michael braun university of applied sciences darmstadt...
TRANSCRIPT
![Page 1: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/1.jpg)
A survey on designs over finite fields
Michael BraunUniversity of Applied Sciences Darmstadt
ALCOMA15
dedicated to our friend Axel Kohnert (1962–2013)
![Page 2: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/2.jpg)
goals of this talk ...
![Page 3: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/3.jpg)
goals of this talk ...
recall resultsof the past 30 years
![Page 4: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/4.jpg)
goals of this talk ...
recall resultsof the past 30 years
focus onconstructions
![Page 5: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/5.jpg)
goals of this talk ...
recall resultsof the past 30 years
focus onconstructions
computer resultsgroup actions
![Page 6: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/6.jpg)
goals of this talk ...
recall resultsof the past 30 years
focus onconstructions
computer resultsgroup actions
provide list ofknown construtions
![Page 7: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/7.jpg)
goals of this talk ...
recall resultsof the past 30 years
focus onconstructions
computer resultsgroup actions
provide list ofknown construtions
describe somenew results
![Page 8: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/8.jpg)
first, let’s start with a typical slide on q-analogs ...
![Page 9: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/9.jpg)
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 ))
![Page 10: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/10.jpg)
first, let’s start with a typical slide on q-analogs ...
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
)set of k-subsets of V
[Vk
]set of k-subspaces of V
(nk
)binomial coefficient
[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 ))
![Page 11: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/11.jpg)
first, let’s start with a typical slide on q-analogs ...
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
)set of k-subsets of V
[Vk
]set of k-subspaces of V
(nk
)binomial coefficient
[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
![Page 12: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/12.jpg)
take a combinatorial design ...
![Page 13: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/13.jpg)
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
![Page 14: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/14.jpg)
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
... and consider its q-analog
![Page 15: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/15.jpg)
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
... and consider its q-analog
(V,B) is a t-(n, k , λ; q) designiff V ≃ F
nq, B ⊆
[Vk
]blocks
∀T ∈[Vt
]: |{B ∈ B | T ⊆ B}| = λ
![Page 16: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/16.jpg)
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
... and consider its q-analog
(V,B) is a t-(n, k , λ; q) designiff V ≃ F
nq, B ⊆
[Vk
]blocks
∀T ∈[Vt
]: |{B ∈ B | T ⊆ B}| = λ
number of blocks |B| = λ[nt]q[kt ]q
![Page 17: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/17.jpg)
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
... and consider its q-analog
(V,B) is a t-(n, k , λ; q) designiff V ≃ F
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
![Page 18: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/18.jpg)
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
![Page 19: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/19.jpg)
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
![Page 20: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/20.jpg)
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
![Page 21: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/21.jpg)
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
![Page 22: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/22.jpg)
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
![Page 23: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/23.jpg)
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
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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
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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
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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
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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
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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
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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
![Page 30: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/30.jpg)
a first example ...
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a first example ... 1-(4, 2, 3; 2)
![Page 32: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/32.jpg)
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
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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 ...
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... but the incidence matrix makes it easy
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... 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
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... 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
1000
0100
0010
0001
1100
1010
1001
0110
0101
0011
1110
1101
1011
0111
1111
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... 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
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
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... 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
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
![Page 39: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/39.jpg)
now, group actions come into play ...
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now, group actions come into play ... extend action of A=Aut(L(V ))to subsets of L(V ) by
gB := {gB | B ∈ B}
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now, group actions come into play ... extend action of A=Aut(L(V ))to subsets of L(V ) by
gB := {gB | B ∈ B}✘✘✘✘✾isomorphism class of B= orbitA(B) := {gB | g ∈ A}
![Page 42: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/42.jpg)
now, group actions come into play ... extend action of A=Aut(L(V ))to subsets of L(V ) by
gB := {gB | B ∈ B}✘✘✘✘✾isomorphism class of B= orbitA(B) := {gB | g ∈ A}
✄✄✄✄✎
automorphism group of B= stabilizerAB := {g ∈ A | gB = B}
![Page 43: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/43.jpg)
now, group actions come into play ... extend action of A=Aut(L(V ))to subsets of L(V ) by
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 )
![Page 44: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/44.jpg)
now, group actions come into play ... extend action of A=Aut(L(V ))to subsets of L(V ) by
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 )
⇒ candidates for possibleautomorphism groupscan be reduced todifferent conjugacy classes
![Page 45: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/45.jpg)
now, group actions come into play ... extend action of A=Aut(L(V ))to subsets of L(V ) by
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 )
⇒ candidates for possibleautomorphism groupscan be reduced todifferent conjugacy classes
⇒ the normalizer ofautomorphims groups giveshints for classification issues(several papers by Laue)
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our 1-(4, 2, 3; 2) design admitsG ≃ Sym(4) as group of automorphisms
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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
![Page 48: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/48.jpg)
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
![Page 49: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/49.jpg)
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
![Page 50: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/50.jpg)
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
⇒
3 02 13 00 3
condensedG -incidence matrixused to representdesign admitting G
![Page 51: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/51.jpg)
some subgroups of GL(n, q) ...
