Classification of doubly resolvable designs and orthogonal resolutions

Classification of doubly resolvable designs and orthogonal resolutions

ROR ofROR of 2-(9,3,3) 2-(9,3,3)

1 2 3 4 5 6 7 8 9 10

11

12

13 14

15

16

17

18 19

20

21

22

23

24 25

26

27

28

29 30

31

32

33

34 35

36

1 5 9 4 3 8 7 2 6 10

14

18

13 12

17

16

11

15 19

23

27

22

21

26 25

20

24

28

32 36

31

30

35

34 33

29

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 6 8 4 2 9 7 3 5 10

15

17

13 11

18

16

12

14 19

24

26

22

20

27 25

21

23

28

33 35

31

29

36

34 30

32

Mutually orthogonal resolutionsMutually orthogonal resolutions

1

2

m

1 2 3 4 5 6 7 8 9 10 11 12

1 2 … … … 12.m

Classification of doubly resolvable

designs and orthogonal resolutions

Svetlana Topalova, Stela ZhelezovaInstitute of Mathematics and Informatics, BAS, Bulgaria


Definitions

Design resolutions and equidistant constant weight codes

Definition and one-to-one correspondence between them

Some cryptographic applications

Mutually orthogonal resolutions

Definition – what makes them usable in cryptography

History of the study of resolutions and orthogonal resolutions

Classification of doubly resolvable designs

Classification of the sets of mutually orthogonal resolutions

Correctness of the results, parameter range and bounds for higher parameters

DefinitionsDefinitions

2-(v,k,λ) design (BIBD); VV – finite set of v points BB – finite collection of bb blocksblocks : kk-element subsets of VV D = (V, BD = (V, B )) – 2-(v,k,λ) design if any 2-subset of VV is in λλ

blocks of BB IsomorphicIsomorphic designsdesigns – exists a one-to-one

correspondence between the point and block sets of both designs, which does not change the incidence.

AutomorphismAutomorphism – isomorphism of the design to itself.



Design resolutions and equidistant Design resolutions and equidistant constant weight codesconstant weight codes

ResolvabilityResolvability – at least one resolution.

ResolutionResolution – partition of the blocks into parallel classes - each point is in exactly one block of each parallel class.

Isomorphic resolutionsIsomorphic resolutions - exists an automorphism of the design transforming each parallel class of the first resolution into a parallel class of the second one.


q-ary code CCqq(n,M,d)(n,M,d) of length nn with MM words and minimal

distance dd is a set of MM elements of ZZnnq q (words), any two words

are at Hamming distance at least dd.

any two words are at the same distance dd -> equidistant code

all codewords have the same weight -> constant weight code.

One-to-one correspondence between resolutions of 22--(qk; k;(qk; k;))

designs and the maximal distance (r; qk; r-(r; qk; r- ))qq equidistant codes,

r = (qk-1)/(k-1)(qk-1)/(k-1),, q > 1 (Semakov N.V., Zinoviev V.A., Equidistant q-ary codes with maximal distance and resolvable balanced incomplete block designs, 1968)



11 0 0 1 0 0 1 0 0 1 0 0

11 0 0 0 1 0 0 1 0 0 1 0

11 0 0 0 0 1 0 0 1 0 0 1

0 11 0 1 0 0 0 1 0 0 0 1

0 11 0 0 1 0 0 0 1 1 0 0

0 11 0 0 0 1 1 0 0 1 1 0

0 0 11 1 0 0 0 0 1 0 1 0

0 0 11 0 1 0 1 0 0 0 0 1

0 0 11 0 0 1 0 1 0 1 0 0

11 1 1 1

11 2 2 2

11 3 3 3

22 1 2 3

22 2 3 1

22 3 1 2

33 1 3 2

33 2 1 3

33 3 2 1

Incidence matrix of 2-(9,3,1)2-(9,3,1)

Equidistant ternary constant weight ((4,9,34,9,3))

code


points

V V = 9

blocks bb = 12parallel class words


Some cryptographic applications:

Anonimous (2,k)-threshold schemes from resolvable 2-(v,k,1) designs (D.R.

Stinson, Combinatorial Designs: Constructions and Analysis, 2004);

Resolvable 2-(v,3,1) designs for synchronous multiple access to channels,

(C.J. Colbourn, J.H. Dinitz, D.R. Stinson, Applications of Combinatorial

Designs to Communications, Cryptography, and Networking,1999)

Unconditionally secure commitment schemes (C.Blundo, B.Masucci,

D.R.Stinson, R.Wei, Constructions and bounds for unconditionally secure

commitment schemes, 2000)

Design resolutions and equidistantDesign resolutions and equidistantconstant weight codesconstant weight codes


Parallel class, orthogonalorthogonal to a resolution – intersects each parallel class of the resolution in at most one block (blocks are labelled).

