Construction of new large sets of designs over the binary field - PowerPoint PPT Presentation

Construction of new large sets of designs over the binary field Alfred Wassermann Department of Mathematics, Universität Bayreuth, Germany joint work with Michael Kiermaier and Reinhard Laue DARNEC ’15 Istanbul

Outline ◮ Designs over finite fields ◮ Computer construction ◮ Infinite series of large sets

Designs over finite fields

Subspaces ◮ vector space V = F  q ◮ Grassmannian: G q ( , k ) : = { U ≤ F  q : d im U = k }

Subspace lattice of F 4 2 G 2 ( 4 , 4 ) G 2 ( 4 , 3 ) G 2 ( 4 , 2 ) G 2 ( 4 , 1 ) G 2 ( 4 , 0 )

Subspace lattice ◮ Gaussian coefficient: ( q  − 1 )( q  − 1 − 1 ) · · · ( q  − k + 1 − 1 ) �  � = ( q k − 1 )( q k − 1 − 1 ) · · · ( q − 1 ) k q �  � ◮ | G q ( , k ) | = k q

Designs over finite fields ◮ Cameron (1974), Delsarte (1976) ◮ B ⊆ G q ( , k ) : set of k -subspaces (blocks) ◮ ( F  q , B ) : t - ( , k, λ ; q ) design over F q each t-subspace of F  q is contained in exactly λ blocks of B ◮ B set: simple design ◮ B multiset: non-simple design

Designs over finite fields �  − t � ◮ B = G q ( , k ) is a t - ( , k, q ; q ) design: trivial k − t design trivial 1- ( 4 , 2 , 7; 2 ) design

Designs over finite fields �  − t � ◮ B = G q ( , k ) is a t - ( , k, q ; q ) design: trivial k − t design trivial 1- ( 4 , 2 , 7; 2 ) design 1- ( 4 , 2 , 1; 2 ) design

t - ( , k, λ ; q ) designs  ◮ | B | = λ [ t ] q k [ t ] q ◮ Necessary conditions: �  −  � t −  q for  = 0 , . . . , t ∈ Z λ  = λ � k −  � t −  q

Related designs t - ( , k, λ ; q ) design → ◮ dual design: t - ( ,  − k, λ ; q ) ◮ derived design: ( t − 1 ) - (  − 1 , k − 1 , λ ; q ) ◮ residual design: ( t − 1 ) - (  − 1 , k, μ ; q ) , where � k − t + 1 �  − k � � μ = λ · q / 1 1 q

Large sets of q -analogs of designs �  − t ◮ G q ( , k ) is a t - ( , k, � q ; q ) design k − t ◮ Large set LS q [ N ]( t, k,  ) : partition of G q ( , k ) into N disjoint t - ( , k, λ ; q ) designs LS 2 [ 7 ]( 1 , 2 , 4 ) �  − t ◮ Necessary: N · λ = � k − t q

Automorphisms Designs over finite fields: ◮ GL ( , q ) = { M ∈ F  ×  : M invertible} q ◮ σ ∈ GL ( , q ) automorphism: B σ = B

Automorphisms of designs over finite fields ◮ Singer cycle: ◮ take  ∈ F  q as an element of F q  ◮ ( F q  \ {0} , · ) is a cyclic group G of order q  − 1, i.e. ◮ G = 〈 σ 〉 ◮ G ≤ GL ( , q ) is called Singer cycle ◮ Frobenius automorphism: ◮ ϕ : F q  → F q  , U �→ U q ◮ |〈 ϕ 〉| =  ◮ |〈 σ, ϕ 〉| =  · ( q  − 1 )

Computer construction

Brute force approach for construction ◮ incidence matrix between t -subset and k -subsets: � 1 if T  ⊂ K j M t,k = ( m ,j ) , where m ,j = 0 else ◮ solve  λ  λ   for 0/1-vector  M t,k ·  = .  .  .   λ

Designs with prescribed automorphism group Construction of designs with prescribed automorphism group ◮ choose group G acting on V , i.e. G ≤ S  ◮ search for t -designs D = ( V , B ) having G as a group of automorphisms, i.e. for all g ∈ G and K ∈ B = ⇒ K g ∈ B . ◮ construct D = ( V , B ) as union of orbits of G on k-subsets.

