 0 1 1 1 0 0 0 0 0 0 1 1 11 1 1 0 0 0 0 0 0 0 0 0 1 1 1, , , , , , , , , , , , , , ,

0 1 1 0 1 1 0 1 1 1 1 1 0 0 01 0 1 1 0 1 1 1 1 0 1 1 0 1 11 1 0 1 1 1 1 0 1 1 0 1 1 0 11 1 1 1 1 0 1 1 0 1 1 0 1 1 0

1000 1 1 1

0100 1 1 1

0010 1 1 1

0001 1 1 1

1100 1 1 1

1010 1 1 1

1001 1 1 1

0110 1 1 1

0101 1 1 1

0011 1 1 1

1110 1 1 1

1101 1 1 1

1011 1 1 1

0111 1 1 1

1111 1 1 1


0 0 0 0 0 0 0 0 0 1 1 1 1 1 10 0 0 0 0 0 1 1 1 0 0 0 1 1 10 0 0 1 1 1 0 0 0 0 0 0 1 1 11 1 1 0 0 0 0 0 0 0 0 0 1 1 1, , , , , , , , , , , , , , ,

0 1 1 0 1 1 0 1 1 1 1 1 0 0 01 0 1 1 0 1 1 1 1 0 1 1 0 1 11 1 0 1 1 1 1 0 1 1 0 1 1 0 11 1 1 1 1 0 1 1 0 1 1 0 1 1 0

1000 1 1 1

0100 1 1 1

0010 1 1 1

0001 1 1 1

1100 1 1 1

1010 1 1 1

1001 1 1 1

0110 1 1 1

0101 1 1 1

0011 1 1 1

1110 1 1 1

1101 1 1 1

1011 1 1 1

0111 1 1 1

1111 1 1 1

⇒

3 02 13 00 3

condensedG -incidence matrixused to representdesign admitting G

some subgroups of GL(n, q) ...


Singer cycleS(n, q)=〈σ〉≃F

∗qn



∗qn

Frobenius automorphismF(n, q)=〈φ〉≃Aut(Fqn/Fq)



∗qn


✻

��✒

❅❅

❅■⋊

normalizer ofa Singer cycleN(n, q)=〈σ, φ〉



∗qn


✻

��✒

❅❅

❅■⋊


if ℓ divides n then 〈σ, φℓ〉 ≃ N(n/ℓ, qℓ)if n prime then N(n, q) is maximal in GL(n, q)N(n, q) is self-normalizing in GL(n, q)



∗qn


✻

��✒

❅❅

❅■⋊


if ℓ divides n then 〈σ, φℓ〉 ≃ N(n/ℓ, qℓ)if n prime then N(n, q) is maximal in GL(n, q)N(n, q) is self-normalizing in GL(n, q)

Corollaryfor primes n and q

different t-(n, k , λ; q) designsadmitting N(n, q) as a group ofautomorphisms are non-isomorphic

Thomas constructed the first non-trivial designextended by Suzuki to an infinite series for all q


Sr (T ) := 〈b0 . . . br−1 | bi ∈ T , 0 ≤ i < r〉Br := {Sr (T ) | T ∈

[V2

]}



[V2

]}

Theorem (Suzuki, 1990)for n ≥ 7 with (n, 4) = 1 the set B2

defines a 2-(n, 3,[32

]q; q) design admitting

N(n, q) as a group of automorphisms



[V2

]}





under which conditions does Br

define a 2-(n, r + 1,[r+12

]q; q) design?



[V2

]}






define a 2-(n, r + 1,[r+12

]q; q) design?

Theorem (Abe, Yoshiara, 1993)the set Br is no design for q = 2and 4 ≤ r + 1 < n ≤ 15except for the pair r = 3, n = 7



[V2

]}






define a 2-(n, r + 1,[r+12

]q; q) design?

Theorem (Abe, Yoshiara, 1993)the set Br is no design for q = 2and 4 ≤ r + 1 < n ≤ 15except for the pair r = 3, n = 7

conjecture: B3 is the dual of B2 for n = 7 and all q

Theorem (Kramer, Mesner, 1976)there exists a t-(n, k , λ; q) design


iff∃ 0-1 solution x = [. . . , xK , . . .]

t of


iff∃ 0-1 solution x = [. . . , xK , . . .]

t of

·

...

xK

...

=

λ...λ

ζt,k = (ζTK ) with ζTK = ζ(T ,K )


iff∃ 0-1 solution x = [. . . , xK , . . .]

t of

K ∈[Vk

]

...[Vt

]∋ T · · · ζTK

·

...

xK

...

