Michael Braun University of Applied Sciences Darmstadt...
Transcript of Michael Braun University of Applied Sciences Darmstadt...
A survey on designs over finite fields
Michael BraunUniversity of Applied Sciences Darmstadt
ALCOMA15
dedicated to our friend Axel Kohnert (1962–2013)
goals of this talk ...
goals of this talk ...
recall resultsof the past 30 years
goals of this talk ...
recall resultsof the past 30 years
focus onconstructions
goals of this talk ...
recall resultsof the past 30 years
focus onconstructions
computer resultsgroup actions
goals of this talk ...
recall resultsof the past 30 years
focus onconstructions
computer resultsgroup actions
provide list ofknown construtions
goals of this talk ...
recall resultsof the past 30 years
focus onconstructions
computer resultsgroup actions
provide list ofknown construtions
describe somenew results
first, let’s start with a typical slide on q-analogs ...
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 ))
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 ))
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
take a combinatorial design ...
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
B., Kerber, Laue, SystematicConstruction of q-Analogs ofDesigns, 2005
Suzuki, 2-Designsover GF (q),1992
Thomas, Designsover FiniteFields, 1987
Suzuki, 2-Designsover GF (2m),1990
Ray-Chaudhuri, Schram, Designson Vectorspaces ConstructedUsing Quadratic Forms, 1992
Thomas, Designs and PartialGeometries over Finite Fields,1996
Abe, Yoshiara, On Suzuki’sConstruction of 2-Designsover GF (q), 1993
Itoh, A New Family of 2-Designsover GF (q) Admitting SLm(qℓ),1998
Tits, Sur les AnaloguesAlgebriques des GroupesSemi-Simples Complexes, 1957
Miyakawa, Munemasa, Yoshiara,On a Class of Small 2-Designsover GF (q), 1995
Kiermaier, Laue, Derivedand Residual SubspaceDesigns, 2014
B., Kiermaier, Nakic, On theAutomorphism Group of a Binaryq-Analog of the Fano Plane, 2015
Fazeli, Lovett, Vardy, Non-trivial t-Designs over FiniteFields Exist for All t, 2013
B., Etzion, Ostergard, Vardy, Wasser-mann, On the Existence of q-Analogsof Steiner Systems, 2013
Ray-Chaudhuri, Singhi,q-Analogues of t-Designsand Their Existence, 1989
Etzion, Vardy, On q-Analogsof Steiner Systems andCovering Designs, 2011
Nakic, Pavcevic, TacticalDecompositions of Designsover Finite Fields, 2013
Cameron, Generalisation ofFisher’s Inequality to Fields withmore than One Element, 1974
B., New 3-Designsover the BinaryField, 2013
Schwartz, Etzion, Codesand Anticodes in theGrassman Graph, 2002
Metsch, Bose-Burton TypeTheorems for Finite Projective,Affine and Polar Spaces, 1999
B., Kohnert, Ostergard, Wasser-mann, Large Sets of t-Designsover Finite Fields, 2014
B., A Note on the Existence ofNon-Simple Designs over FiniteFields, 2012
Cameron, LocallySymmetric Designs,1974
B., Some NewDesigns overFinite Fields, 2005
Heden, Sissokho, On theExistence of a (2, 3)-Spreadin V (7, 2), 2011
Suzuki, On theInequalities of t-Designsover a Finite Field, 1990
Kiermaier, Pavcevic,Intersection Numbers forSubspace Designs, 2014
B., q-Analogs of t-WiseBalanced Designs overFinite Fields, 2014
B., Kiermaier, Kohnert,Laue, Large Sets ofSubspace Designs, 2014
Delsarte, Association Schemesand t-Designs in RegularSemilattices, 1976
a first example ...
a first example ... 1-(4, 2, 3; 2)
a first example ... 1-(4, 2, 3; 2)
1 11 01 00 0
1 11 00 01 0
1 10 01 01 0
1 01 11 00 0
1 01 10 01 0
0 01 11 01 0
1 01 01 10 0
1 00 01 11 0
0 01 01 11 0
1 01 00 01 1
1 00 01 01 1
0 01 01 01 1
1 11 10 10 1
1 10 11 10 1
0 11 11 10 1
a first example ... 1-(4, 2, 3; 2)
1 11 01 00 0
1 11 00 01 0
1 10 01 01 0
1 01 11 00 0
1 01 10 01 0
0 01 11 01 0
1 01 01 10 0
1 00 01 11 0
0 01 01 11 0
1 01 00 01 1
1 00 01 01 1
0 01 01 01 1
1 11 10 10 1
1 10 11 10 1
0 11 11 10 1
verification is quite uncomfortable ...