Orthogonal resolutionsOrthogonal resolutions – all classes of the first resolution are orthogonal to the parallel classes of the second one.

Doubly resolvable design (DRDDRD) – has at least two orthogonal resolutions

RORROR – resolution, orthogonal to at least one other resolution.

Mutually orthogonal resolutionsMutually orthogonal resolutionsDefinitionsDefinitions


m-MORm-MOR – set of m mutually orthogonal resolutions.

m-MORsm-MORs – sets of m mutually orthogonal resolutions.

Isomorphic m-MORsIsomorphic m-MORs – if there is an automorphism of the design transforming the first one into the second one.

Maximal m-MORMaximal m-MOR – if no more resolutions can be added to it.

Mutually orthogonal resolutionsMutually orthogonal resolutionsDefinitionsDefinitions


2-MOR of2-MOR of 2-(9,3,3) 2-(9,3,3)

One resolution has r=12 parallel classesOne resolution has r=12 parallel classes

All 2r=24 parallel classesAll 2r=24 parallel classes

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

1 9 15 4 3 27 7 6 33 10 18 35 13 2 32 16 24 29 19 11 30 22 12 20 25 5 14 28 23 36 31 8 26 34 17 21

Mutually orthogonal resolutionsMutually orthogonal resolutions


Colbourn C.J. and Dinitz J.H. (Eds.), The CRC Handbook of Combinatorial Designs, 2007

Kaski P. and Östergärd P., Classification algorithms for codes and designs, 2006.

Kaski P. and Östergärd P., Classification of resolvable balanced incomplete block designs - the unitals on 28 points, 2008.

Kaski P., Östergärd P., Enumeration of 2-(9,3,) designs and their resolutions, 2002

Kaski P., Morales L., Östergärd P., Rosenblueth D., Velarde C., Classification of resolvable 2-(14,7,12) and 3-(14,7,5) designs, 2003

Morales L., Velarde C., A complete classification of (12,4,3)-RBIBDs, 2001

Morales L., Velarde C., Enumeration of resolvable 2-(10,5,16) and 3-(10,5,6) designs, 2005

Östergärd P., Enumeration of 2-(12,3,2) designs, 2000

Design resolutions and equidistant Design resolutions and equidistant

constant weight codes -constant weight codes - HistoryHistory


Abel R.J.R., Lamken E.R., Wang J., A few more Kirkman squares and doubly near resolvable BIBDS with block size 3, 2008

C.J. Colbourn, A. Rosa, Orthogonal resolutions of triple systems, 1995 J.H. Dinitz, D.R. Stinson, Room squares and related designs, 1992 E.R. Lamken, S.A. Vanstone, Designs with mutually orthogonal

resolutions, 1986. E.R. Lamken, The existence of doubly resolvable - ???BIBDs, 1995 E.R. Lamken, Constructions for resolvable and near resolvable (v,k,k-1)-

BIBDs, 1990.

Mutually orthogonal resolutionsMutually orthogonal resolutionsHistoryHistory


Colbourn C.J., Lamken E., Ling A., Mills W., The existence of Kirkman squares - doubly resolvable (v,3,1)-BIBDs, 2002

Cohen M., Colbourn Ch., Ives Lee A., A.Ling, Kirkman triple systems of order 21 with nontrivial automorphism group, 2002

Kaski P., Östergräd P., Topalova S., Zlatarski R., Steiner Triple Systems of Order 19 and 21 with Subsystems of Order 7, 2008

Topalova S., STS(21) with Automorphisms of order 3 with 3 fixed points and 7 fixed blocks, 2004

Mutually orthogonal resolutionsMutually orthogonal resolutionsHistoryHistory


11 0 0 1 0 0 1 0 0 1 0 0

11 0 0 0 1 0 0 1 0 0 1 0

11 0 0 0 0 1 0 0 1 0 0 1

0 11 0 1 0 0 0 1 0 0 0 1

0 11 0 0 1 0 0 0 1 1 0 0

0 11 0 0 0 1 1 0 0 0 1 0

0 0 11 1 0 0 0 0 1 0 1 0

0 0 11 0 1 0 1 0 0 0 0 1

0 0 11 0 0 1 0 1 0 1 0 0

ClassificationClassification of doubly resolvable designsof doubly resolvable designs

11 1 1 1

11 2 2 2

11 3 3 3

22 1 2 3

22 2 3 1

22 3 1 2

33 1 3 2

33 2 1 3

33 3 2 1

Incident matrix

of 2-(9,3,1)2-(9,3,1)

points

VV = 9

blocks bb = 12parallel class

Corresponding ternary equidistant code – (4,9,3)(4,9,3)

words

backtrack search word by word;