=

λ...λ

ζt,k = (ζTK ) with ζTK = ζ(T ,K )

Theorem (Kramer, Mesner, 1976)there exists a t-(n, k , λ; q) designadmitting G ≤ Aut(L(V ))as a group of automorphismsiff∃ 0-1 solution x = [. . . , xK , . . .]

t of

G (K ) ∈ G\\[Vk

]

...

G\\[Vt

]∋ G (T ) · · · ζGTK

·

...

xK

...

=

λ...λ

ζGt,k = (ζGTK ) with ζGTK =∑

K ′∈G(K)

ζ(T ,K ′)

we consider Suzuki’s design...

we consider Suzuki’s design...2-(7, 3, 7; 2)


plain incidence matrix ζ2,32667 × 11811



prescribed group of automorphismsnormalizer of a singer cycle N(7, 2)of order 889




ζN(7,2)2,3 =

5 3 1 3 2 3 1 2 2 3 2 1 1 2 01 2 0 1 3 2 5 3 3 3 2 1 2 2 11 2 0 3 2 2 1 2 2 1 3 5 4 3 0




ζN(7,2)2,3 =

5 3 1 3 2 3 1 2 2 3 2 1 1 2 01 2 0 1 3 2 5 3 3 3 2 1 2 2 11 2 0 3 2 2 1 2 2 1 3 5 4 3 0

↑ ↑ ↑




ζN(7,2)2,3 =

5 3 1 3 2 3 1 2 2 3 2 1 1 2 01 2 0 1 3 2 5 3 3 3 2 1 2 2 11 2 0 3 2 2 1 2 2 1 3 5 4 3 0

↑ ↑ ↑

19 solutions (non-isomorphic)

a particular class of designs ...

a particular class of designs ...t-(n, k , λ; q) design B transitiveiff |AB\\

[V1

]| = 1


[V1

]| = 1

Theorem (Miyakawa, Munemasa, Yoshiara, 1995)B non-trivial transitive t-(n, k , λ; q) design, t ≥ 2– if n = 6 then q ≡ 1 mod 3 and S(6, q) 6≤ AB ≤ N(6, q)– if n prime then either AB = S(n, q) or AB = N(n, q)


[V1

]| = 1


⇒ Suzuki’s design is transitive


[V1

]| = 1


⇒ Suzuki’s design is transitive

Theorem (Miyakawa, Munemasa, Yoshiara, 1995)[λ, number of non-isom. 2-(7, 3, λ; q) designs B with AB = N(n, q)]

q = 2: [3, 2], [5, 14], [7, 19], [10, 30], [12, 90], [14, 55],q = 2: [λ, 0] for λ = 1, 2, 4, 6, 8, 9, 11, 13, 15q = 3: [5, 22], [λ, 0] for λ = 1, 2, 3, 4

further computer results for the binary field ...

further computer results for the binary field ...

Theorem (B., Kerber, Laue, S. Braun, 2005–2015)t-(n, k, λ; q) G |ζG

t,k| λ

3-(8, 4, λ; 2) N(8, 2) 53×109 11N(4, 22) 105×217 11, 15

2-(11, 3, λ; 2) N(11, 2) 31×2263 245, 2522-(10, 3, λ; 2) N(10, 2) 20×633 15, 30, 45, 60, 75, 90, 105, 1202-(9, 4, λ; 2) N(9, 2) 11×725 21, 63, 84, 126, 147, 189, 210, 252, 273, 315,

336, 378, 399, 441, 462, 504, 525, 567, 588, 630,651, 693, 714, 756, 777, 819, 840, 882, 903, 945,966, 1008, 1029, 1071, 1092, 1134, 1155,1197, 1218, 1260, 1281, 1323

2-(9, 3, λ; 2) N(9, 2) 11×177 63N(3, 23) 31×529 21, 22, 42, 43, 63N(8, 2)×1 28×408 7, 12, 19, 24, 31, 36, 43, 48, 55, 60M(3, 23) 40×460 49

2-(8, 4, λ; 2) N(4, 22) 15×217 21, 35, 56, 70, 91, 105, 126, 140, 161, 175, 196,210, 231, 245, 266, 280, 301, 315

N(7, 2)×1 13×231 7, 14, 49, 56, 63, 98, 105, 112, 147, 154, 161,196, 203, 210, 245, 252, 259, 294, 301, 308

2-(8, 3, λ; 2) N(4, 22) 15×105 212-(7, 3, λ; 2) N(7, 2) 3×15 3, 5, 7, 10, 12, 14,

S(7, 2) 21 × 93 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 152-(6, 3, λ; 2) 〈σ7〉 77×155 3, 6