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some subgroups of GL(n, q) ...
Singer cycleS(n, q)=〈σ〉≃F
∗qn
![Page 53: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/53.jpg)
some subgroups of GL(n, q) ...
Singer cycleS(n, q)=〈σ〉≃F
∗qn
Frobenius automorphismF(n, q)=〈φ〉≃Aut(Fqn/Fq)
![Page 54: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/54.jpg)
some subgroups of GL(n, q) ...
Singer cycleS(n, q)=〈σ〉≃F
∗qn
Frobenius automorphismF(n, q)=〈φ〉≃Aut(Fqn/Fq)
✻
���✒
❅❅
❅■⋊
normalizer ofa Singer cycleN(n, q)=〈σ, φ〉
![Page 55: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/55.jpg)
some subgroups of GL(n, q) ...
Singer cycleS(n, q)=〈σ〉≃F
∗qn
Frobenius automorphismF(n, q)=〈φ〉≃Aut(Fqn/Fq)
✻
���✒
❅❅
❅■⋊
normalizer ofa Singer cycleN(n, q)=〈σ, φ〉
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)
![Page 56: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/56.jpg)
some subgroups of GL(n, q) ...
Singer cycleS(n, q)=〈σ〉≃F
∗qn
Frobenius automorphismF(n, q)=〈φ〉≃Aut(Fqn/Fq)
✻
���✒
❅❅
❅■⋊
normalizer ofa Singer cycleN(n, q)=〈σ, φ〉
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
![Page 57: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/57.jpg)
Thomas constructed the first non-trivial designextended by Suzuki to an infinite series for all q
![Page 58: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/58.jpg)
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
]}
![Page 59: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/59.jpg)
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
]}
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
![Page 60: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/60.jpg)
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
]}
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
under which conditions does Br
define a 2-(n, r + 1,[r+12
]q; q) design?
![Page 61: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/61.jpg)
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
]}
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
under which conditions does Br
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
![Page 62: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/62.jpg)
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
]}
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
under which conditions does Br
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
![Page 63: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/63.jpg)
Theorem (Kramer, Mesner, 1976)there exists a t-(n, k , λ; q) design
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Theorem (Kramer, Mesner, 1976)there exists a t-(n, k , λ; q) design
iff∃ 0-1 solution x = [. . . , xK , . . .]
t of
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Theorem (Kramer, Mesner, 1976)there exists a t-(n, k , λ; q) design
iff∃ 0-1 solution x = [. . . , xK , . . .]
t of
·
...
xK
...
=
λ...λ
ζt,k = (ζTK ) with ζTK = ζ(T ,K )
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Theorem (Kramer, Mesner, 1976)there exists a t-(n, k , λ; q) design
iff∃ 0-1 solution x = [. . . , xK , . . .]
t of
K ∈[Vk
]
...[Vt
]∋ T · · · ζTK
·
...
xK
...
=
λ...λ
ζt,k = (ζTK ) with ζTK = ζ(T ,K )
![Page 67: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/67.jpg)
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 ′)
![Page 68: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/68.jpg)
we consider Suzuki’s design...
![Page 69: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/69.jpg)
we consider Suzuki’s design...2-(7, 3, 7; 2)
![Page 70: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/70.jpg)
we consider Suzuki’s design...2-(7, 3, 7; 2)
plain incidence matrix ζ2,32667 × 11811
![Page 71: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/71.jpg)
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
![Page 72: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/72.jpg)
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
![Page 73: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/73.jpg)
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
↑ ↑ ↑
![Page 74: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/74.jpg)
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
↑ ↑ ↑
19 solutions (non-isomorphic)
![Page 75: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/75.jpg)
a particular class of designs ...
