Combinatorial Designs Man versus Machine

Transcription

1 Combinatorial Designs Man versus Machine Anamari Nakić Department of Energy and Power Systems Faculty of Electrical Engineering and Computing University of Zagreb Zagreb / 47

2 The theory of combinatrial designs 1850 Thomas Kirkman Schoolgirls problem Statistical experiments Balanced incomplete block designs Electrical engineering Generalized balanced tournament designs Fiber-optic communication Optical orthogonal codes Software testing Covering arrays Interconnection computer networks Cyclic projective planes Cryptographic communication Orthogonal arrays Theory of error-correcting codes Steiner systems Agricultural engineering Latin squares Algebra Number theory Geometry 2 / 47

3 The error-correcting coding theory Binary Golay code (24, 12, 8) error-correcting code Application NASA deep space missions Radio communications Binary Golay code can be constructed from Steiner system 5-(24, 8, 1) 3 / 47

4 The theory of combinatrial designs 1850 Thomas Kirkman Schoolgirls problem Statistical experiments Balanced incomplete block designs Electrical engineering Generalized balanced tournament designs Fiber-optic communication Optical orthogonal codes Software testing Covering arrays Interconnection computer networks Cyclic projective planes Cryptographic communication Orthogonal arrays Theory of error-correcting codes Steiner systems Agricultural engineering Latin squares Algebra Number theory Geometry 4 / 47

5 Steiner triple systems - existence and enumeration 5 / 47

6 Steiner triple systems - existence and enumeration 6 / 47

7 Steiner triple systems - existence and enumeration 7 / 47

8 Steiner triple systems - existence and enumeration Figure: Steiner system 2-(7, 3, 1) 8 / 47

9 Steiner triple systems - existence and enumeration 9 / 47

10 Steiner triple systems - existence and enumeration / 47

11 Steiner triple systems - existence and enumeration V B / 47

12 Steiner triple systems - existence and enumeration Figure: Steiner system 2-(7, 3, 1) Definition A Steiner system 2-(v, 3, 1) is a finite incidence structure (V, B), where V is a set of v elements called points, B is a set of 3-subsets of V called blocks, every set of 2 points is contained in precisely 1 block 12 / 47

13 The existence of Steiner triple systems Necessary conditions v 1 2 is integer v(v 1) 6 is integer Theorem (Kirkman) A 2-(v, 3, 1) Steiner triple system exists if and only if v 1, 3 (mod 6) 13 / 47

14 Steiner triple systems - existence and enumeration v No Figure: Dave vs HAL in 2001: A Space Odyssey Designs (V, B) and (V, B ) are isomorphic if there exists a bijection α : V V such that αb = B 14 / 47

15 Steiner triple systems - automorphisms Automorphism of (V, B) is a permutation π : V V such that π(b) = B Set of all automorphisms of a design (V, B) is a group 15 / 47

16 Steiner triple systems - automorphisms Automorphism of (V, B) is a permutation π : V V such that π(b) = B Set of all automorphisms of a design (V, B) is a group 16 / 47

17 Steiner triple systems - automorphisms Automorphism of (V, B) is a permutation π : V V such that π(b) = B Set of all automorphisms of a design (V, B) is a group π = ( ) / 47

18 Steiner triple systems - generalization Definition A Steiner system 2-(v, 3, 1) is a finite incidence structure (V, B), where V is a set of v elements called points, B is a set of 3-subsets of V called blocks, every set of 2 points is contained in precisely 1 block 18 / 47

19 Steiner triple systems - generalization Definition A Steiner system 2-(v, k, 1) is a finite incidence structure (V, B), where V is a set of v elements called points, B is a set of k-subsets of V called blocks, every set of 2 points is contained in precisely 1 block 19 / 47

20 Steiner triple systems - generalization Definition A Steiner system t-(v, k, 1) is a finite incidence structure (V, B), where V is a set of v elements called points, B is a set of k-subsets of V called blocks, every set of t points is contained in precisely 1 block (24, 12, 2) binary Golay code Steiner system 5-(24, 8, 1) 20 / 47