... also for q = 3, 4 and 5

t-(n, k , λ; q) G |ζGt,k | λ

2-(8, 3, λ; 3) N(8, 3) 41 × 977 52, 104, 1562-(7, 3, λ; 3) N(7, 3) 13 × 121 5, . . . , 602-(6, 3, λ; 3) 〈σ2, φ2〉 25 × 76 20

〈σ17, φ〉 93 × 234 16S(5, 3)×1 51 × 150 8, 16, 20〈σ2〉×1 91 × 280 8, 12, 16, 20

2-(6, 3, λ; 4) 〈σ3, φ〉 51 × 161 15, 35〈σ3, φ〉×1 57 × 229 10, 25, 30, 35

2-(6, 3, λ; 5) N(6, 5) 53 × 248 78

further constructions ....


t-(n, k , λ; q)


t-(n, k , λ; q)❅

❅❅

❅❅❅■

supplemented

t-(n, k ,[n−tk−t

]q−λ; q)


t-(n, k , λ; q)❅

❅❅

❅❅❅■

supplemented


]q−λ; q)

(t − 1)-(n − 1, k − 1, λ; q)

❍❍❍❍❍❍❥derived


t-(n, k , λ; q)❅

❅❅

❅❅❅■

supplemented


]q−λ; q)

(t − 1)-(n − 1, k − 1, λ; q)


Theorem (Suzuki, 1989)(V ,B) t-(n, k , λ; q) designi + j ≤ t, P ∈

[Vi

], H ∈

[Vn−j

]

⇒ |{B ∈ B | P ≤ B ≤ H}| = λ[n−j−i

k−i ]q[n−tk−t]q


t-(n, k , λ; q)❅

❅❅

❅❅❅■

supplemented


]q−λ; q)

(t − 1)-(n − 1, k − 1, λ; q)



[Vi

], H ∈

[Vn−j

]

⇒ |{B ∈ B | P ≤ B ≤ H}| = λ[n−j−i


✟✟✟✟✟✟✙(i= t−1, j=0) reduced

(t − 1)-(n, k , λ qn−t+1−1qk−t+1−1

; q)


t-(n, k , λ; q)❅

❅❅

❅❅❅■

supplemented


]q−λ; q)

(t − 1)-(n − 1, k − 1, λ; q)



[Vi

], H ∈

[Vn−j

]

⇒ |{B ∈ B | P ≤ B ≤ H}| = λ[n−j−i


✟✟✟✟✟✟✙(i= t−1, j=0) reduced

(t − 1)-(n, k , λ qn−t+1−1qk−t+1−1

; q)

t-(n, n − k , λ[n−tk

]q/[n−tk−t

]q; q)

✂✂✂✂✂✂✂✂✂✂✌

dual(i=0, j= t, ·⊥)


t-(n, k , λ; q)❅

❅❅

❅❅❅■

supplemented


]q−λ; q)

(t − 1)-(n − 1, k − 1, λ; q)



[Vi

], H ∈

[Vn−j

]

⇒ |{B ∈ B | P ≤ B ≤ H}| = λ[n−j−i


✟✟✟✟✟✟✙(i= t−1, j=0) reduced

(t − 1)-(n, k , λ qn−t+1−1qk−t+1−1

; q)

t-(n, n − k , λ[n−tk

]q/[n−tk−t

]q; q)

✂✂✂✂✂✂✂✂✂✂✌

dual(i=0, j= t, ·⊥)

(t − 1)-(n − 1, k , λ qn−k−1qk−t+1−1

; q)

❇❇❇❇❇❇❇❇❇❇◆

residual(i= t−1, j=1)

Theorem (Kiermaier, Laue, 2014)the existence of designs for derivedand residual parameters impliesthe existence of a designfor the reduced parameters


2-(9, 3, 21; 2)

by computer

2-(9, 4, 441; 2)

✻

❄


2-(9, 3, 21; 2)

by computer

2-(9, 4, 441; 2)

✻

❄

3-(10, 4, 21; 2)✟✟✟✟✟✙

der

❍❍❍❍❍❨

res


2-(9, 3, 21; 2)

by computer

2-(9, 4, 441; 2)

✻

❄

3-(10, 4, 21; 2)✟✟✟✟✟✙

der

❍❍❍❍❍❨

res

✟✟✟✟✟✯red

2-(10, 4, 1785; 2)


2-(9, 3, 21; 2)

by computer

2-(9, 4, 441; 2)

✻

❄

3-(10, 4, 21; 2)✟✟✟✟✟✙

der

❍❍❍❍❍❨

res

✟✟✟✟✟✯red

2-(10, 4, 1785; 2)