... but the incidence matrix makes it easy
... 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
... 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
... 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
... 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
now, group actions come into play ...
now, group actions come into play ... extend action of A=Aut(L(V ))to subsets of L(V ) by
gB := {gB | B ∈ B}
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}
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}
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 )
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
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)
our 1-(4, 2, 3; 2) design admitsG ≃ Sym(4) as group of automorphisms
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
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
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
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
some subgroups of GL(n, q) ...
some subgroups of GL(n, q) ...
Singer cycleS(n, q)=〈σ〉≃F
∗qn
some subgroups of GL(n, q) ...
Singer cycleS(n, q)=〈σ〉≃F
∗qn
Frobenius automorphismF(n, q)=〈φ〉≃Aut(Fqn/Fq)
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〉
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)
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
Thomas constructed the first non-trivial designextended by Suzuki to an infinite series for all q
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
]}
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
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?
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
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
Theorem (Kramer, Mesner, 1976)there exists a t-(n, k , λ; q) design
Theorem (Kramer, Mesner, 1976)there exists a t-(n, k , λ; q) design
iff∃ 0-1 solution x = [. . . , xK , . . .]
t of
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 )
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 )
Theorem (Kramer, Mesner, 1976)there exists a t-(n, k , λ; q) designadmitting G ≤ Aut(L(V ))as a group of automorphismsiff∃ 0-1 solution x = [. . . , xK , . . .]
t of
G (K ) ∈ G\\[Vk
]
...
G\\[Vt
]∋ G (T ) · · · ζGTK
·
...
xK
...
=
λ...λ
ζGt,k = (ζGTK ) with ζGTK =∑
K ′∈G(K)
ζ(T ,K ′)
we consider Suzuki’s design...
we consider Suzuki’s design...2-(7, 3, 7; 2)
we consider Suzuki’s design...2-(7, 3, 7; 2)
plain incidence matrix ζ2,32667 × 11811
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
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
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
↑ ↑ ↑
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)
a particular class of designs ...
a particular class of designs ...t-(n, k , λ; q) design B transitiveiff |AB\\
[V1
]| = 1
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)
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
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
further computer results for the binary field ...
further computer results for the binary field ...
Theorem (B., Kerber, Laue, S. Braun, 2005–2015)t-(n, k, λ; q) G |ζG
t,k| λ
3-(8, 4, λ; 2) N(8, 2) 53×109 11N(4, 22) 105×217 11, 15
2-(11, 3, λ; 2) N(11, 2) 31×2263 245, 2522-(10, 3, λ; 2) N(10, 2) 20×633 15, 30, 45, 60, 75, 90, 105, 1202-(9, 4, λ; 2) N(9, 2) 11×725 21, 63, 84, 126, 147, 189, 210, 252, 273, 315,
336, 378, 399, 441, 462, 504, 525, 567, 588, 630,651, 693, 714, 756, 777, 819, 840, 882, 903, 945,966, 1008, 1029, 1071, 1092, 1134, 1155,1197, 1218, 1260, 1281, 1323
2-(9, 3, λ; 2) N(9, 2) 11×177 63N(3, 23) 31×529 21, 22, 42, 43, 63N(8, 2)×1 28×408 7, 12, 19, 24, 31, 36, 43, 48, 55, 60M(3, 23) 40×460 49
2-(8, 4, λ; 2) N(4, 22) 15×217 21, 35, 56, 70, 91, 105, 126, 140, 161, 175, 196,210, 231, 245, 266, 280, 301, 315
N(7, 2)×1 13×231 7, 14, 49, 56, 63, 98, 105, 112, 147, 154, 161,196, 203, 210, 245, 252, 259, 294, 301, 308
2-(8, 3, λ; 2) N(4, 22) 15×105 212-(7, 3, λ; 2) N(7, 2) 3×15 3, 5, 7, 10, 12, 14,
S(7, 2) 21 × 93 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 152-(6, 3, λ; 2) 〈σ7〉 77×155 3, 6
... also for q = 3, 4 and 5
t-(n, k , λ; q) G |ζGt,k | λ
2-(8, 3, λ; 3) N(8, 3) 41 × 977 52, 104, 1562-(7, 3, λ; 3) N(7, 3) 13 × 121 5, . . . , 602-(6, 3, λ; 3) 〈σ2, φ2〉 25 × 76 20
〈σ17, φ〉 93 × 234 16S(5, 3)×1 51 × 150 8, 16, 20〈σ2〉×1 91 × 280 8, 12, 16, 20
2-(6, 3, λ; 4) 〈σ3, φ〉 51 × 161 15, 35〈σ3, φ〉×1 57 × 229 10, 25, 30, 35
2-(6, 3, λ; 5) N(6, 5) 53 × 248 78
further constructions ....
