Overview of some Combinatorial Designs

Transcription

1 Bimal Roy Indian Statistical Institute, Kolkata.

2 Outline of the talk 1 Introduction

3 Outline of the talk 1 Introduction

4 Introduction Design theory: Study of combinatorial objects (set systems, arrays, etc.) with certain interesting regular or balance properties. Origin: Recreational mathematics: Kirkman s ½ schoolgirls problem, Euler s officers problem, etc. Design of agricultural experiments.

5 Applications: Design of statistical experiments. Communication (Coding theory, Cryptography, etc.) Algorithm (Derandomization) Tournament scheduling, lotteries, etc.

6 Definition A Design is pair(î,b) such that 1 Î is a set of elements, known as points, 2 B is a collection of subsets, known as blocks, of Î. B may be multiset; when it is not, the design is a simple design. Incidence matrix: A convenient way to represent a design. Definition Let(Î,B) be a design, where Î={Ü ½,...,Ü Ú } andb={ ½,..., }. The incidence matrix of(î,b) is the Ú ¼ ½ matrixm=(ñ, ), Ñ, = ½ if Ü ¼ if Ü /.

7 Example: Let Î={½, ¾,,, } B={½, ¾,,},{¾,, },{,, },{½,, },{½, ¾, } M= ½ ¾ ½ ½ ¼ ¼ ½ ½ ¾ ½ ½ ¼ ¼ ½ ½ ½ ½ ¼ ¼ ¼ ½ ½ ½ ¼ ¼ ¼ ½ ½ ½

8 Outline of the talk 1 Introduction

9 Definition Let à be a subset of positive integers and let λ be a positive integer. A pairwise balanced design (PBD(Ú, Ã,λ) or (Ã,λ)-PBD) of order Ú with block sizes from à is a pair(î,b), where Î is a finite set (the point set) of cardinality Ú andbis a family of subsets (blocks) of Î that satisfy 1 if B, then Ã, 2 every pair of distinct elements of Î occurs in exactly λ blocks ofb. The integer λ is the index of the PBD. The notations PBD(Ú, Ã) and Ã-PBD of order Ú are often used when λ=½.

10 Example: A PBD(½¼,{, }): V={½, ¾,...,½¼}. B={½, ¾,, } {½,,, } {½,,, ½¼} {¾,, } {¾,, } {¾,, ½¼} {,, ½¼} {,, } {,, } {,, } {,, ½¼} {,, }

11 Basic properties: Introduction Suppose à {Ò Z Ò ¾}, and Ú is an integer, then there is exists a PBD(Ú, Ã) only if Ú ½ ¼ Ú(Ú ½) ¼ ÑÓ ( { ½ Ã}) ÑÓ ( { ( ½) Ã}) Generalization of Fisher s inequality: In any nontrivial (Ú,,λ)-PBD Ú. Stanton-Kalbfleisch bound: Let, Ú be integers such that ¾ < Ú. Suppose there is a(ú,{¾,...,ú ½})-PBD in which there exists a block containing exactly points then ½+ ¾ (Ú ). Ú ½ For the previous example Ú= ½¼, = ½¾, and there is block of size. Fisher s inequality ½¼. Stanton-Kalbfleisch bound ½¾.

12 Outline of the talk 1 Introduction

13 Definition Let à and be sets of positive integers and let λ be a positive integer. A group divisible design of index λ and order Ú ((Ã,λ)-GDD) is a triple(î,g,b), where Î is a finite set of cardinality Ú,G is a partition of Î into parts (groups) whose sizes lie in, andbis a family of subsets (blocks) of Î that satisfy 1 if B then Ã, 2 every pair of distinct elements of Î occurs in exactly λ blocks or one group, but not both, 3 G >½.

14 If Ã={ }, then the(ã,λ)-gdd is a(,λ)-gdd. If λ=½, the GDD is a Ã-GDD. Furthermore, a({ }, ½)-GDD is a -GDD. If the GDD has groups ½, ¾,..., Ø, then the list Ì=[ = ½, ¾,..., Ø] is the type of the GDD. If all groups have the same size, then the GDD is uniform. Example: A{, }-GDD of type[,,, ½]: V={½, ¾,...,½¼}.G={½, ¾, },{,, },{,, },{½¼}. B={½,,, ½¼}{¾,,, ½¼} {,,, ½¼}{½,, } {¾,, } {,, } {½,, } {¾,, } {,, }

15 Basic properties Theorem (Hanani) Necessary conditions for existence of a uniform(,λ)-gdd of group size on a points set of cardinality Ú are 1 Ú ¼ ÑÓ, Ú, 2 λ(ú ) ¼ ÑÓ ( ½), 3 λú(ú ) ¼ ÑÓ ( ( ½)). For =, the above conditions are sufficient for existence of a uniform(,λ)-gdd.