Corollary (Kiermaier, Laue, 2014)There exist designs with parameters2-(8, 4, λ; 2) for λ = 63, 84, 147, 168, 189, 252, 273, 2942-(10, 4, λ; 2) for λ = 1785, 1870, 3570, 3655, 53552-(8, 4, 91λ; 3) for λ = 5, 6, 7, . . . , 60

further consequences?


take Suzuki’s design



2-(7, 3,[32

]q; q)



2-(7, 3,[32

]q; q)

❄

dual

2-(7, 4,[42

]q; q)



2-(7, 3,[32

]q; q)

❄

dual

2-(7, 4,[42

]q; q)

3-(8, 4,[32

]q; q)✟✟✟✟✟✙

der

❍❍❍❍❍❨

res



2-(7, 3,[32

]q; q)

❄

dual

2-(7, 4,[42

]q; q)

3-(8, 4,[32

]q; q)✟✟✟✟✟✙

der

❍❍❍❍❍❨

res ❅❅❅❅❅❅❘

red

Corollarythere exists a family of

2-(8, 4, (q6−1)(q3−1)

(q2−1)(q−1); q)

designs for all prime powers q

(t − 1)-(2t + 3, t + 1, λ qt+3−1

q2−1; q) (t − 1)-(2t + 3, t, λ; q)

(t − 1)-(2t + 3, t + 1, λ qt+3−1

q2−1; q) (t − 1)-(2t + 3, t, λ; q)

t-(2t + 4, t + 1, λ; q)

❄res

❳❳❳❳❳❳❳❳③der

(t − 1)-(2t + 4, t + 1, λ qt+5−1

q2−1; q)

✘✘✘✘✘✘✘✘✾red

(t − 1)-(2t + 3, t + 2, λ(qt+2

−1)(qt+3−1)

(q2−1)(q3−1); q)

❅❅❘dual

t-(2t + 4, t + 2, λqt+3

−1

q2−1; q)

��✒res

✻

der(t − 1)-(2t + 4, t + 2, λ(qt+3

−1)(qt+5−1)

(q2−1)(q3−1); q)

❅❅■ red

t-(2t + 5, t + 2, λqt+5

−1

q2−1; q)��✒res

✻

der

(t − 1)-(2t + 5, t + 2, λ(qt+5

−1)(qt+6−1)

(q2−1)(q3−1); q)

❅❅❘red

(t − 1)-(2t + 5, t + 3, λ(qt+3

−1)(qt+5−1)(qt+6

−1)

(q2−1)(q3−1)(q3−1); q)✲dual

t-(2t + 6, t + 3, λ(qt+5

−1)(qt+6−1)

(q2−1)(q3−1); q)��✒res

❅❅■ der

(t − 1)-(2t + 6, t + 3, λ(qt+5

−1)(qt+6−1)(qt+7

−1)

(q2−1)(q3−1)(q4−1); q)

❄red

(t − 1)-(2t + 3, t + 1, λ qt+3−1

q2−1; q) (t − 1)-(2t + 3, t, λ; q)

t-(2t + 4, t + 1, λ; q)

❄res

❳❳❳❳❳❳❳❳③der

(t − 1)-(2t + 4, t + 1, λ qt+5−1

q2−1; q)

✘✘✘✘✘✘✘✘✾red

(t − 1)-(2t + 3, t + 2, λ(qt+2

−1)(qt+3−1)

(q2−1)(q3−1); q)

❅❅❘dual

t-(2t + 4, t + 2, λqt+3

−1

q2−1; q)

��✒res

✻

der(t − 1)-(2t + 4, t + 2, λ(qt+3

−1)(qt+5−1)

(q2−1)(q3−1); q)

❅❅■ red

t-(2t + 5, t + 2, λqt+5

−1

q2−1; q)��✒res

✻

der

(t − 1)-(2t + 5, t + 2, λ(qt+5

−1)(qt+6−1)

(q2−1)(q3−1); q)

❅❅❘red

(t − 1)-(2t + 5, t + 3, λ(qt+3

−1)(qt+5−1)(qt+6

−1)

(q2−1)(q3−1)(q3−1); q)✲dual

t-(2t + 6, t + 3, λ(qt+5

−1)(qt+6−1)

(q2−1)(q3−1); q)��✒res

❅❅■ der

(t − 1)-(2t + 6, t + 3, λ(qt+5

−1)(qt+6−1)(qt+7

−1)

(q2−1)(q3−1)(q4−1); q)

❄red

computerresultsfor t = 3and q = 2

CorollaryThere exist designs with parameters2-(10, 4, 85λ; 2), 2-(10, 5, 765λ; 2),2-(11, 5, 6205λ; 2), 2-(12, 6, 423181λ; 2)for λ = 7, 12, 19, 21, 22, 24, 31, 36, 42, 43, 48, 49, 55, 60, 63

Itoh’s infinite series of designs ...