![Page 76: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/76.jpg)
a particular class of designs ...t-(n, k , λ; q) design B transitiveiff |AB\\
[V1
]| = 1
![Page 77: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/77.jpg)
a particular class of designs ...t-(n, k , λ; q) design B transitiveiff |AB\\
[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)
![Page 78: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/78.jpg)
a particular class of designs ...t-(n, k , λ; q) design B transitiveiff |AB\\
[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)
⇒ Suzuki’s design is transitive
![Page 79: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/79.jpg)
a particular class of designs ...t-(n, k , λ; q) design B transitiveiff |AB\\
[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)
⇒ 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
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further computer results for the binary field ...
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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
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... 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
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further constructions ....
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further constructions ....
t-(n, k , λ; q)
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further constructions ....
t-(n, k , λ; q)❅
❅❅
❅❅❅■
supplemented
t-(n, k ,[n−tk−t
]q−λ; q)
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further constructions ....
t-(n, k , λ; q)❅
❅❅
❅❅❅■
supplemented
t-(n, k ,[n−tk−t
]q−λ; q)
(t − 1)-(n − 1, k − 1, λ; q)
❍❍❍❍❍❍❥derived
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further constructions ....
t-(n, k , λ; q)❅
❅❅
❅❅❅■
supplemented
t-(n, k ,[n−tk−t
]q−λ; q)
(t − 1)-(n − 1, k − 1, λ; q)
❍❍❍❍❍❍❥derived
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
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further constructions ....
t-(n, k , λ; q)❅
❅❅
❅❅❅■
supplemented
t-(n, k ,[n−tk−t
]q−λ; q)
(t − 1)-(n − 1, k − 1, λ; q)
❍❍❍❍❍❍❥derived
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
✟✟✟✟✟✟✙(i= t−1, j=0) reduced
(t − 1)-(n, k , λ qn−t+1−1qk−t+1−1
; q)
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further constructions ....
t-(n, k , λ; q)❅
❅❅
❅❅❅■
supplemented
t-(n, k ,[n−tk−t
]q−λ; q)
(t − 1)-(n − 1, k − 1, λ; q)
❍❍❍❍❍❍❥derived
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
✟✟✟✟✟✟✙(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, ·⊥)
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further constructions ....
t-(n, k , λ; q)❅
❅❅
❅❅❅■
supplemented
t-(n, k ,[n−tk−t
]q−λ; q)
(t − 1)-(n − 1, k − 1, λ; q)
❍❍❍❍❍❍❥derived
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
✟✟✟✟✟✟✙(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)
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Theorem (Kiermaier, Laue, 2014)the existence of designs for derivedand residual parameters impliesthe existence of a designfor the reduced parameters
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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)
✻
❄
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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)
✻
❄
3-(10, 4, 21; 2)✟✟✟✟✟✙
der
❍❍❍❍❍❨
res
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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)
✻
❄
3-(10, 4, 21; 2)✟✟✟✟✟✙
der
❍❍❍❍❍❨
res
✟✟✟✟✟✯red
2-(10, 4, 1785; 2)
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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)
✻
❄
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
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further consequences?
![Page 97: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/97.jpg)
further consequences?
take Suzuki’s design
![Page 98: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/98.jpg)
further consequences?
take Suzuki’s design
2-(7, 3,[32
]q; q)
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further consequences?
take Suzuki’s design
2-(7, 3,[32
]q; q)
❄
dual
2-(7, 4,[42
]q; q)
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further consequences?
take Suzuki’s design
2-(7, 3,[32
]q; q)
❄
dual
2-(7, 4,[42
]q; q)
3-(8, 4,[32
]q; q)✟✟✟✟✟✙
der
❍❍❍❍❍❨
res
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further consequences?
take Suzuki’s design
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
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(t − 1)-(2t + 3, t + 1, λ qt+3−1
q2−1; q) (t − 1)-(2t + 3, t, λ; q)
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(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
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(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
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Itoh’s infinite series of designs ...
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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)
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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
![Page 108: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/108.jpg)
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
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
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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
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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
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further results fromItoh’s construction?
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further results fromItoh’s construction?
q = 2: λ = 42s + 6tq = 3: λ = 312s + 24t
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further results fromItoh’s construction?
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)
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further results fromItoh’s construction?
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)
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Sq(t, k , n) q-Steiner system= t-(n, k , 1; q) design
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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
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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
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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
existence of non-trivialq-Steiner systems (t > 1)was long-standing issue ...