further constructions ....
t-(n, k , λ; q)
further constructions ....
t-(n, k , λ; q)❅
❅❅
❅❅❅■
supplemented
t-(n, k ,[n−tk−t
]q−λ; q)
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
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
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)
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, ·⊥)
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)
Theorem (Kiermaier, Laue, 2014)the existence of designs for derivedand residual parameters impliesthe existence of a designfor the reduced parameters
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)
✻
❄
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
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)
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
further consequences?
further consequences?
take Suzuki’s design
further consequences?
take Suzuki’s design
2-(7, 3,[32
]q; q)
further consequences?
take Suzuki’s design
2-(7, 3,[32
]q; q)
❄
dual
2-(7, 4,[42
]q; q)
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
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
(t − 1)-(2t + 3, t + 1, λ qt+3−1
q2−1; q) (t − 1)-(2t + 3, t, λ; q)
(t − 1)-(2t + 3, t + 1, λ qt+3−1
q2−1; q) (t − 1)-(2t + 3, t, λ; q)
t-(2t + 4, t + 1, λ; q)
❄res
❳❳❳❳❳❳❳❳③der
(t − 1)-(2t + 4, t + 1, λ qt+5−1
q2−1; q)
✘✘✘✘✘✘✘✘✾red
(t − 1)-(2t + 3, t + 2, λ(qt+2
−1)(qt+3−1)
(q2−1)(q3−1); q)
❅❅❘dual
t-(2t + 4, t + 2, λqt+3
−1
q2−1; q)
��✒res
✻
der(t − 1)-(2t + 4, t + 2, λ(qt+3
−1)(qt+5−1)
(q2−1)(q3−1); q)
❅❅■ red
t-(2t + 5, t + 2, λqt+5
−1
q2−1; q)��✒res
✻
der
(t − 1)-(2t + 5, t + 2, λ(qt+5
−1)(qt+6−1)
(q2−1)(q3−1); q)
❅❅❘red
(t − 1)-(2t + 5, t + 3, λ(qt+3
−1)(qt+5−1)(qt+6
−1)
(q2−1)(q3−1)(q3−1); q)✲dual
t-(2t + 6, t + 3, λ(qt+5
−1)(qt+6−1)
(q2−1)(q3−1); q)��✒res
❅❅■ der
(t − 1)-(2t + 6, t + 3, λ(qt+5
−1)(qt+6−1)(qt+7
−1)
(q2−1)(q3−1)(q4−1); q)
❄red
(t − 1)-(2t + 3, t + 1, λ qt+3−1
q2−1; q) (t − 1)-(2t + 3, t, λ; q)
t-(2t + 4, t + 1, λ; q)
❄res
❳❳❳❳❳❳❳❳③der
(t − 1)-(2t + 4, t + 1, λ qt+5−1
q2−1; q)
✘✘✘✘✘✘✘✘✾red
(t − 1)-(2t + 3, t + 2, λ(qt+2
−1)(qt+3−1)
(q2−1)(q3−1); q)
❅❅❘dual
t-(2t + 4, t + 2, λqt+3
−1
q2−1; q)
��✒res
✻
der(t − 1)-(2t + 4, t + 2, λ(qt+3
−1)(qt+5−1)
(q2−1)(q3−1); q)
❅❅■ red
t-(2t + 5, t + 2, λqt+5
−1
q2−1; q)��✒res
✻
der
(t − 1)-(2t + 5, t + 2, λ(qt+5
−1)(qt+6−1)
(q2−1)(q3−1); q)
❅❅❘red
(t − 1)-(2t + 5, t + 3, λ(qt+3
−1)(qt+5−1)(qt+6
−1)
(q2−1)(q3−1)(q3−1); q)✲dual
t-(2t + 6, t + 3, λ(qt+5
−1)(qt+6−1)
(q2−1)(q3−1); q)��✒res
❅❅■ der
(t − 1)-(2t + 6, t + 3, λ(qt+5
−1)(qt+6−1)(qt+7
−1)
(q2−1)(q3−1)(q4−1); q)
❄red
computerresultsfor t = 3and q = 2
CorollaryThere exist designs with parameters2-(10, 4, 85λ; 2), 2-(10, 5, 765λ; 2),2-(11, 5, 6205λ; 2), 2-(12, 6, 423181λ; 2)for λ = 7, 12, 19, 21, 22, 24, 31, 36, 42, 43, 48, 49, 55, 60, 63
Itoh’s infinite series of designs ...