Theorem (Itoh, 1998)

there is a family of 2-(nℓ, 3, q3 qn−5−1q−1 ; q) designs

admitting SL(ℓ, qn) as group of automorphismsfor q ≥ 2, ℓ ≥ 3, n ≡ 5 mod 6(q − 1)





ζSL(ℓ,qn)2,3 =

a

ζS(n,q)2,3

. . . 0 0

a

0 X Y Z





ζSL(ℓ,qn)2,3 =

a

ζS(n,q)2,3

. . . 0 0

a

0 X Y Z

2-(n, 3, λ; q) designs admitting S(n, q)can be extended to2-(nℓ, 3, λ; q) designs admitting SL(ℓ, qn)if λ is appropriate

extension possible if

λ = q(q + 1)(q3 − 1)s + q(q2 − 1)t

for some integer s ≥ 0, t ∈ {0, 1} if 3|n and t = 0 otherwise

extension possible if

λ = q(q + 1)(q3 − 1)s + q(q2 − 1)t

for some integer s ≥ 0, t ∈ {0, 1} if 3|n and t = 0 otherwise

for ℓ ≥ 3, n ≡ 5 mod 6(q − 1)Suzuki’s supplemented designhas index value λ of required form

further results fromItoh’s construction?


q = 2: λ = 42s + 6tq = 3: λ = 312s + 24t


q = 2: λ = 42s + 6tq = 3: λ = 312s + 24t

computer constructionsof designs admitting S(n, q)2-(8, 3, 21; 2)

supp⇒ 2-(8, 3, 42; 2)

2-(9, 3, 42; 2)2-(9, 3, 43; 2)

supp⇒ 2-(9, 3, 84; 2)

2-(10, 3, 45; 2)supp⇒ 2-(10, 3, 210; 2)

2-(13, 3, 42s ; 2) for 1 ≤ s ≤ 252-(8, 3, 52; 3)

supp⇒ 2-(8, 3, 312; 3)


q = 2: λ = 42s + 6tq = 3: λ = 312s + 24t

computer constructionsof designs admitting S(n, q)2-(8, 3, 21; 2)

supp⇒ 2-(8, 3, 42; 2)

2-(9, 3, 42; 2)2-(9, 3, 43; 2)

supp⇒ 2-(9, 3, 84; 2)

2-(10, 3, 45; 2)supp⇒ 2-(10, 3, 210; 2)

2-(13, 3, 42s ; 2) for 1 ≤ s ≤ 252-(8, 3, 52; 3)

supp⇒ 2-(8, 3, 312; 3)

Corollarythere exist families of designsadmitting SL(ℓ, qn) for ℓ ≥ 32-(8ℓ, 3, 42; 2)2-(9ℓ, 3, 42s; 2) for 1 ≤ s ≤ 22-(10ℓ, 3, 210; 2)2-(13ℓ, 3, 42s; 2) for 1 ≤ s ≤ 252-(8ℓ, 3, 312; 3)

Sq(t, k , n) q-Steiner system= t-(n, k , 1; q) design


TheoremSq(1, k , n) exists if and only if k divides n



trivial q-Steiner systems



trivial q-Steiner systems

existence of non-trivialq-Steiner systems (t > 1)was long-standing issue ...

Theorem (B., Etzion, Ostergard, Vardy, Wassermann, 2012)q-Steiner systems do exist


S2(2, 3, 13)


S2(2, 3, 13)

automorphism group N(13, 2)of order 13 · (213−1) = 106483

consisting of 15 orbitsof full length 106483

taken out of 25572possible orbits

≥ 1050 disjoint solutionsall non-isomorphic

solved with Knuth’s dancing links

S2(2, 3, 7)?

S2(2, 3, 7)?

✫✪✬✩

✁✁✁✁✁✁✁

❆❆

❆❆

❆❆❆

r rrrr

rr q-analog of theFano plane S(2, 3, 7)

S2(2, 3, 7)?

✫✪✬✩

✁✁✁✁✁✁✁

❆❆

❆❆

❆❆❆

r rrrr


it would have size 381

S2(2, 3, 7)?

✫✪✬✩

✁✁✁✁✁✁✁

❆❆

❆❆

❆❆❆

r rrrr



replacing “exactly”by “at most”yields q-packing(subspace codes)

S2(2, 3, 7)?

✫✪✬✩

✁✁✁✁✁✁✁

❆❆

❆❆

❆❆❆

r rrrr




PPPPPq best known size is 329(B., Reichelt, 2014)(Liu, Honold, 2014)

S2(2, 3, 7)?