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Theorem (B., Etzion, Ostergard, Vardy, Wassermann, 2012)q-Steiner systems do exist
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Theorem (B., Etzion, Ostergard, Vardy, Wassermann, 2012)q-Steiner systems do exist
S2(2, 3, 13)
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Theorem (B., Etzion, Ostergard, Vardy, Wassermann, 2012)q-Steiner systems do exist
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
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S2(2, 3, 7)?
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S2(2, 3, 7)?
✫✪✬✩
✁✁✁✁✁✁✁
❆❆
❆❆
❆❆❆
r rrrr
rr q-analog of theFano plane S(2, 3, 7)
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S2(2, 3, 7)?
✫✪✬✩
✁✁✁✁✁✁✁
❆❆
❆❆
❆❆❆
r rrrr
rr q-analog of theFano plane S(2, 3, 7)
it would have size 381
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S2(2, 3, 7)?
✫✪✬✩
✁✁✁✁✁✁✁
❆❆
❆❆
❆❆❆
r rrrr
rr q-analog of theFano plane S(2, 3, 7)
it would have size 381
replacing “exactly”by “at most”yields q-packing(subspace codes)
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S2(2, 3, 7)?
✫✪✬✩
✁✁✁✁✁✁✁
❆❆
❆❆
❆❆❆
r rrrr
rr q-analog of theFano plane S(2, 3, 7)
it would have size 381
replacing “exactly”by “at most”yields q-packing(subspace codes)
PPPPPq best known size is 329(B., Reichelt, 2014)(Liu, Honold, 2014)
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S2(2, 3, 7)?
✫✪✬✩
✁✁✁✁✁✁✁
❆❆
❆❆
❆❆❆
r rrrr
rr q-analog of theFano plane S(2, 3, 7)
it would have size 381
replacing “exactly”by “at most”yields q-packing(subspace codes)
PPPPPq best known size is 329(B., Reichelt, 2014)(Liu, Honold, 2014)
large gap
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what if S2(2, 3, 7) does exist?
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what if S2(2, 3, 7) does exist?what is the automorphism group?
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what if S2(2, 3, 7) does exist?what is the automorphism group?
exclude groups step by step
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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 ))
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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 ))
rG
1) choose representative G
of some conjugacy class of A
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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 ))
rG
1) choose representative G
of some conjugacy class of A
2) solve ζGt,kx = [1, . . . , 1]t
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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 ))
rG
1) choose representative G
of some conjugacy class of A
2) solve ζGt,kx = [1, . . . , 1]t
s3) no solution
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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 ))
rG
1) choose representative G
of some conjugacy class of A
2) solve ζGt,kx = [1, . . . , 1]t
s3) no solution ❅❅
❅❅✁✁✁
✟✟✟✟✟✟
⇒ subgroups S withG ≤ S ≤ A
and their conjugacy classescannot occur asgroups of automorphisms
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systematic elimination ofsubgroups (p-groups first)and conjugacy classes yields...
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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
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we consider collectionsof disjoint designs ...
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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
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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
N = 2: halving
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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
N = 2: halving
t-(n, k ,[n−tk−t
]q/2; q)
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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
N = 2: halving
t-(n, k ,[n−tk−t
]q/2; q)
❄supp
t-(n, k ,[n−tk−t
]q/2; q)
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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
N = 2: halving
t-(n, k ,[n−tk−t
]q/2; q)
❄supp
t-(n, k ,[n−tk−t
]q/2; q)
❍❍❍❥
✟✟✟✯ LSq[2](t, k , n)
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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
N = 2: halving
t-(n, k ,[n−tk−t
]q/2; q)
❄supp
t-(n, k ,[n−tk−t
]q/2; q)
❍❍❍❥
✟✟✟✯ LSq[2](t, k , n)
examplesLS3[2](2, 3, 6)LS5[2](2, 3, 6)
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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
N = 2: halving
t-(n, k ,[n−tk−t
]q/2; q)
❄supp
t-(n, k ,[n−tk−t
]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
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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
N = 2: halving
t-(n, k ,[n−tk−t
]q/2; q)
❄supp
t-(n, k ,[n−tk−t
]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)
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consider generalization of large sets ...