21 New directions - Steiner systems in vector spaces Definition A Steiner system t-(v, k, 1) q is a finite incidence structure (V, B), where V is a vector space of dimension v over finite field F q, B is a set of k-dimensional subspaces of V called blocks, every t-dimensional subspace of V is contained in precisely 1 block 21 / 47

22 Steiner systems in vector spaces Peter Cameron s Blog (2014) I think that this is the most important area to attack next Examples 1-(v, k, 1) q designs 2-(13, 3, 1) 2 designs??? M Braun, T Etzion, P Ostergard, A Vardy, A Wassermann Existence of q-analogs of Steiner systems arxiv: (2013) 22 / 47

23 Open problem: the existence of 2-(7, 3, 1) q Question: does 2-(7, 3, 1) q exist? Theorem (Braun, Kiermaier, Nakić, 2015) If a 2-(7, 3, 1) 2 designs exists, it is either rigid or its automorphism group is cyclic of order 2, 3 or 4 M Braun, M Kiermaier, A Nakić On the automorphism group of a binary q-analog of the Fano plane European Journal of Combinatorics, to appear (2015) 23 / 47

24 Combinatorial designs - existence and enumeration Definition A Steiner system t-(v, k, 1) is a finite incidence structure (V, B), where V is a set of v elements called points, B is a set of k-subsets of V called blocks, every set of t points is contained in precisely 1 block 24 / 47

25 Combinatorial designs - existence and enumeration Definition A t-(v, k, λ) design is a finite incidence structure (V, B), where V is a set of v elements called points, B is a set of k-subsets of V called blocks, every set of t points is contained in precisely λ blocks 25 / 47

26 The existence of 2-(v, k, λ) designs Necessary conditions (integrality) λ v 1 is integer k 1 v(v 1) λ is integer k(k 1) Figure: Gary Kasparov vs Deep Blue Theorem (Wilson, 1975) For every fixed k and λ, there is a constant C(k, λ), so that if v > C(k, λ) and integrality conditions are satisfied, there exists a 2-(v, k, λ) design R Wilson An existence theory for pairwise balanced designs, III: Proof of the existence conjectures J Comb Theory A 18(1), (1975) P Keevash The existence of designs arxiv: (2014) 26 / 47

27 Combinatorial designs - existence and enumeration Handbook of Combinatorial Designs, eds C Colbourn, J Dinitz, 2007 t (v, k, λ) No 2 (7, 3, 1) 1 2 (7, 3, 2) 4 2 (7, 3, 3) 10 2 (7, 3, 9) (8, 3, 6) (8, 4, 3) 4 2 (8, 4, 12) > (15, 5, 2) 0 2 (39, 13, 6)? 2 (40, 10, 3)? 2 (57, 15, 10)? 2 (55, 10, 4)? 2 (133, 7, 1)? 2 (175, 30, 6)? 2 (211, 7, 1)? 2 (400, 20, 1)? 2 (421, 21, 1)? t (v, k, λ) No 3 (8, 4, 1) 1 3 (10, 4, 1) 1 3 (14, 4, 1) 4 3 (16, 7, 5)? 3 (17, 7, 7)? 3 (19, 9, 140)? 3 (19, 9, 644)? 3 (20, 5, 8) > 1 4 (17, 5, 1)? 4 (23, 7, 1) > 1 4 (32, 5, 5) > 1 5 (12, 6, 1) > 1 5 (18, 6, 1)? 5 (30, 12, 220) > 1 5 (60, 18, 3060) > 1 6 (19, 7, 1)? 7 (33, 8, 10) > 1 Designs (V, B) and (V, B ) are isomorphic if there exists a bijection α : V V such that αb = B 27 / 47

28 Computational construction of designs 28 / 47

29 Computational methods for construction of designs Exhaustive search The incidence matrix of (V, B) Design Number of matrices 2 (7, 3, 1) (15, 3, 1) (16, 7, 5) (175, 30, 6) / 47