✫✪✬✩

✁✁✁✁✁✁✁

❆❆

❆❆

❆❆❆

r rrrr




PPPPPq best known size is 329(B., Reichelt, 2014)(Liu, Honold, 2014)

large gap

what if S2(2, 3, 7) does exist?

what if S2(2, 3, 7) does exist?what is the automorphism group?


exclude groups step by step



consider only one subgroupof each conjugacy class

r{1}

rA = Aut(L(V ))




r{1}

rA = Aut(L(V ))

rG

1) choose representative G

of some conjugacy class of A




r{1}

rA = Aut(L(V ))

rG



2) solve ζGt,kx = [1, . . . , 1]t




r{1}

rA = Aut(L(V ))

rG



2) solve ζGt,kx = [1, . . . , 1]t

s3) no solution




r{1}

rA = Aut(L(V ))

rG



2) solve ζGt,kx = [1, . . . , 1]t

s3) no solution ❅❅

❅❅✁✁✁

✟✟✟✟✟✟

⇒ subgroups S withG ≤ S ≤ A

and their conjugacy classescannot occur asgroups of automorphisms

systematic elimination ofsubgroups (p-groups first)and conjugacy classes yields...

systematic elimination ofsubgroups (p-groups first)and conjugacy classes yields...

Theorem (B., Kiermaier, Nakic, 2014)the automorphism group of S2(2, 3, 7) is either trivial orgenerated by one of the following matrices

0 1 0 0 0 0 0

1 0 0 0 0 0 0

0 0 0 1 0 0 0

0 0 1 0 0 0 0

0 0 0 0 0 1 0

0 0 0 0 1 0 0

0 0 0 0 0 0 1

0 1 0 0 0 0 0

1 1 0 0 0 0 0

0 0 0 1 0 0 0

0 0 1 1 0 0 0

0 0 0 0 0 1 0

0 0 0 0 1 1 0

0 0 0 0 0 0 1

0 1 0 0 0 0 0

1 1 0 0 0 0 0

0 0 0 1 0 0 0

0 0 1 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

1 1 0 0 0 0 0

0 1 1 0 0 0 0

0 0 1 0 0 0 0

0 0 0 1 1 0 0

0 0 0 0 1 1 0

0 0 0 0 0 1 1

0 0 0 0 0 0 1

order 2 order 3 order 3 order 4

we consider collectionsof disjoint designs ...


an LSq[N](t, k , n) large set

is a partition of[Vk

]into N

disjoint t-(n, k ,[n−tk−t

]q/N; q)

designs




]into N


]q/N; q)

designs

N = 2: halving




]into N


]q/N; q)

designs

N = 2: halving


]q/2; q)




]into N


]q/N; q)

designs

N = 2: halving


]q/2; q)

❄supp


]q/2; q)




]into N


]q/N; q)

designs

N = 2: halving


]q/2; q)

❄supp


]q/2; q)

❍❍❍❥

✟✟✟✯ LSq[2](t, k , n)




]into N


]q/N; q)

designs

N = 2: halving


]q/2; q)

❄supp


]q/2; q)

❍❍❍❥

✟✟✟✯ LSq[2](t, k , n)

examplesLS3[2](2, 3, 6)LS5[2](2, 3, 6)




]into N


]q/N; q)

designs

N = 2: halving


]q/2; q)

❄supp


]q/2; q)

❍❍❍❥

✟✟✟✯ LSq[2](t, k , n)

examplesLS3[2](2, 3, 6)LS5[2](2, 3, 6)

Theorem (B., Kohnert, Ostergard, Wassermann, 2014)large sets LSq[N](t, k , n) for N > 2 do exist




]into N


]q/N; q)

designs

N = 2: halving


]q/2; q)

❄supp


]q/2; q)

❍❍❍❥

✟✟✟✯ LSq[2](t, k , n)

examplesLS3[2](2, 3, 6)LS5[2](2, 3, 6)

Theorem (B., Kohnert, Ostergard, Wassermann, 2014)large sets LSq[N](t, k , n) for N > 2 do exist

LS2[3](2, 3, 8)

consider generalization of large sets ...


