Sets of mutually orthogonal resolutions of BIBDs

1
Sets of mutually orthogonal resolutions of BIBDs

• Svetlana Topalova, Stela Zhelezova
• Institute of Mathematics and Informatics, BAS,
  Bulgaria

2
Sets of mutually orthogonal resolutions of BIBDs

• Introduction
• History
• m-MORs construction and classification
• m-MORs of multiple designs with v/k 2
• m-MORs of true m-fold multiple designs with v/k gt
  2.

3
Sets of mutually orthogonal resolutions of BIBDs
Introduction

• 2-(v,k,?) design (BIBD)
• V finite set of v points
• B finite collection of b blocks k-element
  subsets of V
• D (V, B ) 2-(v,k,?) design if any 2-subset
  of V is in ? blocks of B

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Sets of mutually orthogonal resolutions of BIBDs
Introduction

• Isomorphic designs exists a one-to-one
  correspondence between the point and block sets
  of both designs, which does not change the
  incidence.
• Automorphism isomorphism of the design to
  itself.
• Resolvability at least one resolution.
• Resolution partition of the blocks into
  parallel classes - each point is in exactly one
  block of each parallel class.

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Sets of mutually orthogonal resolutions of BIBDs
Introduction

• Equal blocks incident with the same set of
  points.
• Equal designs if each block of the first design
  is equal to a block of the second one.
• Equal parallel classes if each block of the
  first parallel class is equal to a block of the
  second one.
• 2-(v,k,m?) design m-fold multiple of 2-(v,k,?)
  designs if there is a partition of its blocks
  into m subcollections, which form m 2-(v,k,?)
  designs.
• True m-fold multiple of 2-(v,k,?) design if the
  2-(v,k,?) designs are equal.

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Sets of mutually orthogonal resolutions of BIBDs
Introduction

• Isomorphic resolutions - exists an automorphism
  of the design transforming each parallel class of
  the first resolution into a parallel class of the
  second one.
• One-to-one correspondence between resolutions of
  2-(qk k?) designs and the (r qk r- ?)q
  equidistant codes, r ?(qk-1)/(k-1), q gt 1
  (Semakov and Zinoviev)
• Parallel class, orthogonal to a resolution
  intersects each parallel class of the resolution
  in at most one block.

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Sets of mutually orthogonal resolutions of BIBDs
Introduction

• Orthogonal resolutions all classes of the first
  resolution are orthogonal to the parallel classes
  of the second one.
• Doubly resolvable design (DRD) has at least two
  orthogonal resolutions
• ROR resolution, orthogonal to at least one
  other resolution.

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Sets of mutually orthogonal resolutions of BIBDs
Introduction

• m-MOR set of m mutually orthogonal resolutions.
• m-MORs sets of m mutually orthogonal
  resolutions.
• Isomorphic m-MORs if there is an automorphism
  of the design transforming the first one into the
  second one.
• Maximal m-MOR if no more resolutions can be
  added to it.

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Sets of mutually orthogonal resolutions of BIBDs
History

• Mathon R., Rosa A., 2-(v,k,?) designs of small
  order, The CRC Handbook of Combinatorial Designs,
  2007
• Abel R.J.R., Lamken E.R., Wang J., A few more
  Kirkman squares and doubly near resolvable BIBDS
  with block size 3, Discrete Mathematics 308, 2008
• Colbourn C.J. and Dinitz J.H. (Eds.), The CRC
  Handbook of Combinatorial Designs, 2007
• Semakov N.V., Zinoviev V.A., Equidistant q-ary
  codes with maximal distance and resolvable
  balanced incomplete block designs, Problems
  Inform.Transmission vol. 4, 1968
• Topalova S., Zhelezova S., On the classification
  of doubly resolvable designs, Proc. IV Intern.
  Workshop OCRT, Pamporovo, Bulgaria, 2005
• Zhelezova S.,PCIMs in constructing doubly
  resolvable designs, Proc. V Intern. Workshop
  OCRT, White Lagoon, Bulgaria, 2007

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Sets of mutually orthogonal resolutions of BIBDs
m-MORs construction and classification

• Start with a DRD.
• Block by block construction of the m resolutions.
• Construction of a resolution Rm
  lexicographically greater and orthogonal to the
  resolutions R1, R2, , Rm-1.
• Isomorphism test
• Output a new m-MOR if it is maximal.

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Sets of mutually orthogonal resolutions of BIBDs
m-MORs construction and classification
q v k ? b r DRDs RORs 2-MORs 3-MORs 4-MORs No
2 6 3 8 40 20 1 1 1/1 - - 236
2 6 3 12 60 30 1 1 0/1 1/1 - 596
2 6 3 16 80 40 1 1 0 / 485 0 / 485 485 / 485 1078
2 8 4 6 28 14 1 1 1 / 1 - - 101
2 8 4 9 42 21 1 1 0 / 1 1 / 1 - 278
2 8 4 12 56 28 4 4 7 / 17 0 / 60 60 /60 524
2 10 5 16 72 36 5 5 5 / 5 - - 891
2 10 5 24 108 54 6 6 2 / 7 5 / 5 - -
2 12 6 10 44 22 1 1 1 / 1 - - 319
2 12 6 15 66 33 1 1 0 / 1 1 / 1 - 743
2 12 6 20 88 44 546 546 691 / 718 0 / 27 27 / 27 -
2 16 8 14 60 30 5 5 5 / 5 - - 618
2 16 8 21 90 45 5 5 0 / 5 5 / 5 - -
2 20 10 18 76 38 3 3 3 / 3 - - 1007
3 9 3 3 36 12 3 5 2 / 7 5 / 5 - 66
3 9 3 4 48 16 38 83 388 / 495 333 / 334 1 / 1 145
4 12 3 2 44 11 20 70 319 / 321 1 / 2 1 / 1 55
4 16 4 2 40 10 1 1 0 / 1 1 / 1 - 44
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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of multiple designs