30 Computational methods for construction of designs Nonexhaustive search Automorphism of a design (V, B) is a mapping π : V V such that π(b) = B Set of all automorphisms of a design (V, B) is a group Construction of designs with prescribed automorphism group: The Kramer-Mesner method ES Kramer, DM Mesner t-designs on hypergraphs Discret Math 15, (1976) The method of tactical decomposition Z Janko, T van Trung Construction of a new symmetric block design for (78, 22, 6) with the help of tactical decompositions J Comb Theory A 40, (1985) 30 / 47

31 Open problem: the existence of a 3-(16, 7, 5) design Theorem (Nakić, 2015) If a 3-(16, 7, 5) design exists, then it is either rigid or its full automorphism group is a 2-group Figure: Jeopardy! Pros vs Watson Z Eslami On the possible automorphisms of a 3-(16, 7, 5) design Ars Combinatoria 95, (2010) A Nakić Non-existence of a simple 3-(16, 7, 5) design with an automorphism of order 3 Discrete Mathematics 338(4), (2015) 31 / 47

32 Example: Tactical decomposition of 2-(8, 4, 3) design Incidence matrix B 1 B 2 B 3 B 4 B 5 B 6 B 7 B 8 B 9 B 10 B 11 B 12 B 13 B 14 p p p p p p p p / 47

33 Example: Tactical decomposition of 2-(8, 4, 3) design Incidence matrix B 1 B 2 B 3 B 4 B 5 B 6 B 7 B 8 B 9 B 10 B 11 B 12 B 13 B 14 p p p p p p p p [ρ ij ] = [ ] [κ ij ] = [ ] / 47

34 Tactical decomposition of designs Definition A tactical decomposition of a design (V, B) is any partition V = V 1 V m, B = B 1 B n with the property that there exist nonnegative integers ρ ij and κ ij such that each point of V i lies in precisely ρ ij blocks of B j, and each block of B j contains precisely κ ij points from V i Matrices R = [ρ ij ] and K = [κ ij ] are called tactical decomposition matrices Orbits of V and orbits of B under an action of G form a tactical decomposition of (V, B) Figure: Scrabble Showdown: Quackle vs David Boys 34 / 47

35 The method of tactical decomposition for 2-designs [ρ ij ] = [ ] [κ ij ] = [ ] / 47

36 Tactical decomposition of 2-(v, k, λ) designs (V, B) is a 2-(v, k, λ 2 ) design with tactical decomposition V = V 1 V m, B = B 1 B n Figure: Terminator n j=1 ρ lj κ rj = { λ1 + ( V r 1) λ 2, l = r, V r λ 2, l r [ρ ij ] = ρ l1 ρ ln [κ ij ] = κ r1 κ rn 36 / 47

37 The method of tactical decomposition for 2-(8, 4, 3) { (4 1) 3, l = r, ρ lj κ rj = j=1 4 3, l r [ρ ij ] = [ ] [κ ij ] = [ ] / 47

38 Tactical decomposition of t-(v, k, λ) designs Theorem (Krčadinac, Nakić, Pavčević, 2014) Let D = (V, B) be a t-(v, k, λ t) design with tactical decomposition V = V 1 V m, B = B 1 B n Then the coefficients of R = [ρ ij ] and K = [κ ij ] satisfy n ρ i1 jκ m 1 1 i 1 j κ m 2 i 2 j κms i = sj j=1 m 1 m 2 ω 1 =1 ω 2 =1 m s ω s=1 λ ω1 + +ω s { m1 ω 1 } s ( V i1 1) ω1 1 j=2 { mj ω j } ( V ij ) ωj V Krčadinac, A Nakić, M O Pavčević Equations for coefficients of tactical decomposition matrices for t-designs Des Codes Cryptogr 72(2), (2014) 38 / 47