S ⊆[Vk

]is (N, t)-partitionable iff

there is a partition B1, . . . ,BN of Ssuch that ∀T ∈

[Vt

]and ∀i , j

|{B ∈ Bi |T ⊆B}|= |{B ∈ Bj |T ⊆B}|


S ⊆[Vk



[Vt

]and ∀i , j

|{B ∈ Bi |T ⊆B}|= |{B ∈ Bj |T ⊆B}|

✏✏✏✏✮number can be differentfor different T


S ⊆[Vk



[Vt

]and ∀i , j

|{B ∈ Bi |T ⊆B}|= |{B ∈ Bj |T ⊆B}|


large set LSq[N](t, k , n)

for S =[Vk

]

✡✡✡✢


S ⊆[Vk



[Vt

]and ∀i , j

|{B ∈ Bi |T ⊆B}|= |{B ∈ Bj |T ⊆B}|



for S =[Vk

]

✡✡✡✢

idea for recursive construction of large sets


S ⊆[Vk



[Vt

]and ∀i , j

|{B ∈ Bi |T ⊆B}|= |{B ∈ Bj |T ⊆B}|



for S =[Vk

]

✡✡✡✢

idea for recursive construction of large sets– union and “joins” of (N, ∗)-partitionable sets– are also (N, ∗)-partitionable


S ⊆[Vk



[Vt

]and ∀i , j

|{B ∈ Bi |T ⊆B}|= |{B ∈ Bj |T ⊆B}|



for S =[Vk

]

✡✡✡✢

idea for recursive construction of large sets– union and “joins” of (N, ∗)-partitionable sets– are also (N, ∗)-partitionable– decompose

[Vk

]into union of joins

– of (N, ∗)-partitionable sets


S ⊆[Vk



[Vt

]and ∀i , j

|{B ∈ Bi |T ⊆B}|= |{B ∈ Bj |T ⊆B}|



for S =[Vk

]

✡✡✡✢


[Vk


– of (N, ∗)-partitionable sets⇒

[Vk

]is (N, ∗)-partitionable


S ⊆[Vk



[Vt

]and ∀i , j

|{B ∈ Bi |T ⊆B}|= |{B ∈ Bj |T ⊆B}|



for S =[Vk

]

✡✡✡✢


[Vk


– of (N, ∗)-partitionable sets⇒

[Vk

]is (N, ∗)-partitionable ⇒ large set

identify subspaces withcanonical generator matrices(columns form base)


Γ ⋆∆[Γ X

∆

]


Γ ⋆∆[Γ X

∆

]

e

e

Γ ⋆e ∆

Γ X Y

Id

∆


Γ ⋆∆[Γ X

∆

]

e

e

Γ ⋆e ∆

Γ X Y

Id

∆

e

Γ ⋆e ∆

Γ X

∗

∆


Γ ⋆∆[Γ X

∆

]

e

e

Γ ⋆e ∆

Γ X Y

Id

∆

e

Γ ⋆e ∆

Γ X

∗

∆

X ,Y contain 0 at all pivot row positions of Γ, Id identity matrix


Γ ⋆∆[Γ X

∆

]

e

e

Γ ⋆e ∆

Γ X Y

Id

∆

e

Γ ⋆e ∆

Γ X

∗

∆


S ⋆ T := {Γ ⋆∆ | Γ ∈ S,∆ ∈ T }


Γ ⋆∆[Γ X

∆

]

e

e

Γ ⋆e ∆

Γ X Y

Id

∆

e

Γ ⋆e ∆

Γ X

∗

∆


S ⋆ T := {Γ ⋆∆ | Γ ∈ S,∆ ∈ T }

Lemma (B., Kiermaier, Kohnert, Laue, 2014)1) S,T ⊆

[Vk

](N, t)-partitionable, disjoint

1) ⇒ S ∪ T (N, t)-partitionable


Γ ⋆∆[Γ X

∆

]

e

e

Γ ⋆e ∆

Γ X Y

Id

∆

e

Γ ⋆e ∆

Γ X

∗

∆


S ⋆ T := {Γ ⋆∆ | Γ ∈ S,∆ ∈ T }


[Vk


1) ⇒ S ∪ T (N, t)-partitionable2) S ⊆

[Vk

](N, t)-partitionable, T ⊆

[Ws

](N, r)-partitionable

2) ⇒ S ⋆ T (N, t + r +1)-partitionable


Γ ⋆∆[Γ X

∆

]

e

e

Γ ⋆e ∆

Γ X Y

Id

∆

e

Γ ⋆e ∆

Γ X

∗

∆


S ⋆ T := {Γ ⋆∆ | Γ ∈ S,∆ ∈ T }


[Vk


1) ⇒ S ∪ T (N, t)-partitionable2) S ⊆

[Vk

](N, t)-partitionable, T ⊆

[Ws

](N, r)-partitionable

2) ⇒ S ⋆ T (N, t + r +1)-partitionable

also valid for ⋆e and ⋆e

there exist decompositions of[Vk

]

into unions of joins corresponding togeneralized q-Vandermonde identities


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)

(N, 2) (N, 2)


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)

(N, 2) (N, 2)

✲der❄

LSq [N](1,2,5)


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)

(N, 2) (N, 2)

✲der❄

LSq [N](1,2,5)

(N, 1) (N, 1)


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)

(N, 2) (N, 2)