E test;

ORE test;


1 0 0 1 0 0 1 0 0 1 0 0

1 0 0 0 1 0 0 1 0 0 1 0

1 0 0 0 0 1 0 0 1 0 0 1

0 1 0 1 0 0 0 1 0 0 0 1

0 1 0 0 1 0 0 0 1 1 0 0

0 1 0 0 0 1 1 0 0 1 1 0

0 0 1 1 0 0 0 0 1 0 1 0

0 0 1 0 1 0 1 0 0 0 0 1

0 0 1 0 0 1 0 1 0 1 0 0

partial solution R R nn

VV

VVnn

resolution RR

empty block RORROR – each of two orthogonal resolutions (different partitions of the blocks with at most one common block)

RORn – partial solution of ROR

words < n n e e - E test after each one

words > n n ee - ORE test followed by E test

words > n n oo – ORE test

n n ee ≈ 10, V/2 ≤ n n oo ≤ 2V/3

ClassificationClassification of doubly resolvable designsof doubly resolvable designs

Classification of of RORs and DRDsClassification of of RORs and DRDs

q v k Nr DRDs* RORs * No

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

3

3

3

4

4

6

8

8

8

8

8

8

8

8

10

10

12

12

12

12

16

16

20

9

9

9

12

16

3

4

4

4

4

4

4

4

4

5

5

6

6

6

6

8

8

10

3

3

3

3

4

n.4

6

9

12

15

18

21

24

27

16

24

10

15

20

25

14

21

18

3

4

5

2

2

1

4

10

31

82

240

650

1803

4763

27121734

73534

545

128284

546

1

1895

5

4

426

149041

203047732

74700

339592

√√√√√*

*

*

*

*

√*

*

*

*

*

√√

1

1

1

4

4

13

16

44

70

5

6

1

1

546

1

5

5

3

3

38

27269

20

1

1

1

1

4

4

13

16

44

70

5

6

1

1

546

1

5

5

3

5

83

76992

70

1

1 n 13

101

278

524

819

-

-

-

-

891

-

319

743

-

-

618

-

1007

66

145

145

55

44


Start with a DRDDRD.

BBlock by blocklock by block construction of the mm resolutions.

Construction of a resolution Rm– lexicographicallylexicographically

greater and orthogonalgreater and orthogonal to the resolutions R1, R2, …, Rm-

1.

Isomorphism test

Output a new new mm-MOR-MOR – if it is maximalmaximal.

Classification of the sets of mutuallyClassification of the sets of mutuallyorthogonal resolutions (orthogonal resolutions (mm-MORs)-MORs)


q v k b r DRDs RORs 2-MORs 3-MORs 4-MORs No

2

2

2

2

2

2

2

2

2

2

2

2

2

2

3

3

4

4

6

6

6

8

8

8

10

10

12

12

12

16

16

20

9

9

12

16

3

3

3

4

4

4

5

5

6

6

6

8

8

10

3

3

3

4

8

12

16

6

9

12

16

24

10

15

20

14

21

18

3

4

2

2

40

60

80

28

42

56

72

108

44

66

88

60

90

76

36

48

44

40

20

30

40

14

21

28

36

54

22

33

44

30

45

38

12

16

11

10

1

1

1

1

1

4

5

6

1

1

546

5

5

3

3

38

20

1

1

1

1

1

1

4

5

6

1

1

546

5

5

3

5

83

70

1

1/1

0/1

0 / ≥485

1 / 1

0 / 1

7 / 17

5 / 5

2 / 7

1 / 1

0 / 1

691 / ≥718

5 / 5

0 / 5

3 / 3

2 / 7

388 / 495

319 / 321

0 / 1

-

1/1

0 / ≥485

-

1 / 1

0 / 60

-

5 / 5

-

1 / 1

0 / ≥27

-

5 / 5

-

5 / 5

333 / 334

1 / 2

1 / 1

-

-

≥485 / ≥485

-

-

60 /60

-

-

-

-

≥27 / ≥27

-

-

-

-

1 / 1

1 / 1

-

236

596

1078

101

278

524

891

-

319

743

-

618

-

1007

66

145

55

44

Classification of the sets of mutuallyClassification of the sets of mutuallyorthogonal resolutions (orthogonal resolutions (mm-MORs)-MORs)


The resolution generating part of our software – for smallest

parameters - the same number of nonisomorphic resolutions as in

R.Mathon, A.Rosa, 2-(v,k,) designs of small order, 2007.

In cases of known number of resolutions, we obtained the number of

RORs in two different ways.