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)
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
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
extension possible if
λ = q(q + 1)(q3 − 1)s + q(q2 − 1)t
for some integer s ≥ 0, t ∈ {0, 1} if 3|n and t = 0 otherwise
extension possible if
λ = q(q + 1)(q3 − 1)s + q(q2 − 1)t
for some integer s ≥ 0, t ∈ {0, 1} if 3|n and t = 0 otherwise
for ℓ ≥ 3, n ≡ 5 mod 6(q − 1)Suzuki’s supplemented designhas index value λ of required form
further results fromItoh’s construction?
further results fromItoh’s construction?
q = 2: λ = 42s + 6tq = 3: λ = 312s + 24t
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)
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)
Sq(t, k , n) q-Steiner system= t-(n, k , 1; q) design
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
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
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 ...
Theorem (B., Etzion, Ostergard, Vardy, Wassermann, 2012)q-Steiner systems do exist
Theorem (B., Etzion, Ostergard, Vardy, Wassermann, 2012)q-Steiner systems do exist
S2(2, 3, 13)
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
S2(2, 3, 7)?
S2(2, 3, 7)?
✫✪✬✩
✁✁✁✁✁✁✁
❆❆
❆❆
❆❆❆
r rrrr
rr q-analog of theFano plane S(2, 3, 7)
S2(2, 3, 7)?
✫✪✬✩
✁✁✁✁✁✁✁
❆❆
❆❆
❆❆❆
r rrrr
rr q-analog of theFano plane S(2, 3, 7)
it would have size 381
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)
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)
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
what if S2(2, 3, 7) does exist?
what if S2(2, 3, 7) does exist?what is the automorphism group?
what if S2(2, 3, 7) does exist?what is the automorphism group?
exclude groups step by step
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 ))
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
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
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
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
systematic elimination ofsubgroups (p-groups first)and conjugacy classes yields...
systematic elimination ofsubgroups (p-groups first)and conjugacy classes yields...
Theorem (B., Kiermaier, Nakic, 2014)the automorphism group of S2(2, 3, 7) is either trivial orgenerated by one of the following matrices
0 1 0 0 0 0 0
1 0 0 0 0 0 0
0 0 0 1 0 0 0
0 0 1 0 0 0 0
0 0 0 0 0 1 0
0 0 0 0 1 0 0
0 0 0 0 0 0 1
0 1 0 0 0 0 0
1 1 0 0 0 0 0
0 0 0 1 0 0 0
0 0 1 1 0 0 0
0 0 0 0 0 1 0
0 0 0 0 1 1 0
0 0 0 0 0 0 1
0 1 0 0 0 0 0
1 1 0 0 0 0 0
0 0 0 1 0 0 0
0 0 1 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
1 1 0 0 0 0 0
0 1 1 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 1 0 0
0 0 0 0 1 1 0
0 0 0 0 0 1 1
0 0 0 0 0 0 1
order 2 order 3 order 3 order 4
we consider collectionsof disjoint designs ...
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
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
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)
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)
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)
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)
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
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)
consider generalization of large sets ...
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}|
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
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
]
✡✡✡✢
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
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
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
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
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
identify subspaces withcanonical generator matrices(columns form base)
identify subspaces withcanonical generator matrices(columns form base)
Γ ⋆∆[Γ X
∆
]
identify subspaces withcanonical generator matrices(columns form base)
Γ ⋆∆[Γ X
∆
]
e
e
Γ ⋆e ∆
Γ X Y
Id
∆
identify subspaces withcanonical generator matrices(columns form base)
Γ ⋆∆[Γ X
∆
]
e
e
Γ ⋆e ∆
Γ X Y
Id
∆
e
Γ ⋆e ∆
Γ X
∗
∆
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
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 }
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
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
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
there exist decompositions of[Vk
]
into unions of joins corresponding togeneralized q-Vandermonde identities
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
]
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)
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)
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)
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)
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)
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)
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)
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)
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) =
✻
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)
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
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}
finally... the world of combinatorial q-analogs
finally... the world of combinatorial q-analogs
designt-(n, k, λ; q)
(t−1, k−1)-spreadq-Steiner systemt-(n, k, 1; q)
(k−1)-spread1-(n, k, 1; q)
spread1-(n, 2, 1; q)
q-packing designt-(n, k,≤λ; q)
constant dimension codet-(n, k,≤1; q)
partial spread1-(n, k,≤1; q)
arc = dual of1-(n, n−1,≤λ; q)
multiset arc= linear code
q-covering designt-(n, k,≥λ; q)
= complement oft-(n, k,≤
[
n−tk−t
]
q−λ; q)
blocking set = dual of1-(n, n−1,≥λ; q)
large setdisjoint t-(n, k, λ; q)
(k−1)-parallelismdisjoint 1-(n, k, 1; q)
parallelismdisjoint 1-(n, 2, 1; q)
t-wise balanced designt-(n,K , λ; q)
vector space partition1-(n,K , 1; q)