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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}|
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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}|
✏✏✏✏✮number can be differentfor different T
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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}|
✏✏✏✏✮number can be differentfor different T
large set LSq[N](t, k , n)
for S =[Vk
]
✡✡✡✢
![Page 151: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/151.jpg)
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}|
✏✏✏✏✮number can be differentfor different T
large set LSq[N](t, k , n)
for S =[Vk
]
✡✡✡✢
idea for recursive construction of large sets
![Page 152: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/152.jpg)
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}|
✏✏✏✏✮number can be differentfor different T
large set LSq[N](t, k , n)
for S =[Vk
]
✡✡✡✢
idea for recursive construction of large sets– union and “joins” of (N, ∗)-partitionable sets– are also (N, ∗)-partitionable
![Page 153: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/153.jpg)
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}|
✏✏✏✏✮number can be differentfor different T
large set LSq[N](t, k , n)
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
![Page 154: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/154.jpg)
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}|
✏✏✏✏✮number can be differentfor different T
large set LSq[N](t, k , n)
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⇒
[Vk
]is (N, ∗)-partitionable
![Page 155: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/155.jpg)
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}|
✏✏✏✏✮number can be differentfor different T
large set LSq[N](t, k , n)
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⇒
[Vk
]is (N, ∗)-partitionable ⇒ large set
![Page 156: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/156.jpg)
identify subspaces withcanonical generator matrices(columns form base)
![Page 157: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/157.jpg)
identify subspaces withcanonical generator matrices(columns form base)
Γ ⋆∆[Γ X
∆
]
![Page 158: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/158.jpg)
identify subspaces withcanonical generator matrices(columns form base)
Γ ⋆∆[Γ X
∆
]
e
e
Γ ⋆e ∆
Γ X Y
Id
∆
![Page 159: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/159.jpg)
identify subspaces withcanonical generator matrices(columns form base)
Γ ⋆∆[Γ X
∆
]
e
e
Γ ⋆e ∆
Γ X Y
Id
∆
e
Γ ⋆e ∆
Γ X
∗
∆
![Page 160: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/160.jpg)
identify subspaces withcanonical generator matrices(columns form base)
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
![Page 161: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/161.jpg)
identify subspaces withcanonical generator matrices(columns form base)
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
S ⋆ T := {Γ ⋆∆ | Γ ∈ S,∆ ∈ T }
![Page 162: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/162.jpg)
identify subspaces withcanonical generator matrices(columns form base)
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
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
![Page 163: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/163.jpg)
identify subspaces withcanonical generator matrices(columns form base)
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
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)-partitionable2) S ⊆
[Vk
](N, t)-partitionable, T ⊆
[Ws
](N, r)-partitionable
2) ⇒ S ⋆ T (N, t + r +1)-partitionable
![Page 164: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/164.jpg)
identify subspaces withcanonical generator matrices(columns form base)
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
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)-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
![Page 165: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/165.jpg)
there exist decompositions of[Vk
]
into unions of joins corresponding togeneralized q-Vandermonde identities
![Page 166: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/166.jpg)
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
]
![Page 167: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/167.jpg)
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
]
❄LSq[N](2,3,6)
![Page 168: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/168.jpg)
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
]
❄LSq[N](2,3,6)
(N, 2) (N, 2)
![Page 169: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/169.jpg)
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
]
❄LSq[N](2,3,6)
(N, 2) (N, 2)
✲der❄
LSq [N](1,2,5)
![Page 170: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/170.jpg)
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
]
❄LSq[N](2,3,6)
(N, 2) (N, 2)
✲der❄
LSq [N](1,2,5)
(N, 1) (N, 1)
![Page 171: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/171.jpg)
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
]
❄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)
![Page 172: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/172.jpg)
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
]
❄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)
![Page 173: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/173.jpg)
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
]
❄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)
![Page 174: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/174.jpg)
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
]
❄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)
�✒ ❅■join✻
(N, 2)
�✒ ❅■join✻
(N, 2)
![Page 175: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/175.jpg)
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
]
❄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)
�✒ ❅■join✻
(N, 2)
�✒ ❅■join✻
(N, 2)∪ ∪ ∪(N, 2) =
✻
![Page 176: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/176.jpg)
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
]
❄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)
�✒ ❅■join✻
(N, 2)
�✒ ❅■join✻
(N, 2)∪ ∪ ∪(N, 2) =
✻
✻
LSq[N](2,3,10)
![Page 177: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/177.jpg)
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
]
❄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)
�✒ ❅■join✻
(N, 2)
�✒ ❅■join✻
(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
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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
]
❄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)
�✒ ❅■join✻
(N, 2)
�✒ ❅■join✻
(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}
![Page 179: 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](https://reader033.vdocument.in/reader033/viewer/2022043021/5f3cf0bee924b679f97aed6c/html5/thumbnails/179.jpg)
finally... the world of combinatorial q-analogs
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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)