✲der❄

LSq [N](1,2,5)

(N, 1) (N, 1)

✲der❄

LSq [N](0,1,4)


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)

(N, 2) (N, 2)

✲der❄

LSq [N](1,2,5)

(N, 1) (N, 1)

✲der❄

LSq [N](0,1,4)

(N, 0) (N, 0)


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)

(N, 2) (N, 2)

✲der❄

LSq [N](1,2,5)

(N, 1) (N, 1)

✲der❄

LSq [N](0,1,4)

(N, 0) (N, 0)(N,−1) (N,−1)


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)

(N, 2) (N, 2)

✲der❄

LSq [N](1,2,5)

(N, 1) (N, 1)

✲der❄

LSq [N](0,1,4)

(N, 0) (N, 0)(N,−1) (N,−1)�✒ ❅■

join✻

(N, 2)

�✒ ❅■join✻

(N, 2)


(N, 2)


(N, 2)


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)

(N, 2) (N, 2)

✲der❄

LSq [N](1,2,5)

(N, 1) (N, 1)

✲der❄

LSq [N](0,1,4)

(N, 0) (N, 0)(N,−1) (N,−1)�✒ ❅■

join✻

(N, 2)


(N, 2)


(N, 2)


(N, 2)∪ ∪ ∪(N, 2) =

✻


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)

(N, 2) (N, 2)

✲der❄

LSq [N](1,2,5)

(N, 1) (N, 1)

✲der❄

LSq [N](0,1,4)

(N, 0) (N, 0)(N,−1) (N,−1)�✒ ❅■

join✻

(N, 2)


(N, 2)


(N, 2)


(N, 2)∪ ∪ ∪(N, 2) =

✻

✻

LSq[N](2,3,10)


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)

(N, 2) (N, 2)

✲der❄

LSq [N](1,2,5)

(N, 1) (N, 1)

✲der❄

LSq [N](0,1,4)

(N, 0) (N, 0)(N,−1) (N,−1)�✒ ❅■

join✻

(N, 2)


(N, 2)


(N, 2)


(N, 2)∪ ∪ ∪(N, 2) =

✻

✻

LSq[N](2,3,10)

Theorem (B., Kiermaier, Kohnert, Laue, 2014)the existence of LSq[N](2, 3, 6) implies the existence ofLSq[N](2, k , n) for n≥10, n≡2 mod 4, n−3 ≥k≥3, k≡3 mod 4


]


[F10q

3

]=

[F3q

0

]⋆1

[F6q

3

]∪

[F4q

1

]⋆1

[F5q

2

]∪

[F5q

2

]⋆1

[F4q

1

]∪

[F6q

3

]⋆1

[F3q

0

]

❄LSq[N](2,3,6)

(N, 2) (N, 2)

✲der❄

LSq [N](1,2,5)

(N, 1) (N, 1)

✲der❄

LSq [N](0,1,4)

(N, 0) (N, 0)(N,−1) (N,−1)�✒ ❅■

join✻

(N, 2)


(N, 2)


(N, 2)


(N, 2)∪ ∪ ∪(N, 2) =

✻

✻

LSq[N](2,3,10)

Theorem (B., Kiermaier, Kohnert, Laue, 2014)the existence of LSq[N](2, 3, 6) implies the existence ofLSq[N](2, k , n) for n≥10, n≡2 mod 4, n−3 ≥k≥3, k≡3 mod 4⇒ infinite series for N = 2 and q ∈ {3, 5}

finally... the world of combinatorial q-analogs

finally... the world of combinatorial q-analogs

designt-(n, k, λ; q)

(t−1, k−1)-spreadq-Steiner systemt-(n, k, 1; q)

(k−1)-spread1-(n, k, 1; q)

spread1-(n, 2, 1; q)

q-packing designt-(n, k,≤λ; q)

constant dimension codet-(n, k,≤1; q)

partial spread1-(n, k,≤1; q)

arc = dual of1-(n, n−1,≤λ; q)

multiset arc= linear code

q-covering designt-(n, k,≥λ; q)

= complement oft-(n, k,≤

[

n−tk−t

]

q−λ; q)

blocking set = dual of1-(n, n−1,≥λ; q)

large setdisjoint t-(n, k, λ; q)

(k−1)-parallelismdisjoint 1-(n, k, 1; q)

parallelismdisjoint 1-(n, 2, 1; q)

t-wise balanced designt-(n,K , λ; q)

vector space partition1-(n,K , 1; q)

Documents

Michael Braun University of Applied Sciences Darmstadt ...alcoma15.uni-bayreuth.de/files/slides/invited/braun.pdf · A survey on designs over ﬁnite ﬁelds Michael Braun University