The method of Kramer and Mesner Definition ◮ K ⊂ V and | K | = k : K G : = { K g | g ∈ G } ◮ T ⊂ V and | T | = t : T G : = { T g | g ∈ G } ◮ Let � V � K G 1 ∪ K G 2 ∪ . . . ∪ K G n ⊆ k and � V � T G 1 ∪ T G 2 ∪ . . . ∪ T G m = t ◮ M G t,k = ( m ,j ) where m ,j : = | { K ∈ K G j | T  ⊂ K } |

The method of Kramer and Mesner Theorem (Kramer and Mesner, 1976) The union of orbits corresponding to the 1 s in a {0 , 1} vector which solves  λ  λ   M G t,k ·  = .   . .   λ is a t- ( , k, λ ) design having G as an automorphism group.

Known large sets for t ≥ 2 ◮ LS 2 [ 3 ]( 2 , 3 , 8 ) : Braun, Kohnert, Östergård, W. (2014) ◮ Three disjoint 2- ( 8 , 3 , 21; 2 ) designs ◮ Group: 〈 σ 〉 in GL ( 8 , 2 ) of order 255 ◮ LS 3 [ 2 ]( 2 , 3 , 6 ) : Braun (2005) ◮ T wo disjoint 2- ( 6 , 3 , 20; 3 ) designs ◮ LS 5 [ 2 ]( 2 , 3 , 6 ) : Braun, Kiermaier, Kohnert, Laue (2014) ◮ T wo disjoint 2- ( 6 , 3 , 78; 5 ) designs

A new large set ◮ LS 2 [ 3 ]( 2 , 4 , 8 ) ◮ Three disjoint 2- ( 8 , 4 , 217; 2 ) designs ◮ Group: 〈 σ 5 , ϕ 2 〉 in GL ( 8 , 2 ) of order 204

Related large sets Theorem (Kiermaier, Laue 2015) ◮ derived large set: LS q [ N ]( t − 1 , k − 1 ,  − 1 ) LS q [ N ]( t, k,  ) → ◮ q-analog of Van Trung, Van Leyenhorst, Driessen: LS q [ N ]( t, k − 1 ,  − 1 ) LS q [ N ]( t, k,  − 1 ) and → LS q [ N ]( t, k,  )

Related large sets LS q [ 3 ]( 2 , 3 , 8 ) LS q [ 3 ]( 2 , 4 , 8 ) LS q [ 3 ]( 2 , 5 , 8 ) LS q [ 3 ]( 2 , 4 , 9 ) LS q [ 3 ]( 2 , 5 , 9 ) LS q [ 3 ]( 2 , 5 , 10 )

Admissibility and realizability of LS 2 [ 3 ]( 2 , k,  ) v - 6 - 7 3 4 8 - 4 9 - - 5 10 - - - 11 - - - - 12 - - - - 13 3 4 5 - - 14 - 4 5 - - 15 - - 5 - - - 16 - - - - - - 17 - - - - - - - 18 - - - - - - - 19 3 4 5 ? ? ? 9 10 20 - 4 5 ? ? ? ? 10 21 - - 5 ? ? ? ? ? 11 22 - - - ? ? ? ? ? ? 23 - - - - ? ? ? ? ? ? 24 - - - - - ? ? ? ? ? 25 3 4 5 - - - 9 10 11 ? ? 26 - 4 5 - - - - 10 11 ? ? 27 - - 5 - - - - - 11 ? ? ? 28 - - - - - - - - - ? ? ? 29 - - - - - - - - - - ? ? ? 30 - - - - - - - - - - - ? ? 31 3 4 5 - - - 9 10 11 - - - 15 16 32 - 4 5 - - - - 10 11 - - - - 16 33 - - 5 - - - - - 11 - - - - - 17 34 - - - - - - - - - - - - - - - 35 - - - - - - - - - - - - - - - - 36 - - - - - - - - - - - - - - - - 37 3 4 5 ? ? ? 9 10 11 ? ? ? 15 16 17 - - 38 - 4 5 ? ? ? ? 10 11 ? ? ? ? 16 17 - - 39 - - 5 ? ? ? ? ? 11 ? ? ? ? ? 17 - - - 40

Open problems

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