16 A Ã-GDD of order Ú with group sizes in PBD(Ú, Ã ). A Ã-GDD of order Ú with group sizes in PBD(Ú+ ½, Ã { + ½ }). Add a new element to each group of GDD. PBD(Î,B) GDD(Î,G,B), whereg={{ú} Ú Î}.

17 Outline of the talk 1 Introduction

18 Definition A transversal design of order or groupsize Ò, blocksize, and index λ, denoted TD λ (, Ò), is a triple(î,g,b), where 1 Î is a set of Ò elements; 2 G is a partition of Î into classes (the groups), each of size Ò; 3 B is a collection of -subsets of Î (the blocks); 4 every unordered pair of elements from Î is contained either in exactly one group or in exactly λ blocks, but not both. When λ=½, one writes simply TD(, Ò). A TD(, Ò) is a uniform -GDD of group size Ò. Closely related to mutually orthogonal latin squares (To be discussed).

19 Outline of the talk 1 Introduction

20 Definition Let Ú,,λ be positive intevgers with Ú> ¾. A (Ú,,λ)-balanced incomplete block design (BIBD) is a pair (Î,B) where Î is a Ú-set andbis a collection of -subsets of Î (blocks) such that any ¾-subset of Î is contained in exactly λ blocks. (Ú,,λ)-BIBD PBD(Ú,{ },λ).

21 Example: BIBD with Ú=, =, Ö=, =,λ=½, a. k. a. Fano plane. V={½, ¾,,,,, } B={½,, } {½,, } {¾,, } {½, ¾, } {¾,, } {,, } {,, } Figure: Fano plane 6

22 Basic properties: Every point occurs in exactly Ö= λ(ú ½) ½ blocks, i.e., λ(ú ½) ¼ ÑÓ ( ½). (1) Number of blocks = ÚÖ. Since we have λ( Ú ¾ )= ( ), it follows that ¾ λú(ú ½) ¼ ÑÓ ( ( ½)). (2) Parameters,, Ú, Ö,λ satisfying the above necessary conditions 1 and 2 are admissible parameters. Example (non-existence upon satisfaction of necessary conditions): Ú= ½¼, =,λ=½.. =Ö.½¼ Ö=.. ½¼. ½¼.¾= = ½, Ö= ¾¼. Impossible.

23 Necessary conditions are sufficient for λ = ½: Theorem (Kirkman) A BIBD with =,λ=½ exists iff Ú ½ or ÑÓ ( ). Theorem (Wilson) For sufficiently large Ú, A BIBD with λ=½ exists iff (Ú ½) ¼ and Ú(Ú ½) ¼ ÑÓ ( ½), ÑÓ ( ( ½)). Fisher s inequality: For a BIBD with blocks and Ú>, we have Ú.

24 Symmetric BIBD Definition A BIBD with = Ú (or, equivalently, with Ö= ) is a symmetric BIBD. Example: Fano plane,(,, ½)-BIBD. From -dimensional Projective space È (, Õ). Theorem For every prime power Õ and positive integer ¾, there is a symmetric( Õ +½ ½, Õ ½, Õ ½ )-BIBD Õ ½ Õ ½ Õ ½ (È ½(, Õ)). Î=FÕ +½. Points are ½-dimensional subspaces of Î, blocks are -dimensional subspaces of Î.

25 Theorem From Affine plane (¾, Õ). For every prime power Õ, there is a symmetric(õ ¾, Õ, ½)-BIBD. Let Î=F Õ F Õ. For,, F Õ, let B={ (, ), F Õ } { (, ) F Õ }, where (, )={(Ü, Ý) Î Ý= Ü+ }, (, )={(, Ý) Î Ý F Õ }. Then(Î,B) is the(õ ¾, Õ, ½)-BIBD.

26 Theorem (Bruck-Ryser-Chowla theorem) Suppose that a symmetric ¾ (Ú,,λ)-design exists. Then if Ú is even, λ is a square; if Ú is odd, the equation Þ ¾ =( λ)ü ¾ +( ½) Ú ½ ¾ ¾ λý has a solution in integers Ü, Ý, Þ, not all zero.

27 Example: Ú= ¾¾, =,λ=¾: λ is not a perfect square. Such a BIBD does not exist. Ú=, =,λ=½: Þ ¾ + Ý ¾ = Ü ¾ has a solution (Ü ½, Ý ½, Þ ½ ) (¼, ¼, ¼); ( Ü ½, Ý ½, Þ ½) is also a solution. The process can not continue ad infinitum unless (Ü, Ý, Þ)=(¼, ¼, ¼). So a symmetric BIBD with these parameters does not exist.

28 Construction from Difference sets: Definition Let be an abelian group of order Ú. A(Ú,,λ)-difference set in is a -subset such that each nonzero occurs exactly λ times in the multiset{ü Ý Ü, Ý } of differences from. That is, the number of ordered pairs(ü, Ý) with Ü, Ý and Ü Ý= is λ when ¼ and is for = ¼. Example: (,, ½) difference set{½, ¾, } in(z,+). (½½,, ¾) difference set{½,,,, } in(z ½½,+).

