classification of doubly resolvable designs and orthogonal resolutions
DESCRIPTION
Classification of doubly resolvable designs and orthogonal resolutions. Mutually orthogonal resolutions. ROR of 2-(9,3,3). 1 2 3 4 5 6 7 8 9 10 11 12. 1. 2. - PowerPoint PPT PresentationTRANSCRIPT
Classification of doubly resolvable designs and orthogonal resolutions
ROR ofROR of 2-(9,3,3) 2-(9,3,3)
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36
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34 33
29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 6 8 4 2 9 7 3 5 10
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13 11
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14 19
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21
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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
Classification of doubly resolvable designs and orthogonal resolutions
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.
Classification of doubly resolvable designs and orthogonal resolutions
Classification of doubly resolvable designs and orthogonal resolutions
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.
Classification of doubly resolvable designs and orthogonal resolutions
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)
Design resolutions and equidistant Design resolutions and equidistant constant weight codesconstant weight codes
Classification of doubly resolvable designs and orthogonal resolutions
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
Design resolutions and equidistant Design resolutions and equidistant constant weight codesconstant weight codes
points
V V = 9
blocks bb = 12parallel class words
Classification of doubly resolvable designs and orthogonal resolutions
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
Classification of doubly resolvable designs and orthogonal resolutions
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
Classification of doubly resolvable designs and orthogonal resolutions
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
Classification of doubly resolvable designs and orthogonal resolutions
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
Classification of doubly resolvable designs and 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
Classification of doubly resolvable designs and orthogonal resolutions
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
Classification of doubly resolvable designs and orthogonal resolutions
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
Classification of doubly resolvable designs and orthogonal resolutions
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;
Classification of doubly resolvable designs and orthogonal resolutions
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
Classification of doubly resolvable designs and orthogonal resolutions
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)
Classification of doubly resolvable designs and orthogonal resolutions
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)
Classification of doubly resolvable designs and orthogonal resolutions
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
Classification of doubly resolvable designs and orthogonal resolutions
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
Classification of doubly resolvable designs and orthogonal resolutions
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
Classification of doubly resolvable designs and orthogonal resolutions
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
Classification of doubly resolvable designs and orthogonal resolutions
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
Bounds for higher parametersBounds for higher parameters
Classification of doubly resolvable designs and orthogonal resolutions
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
Bounds for higher parametersBounds for higher parameters
Classification of doubly resolvable designs and orthogonal resolutions
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