39 Tactical decomposition of 3-(v, k, λ) designs Theorem (Krčadinac, Nakić, Pavčević, 2011) { n λ1 + ( V r 1) λ 2, l = r, ρ lj κ rj = j=1 V r λ 2, l r n ρ lj κ rj κ sj = j=1 λ ( V l 1) λ 2 + ( V l 1) ( V l 2) λ 3, for l = r = s, = V r V s λ 3, for l r s l, V s λ 2 + ( V r 1) V s λ 3, otherwise Theorem (Nakić, 2015) If a 3-(16, 7, 5) design exists, then it is either rigid or its full automorphism group is a 2-group 39 / 47

40 New directions - designs in vector spaces Definition A t-(v, k, λ) design is a finite incidence structure (V, B), where V is a set of v elements called points, B is a set of k-subsets of V called blocks, every set of t points is contained in precisely λ blocks 40 / 47

41 New directions - designs in vector spaces Definition A t-(v, k, λ t) design over a finite field is a pair (V, B), where V is a v-dimensional vector space over the finite field F q B is a set of k-dimensional subspaces of V called blocks, every t-dimensional subspace of V is contained in precisely λ blocks P Cameron Locally symmetric designs Geom Dedicata 3, 56 76, (1974) P Delsarte Association schemes and t-designs in regular semilattices J Combin Theory Ser A 20(2), (1976) 41 / 47

42 Tactical decomposition of 2-(v, k, λ) q designs Theorem (Nakić, Pavčević, 2014) If (V, B) is a 2-(v, k, λ) q design with tactical decomposition V = V 1 V m, B = B 1 n, then { n λ1 + ( V r 1) λ 2, l = r, ρ lj κ rj = j=1 V r λ 2, l r A Nakić, MO Pavčević Tactical decompositions of designs over finite fields Des Codes Crypto, DOI /s (2014) Figure: The Duel: Timo Boll vs KUKA Robot 42 / 47

43 Tactical decomposition of 3-(v, k, λ) q designs Theorem (De Boeck, Nakić, 2015) If (V, B) is a 3-(v, k, λ) q design with tactical decomposition V = V 1 V m, B = B 1 n, then n ρ lj κ rj κ sj = j=1 λ 1 + Λ lrs λ 2 + ( V r V s Λ lrs 1) λ 3, for l = r = s, = Λ lrs λ 2 + ( V r V s Λ lrs ) λ 3, otherwise De Boeck, Nakić: Necessary conditions for the existence of 3-designs over finite fields with non-trivial automorphism groups Finished! (2015) 43 / 47

44 Tactical decomposition of 3-(v, k, λ) q designs Theorem (De Boeck, Nakić, 2015) If (V, B) is a 3-(v, k, λ) q design with tactical decomposition V = V 1 V m, B = B 1 n, then n ρ lj κ rj κ sj = j=1 λ 1 + Λ lrs λ 2 + ( V r V s Λ lrs 1) λ 3, for l = r = s, = Λ lrs λ 2 + ( V r V s Λ lrs ) λ 3, otherwise De Boeck, Nakić: Necessary conditions for the existence of 3-designs over finite fields with non-trivial automorphism groups Finished! (2015) Open problem Do Steiner systems 3-(v, k, 1) q exist? 44 / 47

45 Some results regarding parameter Λ lrs 1 Λ lrs = Λ lsr 2 V l Λ lrs = V r Λ rls 3 { m Λ lrs = s=1 V r (q + 1) + qv q 2 1, q 1 l = r, V r (q + 1), l r Figure: Bladerunner 45 / 47

46 Some results regarding parameter Λ lrs Lemma The set of 2-subspaces of F v q is a 2 (v, 2, 1)q design (V, L) Group G P ΓL(Fv q ) acts on (V, L) inducing tactical decomposition V = V 1 V m, L = L 1 L ω with tactical decomposition matrices [ρ L ij ] and [κl ij ] Then ω ρ L lj κl rj κl sj λl 1, for l = r = s, j=1 Λ lrs = ω ρ L lj κl rj κl sj, otherwise j=1 46 / 47

47 Thank you for your attention! But they are useless They can only give you answers Pablo Picasso 47 / 47