For most design parameters we obtained the results for RORs using

two different algorithms: block by block construction and class by

class construction.

Correctness of the resultsCorrectness of the results for the numbers of RORs and DRDsfor the numbers of RORs and DRDs


An affine design has a unique resolution and can not be doubly

resolvable.

Designs with parameters 2-(6,3,4m) are unique and have a unique

resolution (Kaski P. and Östergärd P., Classification algorithms for codes

and designs, Springer, Berlin, 2006).

Using Theorem 1 we verified the number of DRDs of some multiple

designs with v = 2k in another way.

For some parameters we obtained in parallel the maximum m for which

m-MORs exist.

Correctness of the resultsCorrectness of the results for the numbers of RORs and DRDsfor the numbers of RORs and DRDs


Equal blocksEqual blocks – incident with the same set of points.

Equal designsEqual designs – if each block of the first design is equal to a block of the second one.

2-(v,k,m2-(v,k,m)) design – m-fold multiplem-fold multiple of 2-(v,k,) designs if there is a partition of its blocks into mm subcollections, which form mm 2-(v,k,) designs.

True m-fold multipleTrue m-fold multiple of 2-(v,k,) design – if the 2-(v,k,) designs are equal.

Bounds for higher parameters - Bounds for higher parameters - DefinitionsDefinitions


Proposition 1:Proposition 1: Let DD be a 2-(v,k,2-(v,k,)) design and v = 2kv = 2k.

1) D is doubly resolvable iff it is resolvable and each set of kk points is either incident with no block, or with at least two at least two blocksblocks of the design.

2) If D is doubly resolvable and at least one setat least one set of kk points is in mm blocks, and the rest in 00 or moremore than mm blocks, then DD has at leastat least one maximal m-MORone maximal m-MOR, no i-MORs for i > m and no maximal i-MORs for i < m.

Bounds for higher parametersBounds for higher parameters

mm-MORs -MORs of multiple designsof multiple designs with v/k = 2 with v/k = 2


Proposition 2:Proposition 2: Let llq-1,mq-1,m be the number of main class

inequivalent sets of q - 1 MOLSq - 1 MOLS of side mm. Let q = v/kq = v/k and

mm≥≥qq. Let the 2-(v,k,m2-(v,k,m)) design DD be a true m-foldtrue m-fold multiple of a

resolvable 2-(v,k, resolvable 2-(v,k, )) design dd. If llq-1,mq-1,m > 0 > 0, then DD is doubly

resolvable and has at least m-MORs.

m

r

lm

rmq ,11

mm-MORs -MORs of of true m-fold true m-fold multiple designsmultiple designs with v/k > 2 with v/k > 2



Corollary 3:Corollary 3: Let llmm be the number of main class inequivalentmain class inequivalent

Latin squares of side mm. Let v/k = 2v/k = 2 and m m ≥≥ 2 2. Let the 2-2-

(v,k,m(v,k,m)) design DD be a true m-fold multipletrue m-fold multiple of a resolvableresolvable 2-(v,k,) design dd. Then DD is doubly resolvable and has at least

m-MORsm-MORs, no maximal i-MORs for i < m, and if dd

is not doublynot doubly resolvable, no i-MORs for i > m.

m

r

lm

rm1

mm-MORs -MORs of of true m-fold true m-fold multiple designsmultiple designs with v/k > 2 with v/k > 2



v k Nr ROR s DRDs 2-MORs 3-MORs 4-MORs m sm

6666688888

101012

3333344444556

20242832361518212427324025

11111

82240650

18034763

≥27.106

≥5≥1

111114

13164470

≥27.106

≥5≥1

111114

13164470≥5≥1

0/≥ sm

0/≥ sm

0/≥ sm

0/≥ sm

0/≥ sm

/≥8/≥ sm

/≥ sm

/≥ sm

/≥ sm

/≥13.106

/≥ sm

/≥ sm

0/≥ sm

0/≥ sm

0/≥ sm

0/≥ sm

0/≥ sm

≥8/≥ sm

/≥ sm

/≥ sm

/≥ sm

/≥95/≥ sm

/≥ sm

0/≥ sm

0/≥ sm

0/≥ sm

0/≥ sm

0/≥ sm

/≥8/≥ sm

/≥ sm

/≥ sm

/≥ sm

≥95/≥ sm

/≥ sm

5678956789-55

11352 716

2.1015

3.1042

2.1096

831824

33.1010

29.1033

19.1067

-9512

Lower bounds on the number of mLower bounds on the number of m-MORs-MORs

Classification of doubly resolvable designs and orthogonal resolutions

Documents

Transcript of Classification of doubly resolvable designs and orthogonal resolutions