29 LetD V be a(ú,,λ)-difference set, then(v,b) is a symmetric (Ú,,λ) BIBD, whereb={ú+d Ú V}. Example: (Z ½½,{ +{½,,,, } Z ½½ })is a symmetric(½½,, ¾)- BIBD.

30 Outline of the talk 1 Introduction

31 Definition A parallel class or resolution class in a design is a set of blocks that partition the point set; a resolvable balanced incomplete block design is a BIBD(Ú,,λ) whose blocks can be partitioned into parallel classes. Example: Let Ú be an even integer: Î=Z Ú ½ { }. For Z Ú ½, Π ={{, }} {{ + ÑÓ (Ú ½), ÑÓ (Ú ½)} ½ Ú ¾ }. ¾ (Î, Π ) is a(ú, ¾, ½)-RBIBD, where Π are parallel classes.

32 A(, ¾, ½)-RBIBD Î={¼, ½, ¾,,, } Π ¼ ={{, ¼},{½, },{¾, }} Π ½ ={{, ½},{¾, ¼},{, }} Π ¾ ={{, ¾},{, ½},{, ¼}} Π ={{, },{, ¾},{¼, ½}} Π ={{, },{¼, },{½, ¾}} Properties: Necessary conditions for the existence of an RBIBD(Ú,,λ) are Ú ¼ ÑÓ, and λ(ú ½) ¼ ÑÓ ( ½). If Ú and are both powers of the same prime, then the necessary conditions for the existence of an RBIBD(Ú,,λ) are sufficient.

33 Theorem For Õ a prime power, Ñ ¾, and ½ Ñ ½, there is a (Õ Ñ, Õ,λ)-RBIBD ( -flats of Ñ-dimensional affine geometry over F Õ, i.e., (Ñ, Õ)), where λ= (ÕÑ ½ ½)(Õ Ñ ¾ ½)...(Õ Ñ +½ ½). (Õ ½ ½)(Õ ¾ ½)...(Õ ½) Theorem (Bose) If an RBIBD(Ú,,λ) exists, then Ú+ Ö ½, where is the number of blocks.

34 Outline of the talk 1 Introduction

35 Definition Let Ú,, Ø,λ be integers with Ú Ø ¼, and λ ½. A Ø (Ú,,λ)-design or Ëλ(Ø,, Ú) is a tuple(v,b), wherev is a Ú-element set,bis a family of -subsets ofv such that each V V with V =Ø appears exactly λ times among the substs of B. Definition A Steiner system S(Ø,, Ú) is Ë ½ (Ø,, Ú). A Steiner triple system (STS) STS(Ú) is an S(¾,, Ú). A Steiner quadruple system (SQS) SQS(Ú) is an S(,, Ú).

36 Basic properties: The number of blocks of an Ëλ(Ø,, Ú) is =λ( Ú Ø )/( ). Ø Given an -subseti V, where ¼ Ø, the number of blocks of Ëλ(Ø,, Ú) containingi is λ( Ú Ø )/( ). Ø Every Ë λ (Ø,, Ú) is also an -design for Ø. IfDis a Ø-design(V,B) then for Á V with Á =, ¼ Ø, then(v Á,{ Á B}) is an Ë λ (Ø,, Ú ). This is called derived designd Á ofd.

37 Example: some infinite families of simple Ø-designs (Ø ¾) ¾ (Õ ¾ + Õ+ ½, Õ+ ½, ½), (Õ, prime power) - Projective plane. ¾ (Õ ¾, Õ, ½) (Õ, prime power)- Affine plane. (, ¾, ½) - Hadamard design. (Õ Ò + ½, Õ+ ½, ½), (Õ, prime power) -Spherical design. (¾ + ½,, ½¼) for all odd. (¾ Ò + ½, ¾ Ò ½ ½,(¾ Ò ½ )(¾ Ò ¾ ½)(¾ Ò ½ )), (Ò ). (¾ Ò + ¾, ¾ Ò ½ + ½,(¾ Ò ½ )(¾ Ò ¾ ½)), (Ò ). (¾ Ò +, ¾ Ò ½ + ¾, ¾ Ò (¾ Ò ½)(¾ Ò ¾)(¾ Ò ¾ ½)), (Ò ). No infinte family of simple Ø-design is known for Ø.

38 Theorem (Wilson) Given Ø,, Ú there exists Ëλ(Ø,, Ú), provided λ is sufficiently large. Remark: The design may contain repeated blocks. Theorem (Teirlinck) Ø-designs without repeated blocks exist for all Ø.

39 References Introduction C. j. Colbourn and J. H. Dinitz (Ed.), Handbook of Combinatorial Designs, Discrete Mathematics and its Applications, Second Edition, CRC Press, D. R. Stinson, Combinatorial Designs: Constructions and Analysis, Springer, J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Second Edition, Cambridge University Press, 2007.

40 Thank You