• Latin square of order n n x n array, each
  symbol occurs exactly once in each row and
  column.
• m x n latin rectangle m x n array, each symbol
  occurs exactly once in each row and at most once
  in each column.

1 4 2 3
2 3 1 4
4 1 3 2
3 2 4 1
1 4 2 3
2 3 1 4
4 1 3 2
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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of multiple designs

• L latin square of order n E1, E2, E3 sets of
  n elements .
• ? (x1,x2,x3) L(x1,x2) x3 a,b,c
  1,2,3
• (a,b,c)-conjugate of L rows indexed by Ea,
  columns by Eb and symbols by Ec, L(a,b,c)(xa,xb)
  xc for each (x1,x2,x3) ? ?
• E2
  E1

1 4 2 3
2 3 1 4
4 1 3 2
3 2 4 1
1 2 4 3
4 3 1 2
2 1 3 4
3 4 2 1
E1
E2
(1,2,3)-conjugate
(2,1,3)-conjugate
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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of multiple designs

• Equivalent latin squares three bijections from
  the rows, columns and symbols of the first square
  to the rows, columns and symbols, respectively of
  the second one that map first one in the second
  one.
• Main class equivalent latin squares the first
  latin square is equivalent to any conjugate of
  the second one.

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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of multiple designs

• L1 (aij), S1 L2 (bij), S2 i,j1,2,,n n
  order of latin squares.
• Orthogonal latin squares every element in S1 x
  S2 occurs exactly once among the pairs (aij,
  bij).
• Mutually orthogonal set of latin squares (set of
  MOLS) each pair of latin squares in the set is
  orthogonal.

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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of multiple designs

• Equivalent sets of MOLS - three bijections from
  the rows, columns and symbols of the elements of
  the first set to the rows, columns and symbols,
  respectively of the elements of the second one
  that map the first one in the second one.

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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of multiple designs

• M set of m MOLS E1, E2, E3, , Em2 n-sets.
• ? (x1,x2, , xm2) L(x1,x2) xi2, i1,2,
  , m
• a1,a2,, am2 1,2,, m2
• M(a1,a2,, am2) contains the Latin squares Li
  Li(a1, a2) ai2, i 1, 2, ,m for each (x1,
  x2, , xm2) ? ? .

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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of multiple designs with v/k 2

• v / k 2 ? If one block of a parallel class is
  known, the point set of the second one is known
  too
• R1 B1, B2
• R2 B1, B2 B2, B1
• ?R3 ? ? block ? 2 equal blocks

? B2 B2, B1 B1 ? ? block ? 1 equal block
Partition of parallel classes into na
subcollections, each contains equal parallel
classes. min ni m
n1
n2

na
Example 4 equal parallel classes of 3 mutually
orthogonal resolutions, v 2k
2-(6,3,16) 2-(8,4,12) 2-(10,5,32) 2-(12,6,20) 2-(1
6,8,28)
1 2 3 4
R1 1112 2122 3132 4142
R2 1122 2112 3142 4132
R3 1132 2142 3112 4122
Latin rectangle Latin rectangle Latin rectangle Latin rectangle
1 2 3 4
2 1 4 3
3 4 1 2
?
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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of multiple designs with v/k 2

• Proposition 1 Let D be a 2-(v,k,?) design and v
  2k.
• 1) D is doubly resolvable iff it is resolvable
  and each set of k points is either incident with
  no block, or with at least two blocks of the
  design.
• 2) If D is doubly resolvable and at least one set
  of k points is in m blocks, and the rest in 0 or
  more than m blocks, then D has at least one
  maximal m-MOR, no i-MORs for i gt m and no maximal
  i-MORs for i lt m.

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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of true m-fold multiple designs with v/k gt
2
Example true 4-fold multiple, 4 equal parallel
classes of 4 mutually orthogonal resolutions, v
3k
M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4)
1 2 3 4 1 2 3 4
2 1 4 3 3 4 1 2
3 4 1 2 4 3 2 1
4 3 2 1 2 1 4 3
1 2 3 4
R1 111213 212223 313233 414243
R2 112233 211243 314213 413223
R3 113243 214233 311223 412213
R4 114223 213213 312243 411233
?

• Permutation of resolution classes, numbers of
  equal classes, resolutions of the m-MOR -gt
  columns, symbols and rows of all the latin
  squares in M.
• A nontrivial automorphism of the design
  transforms M into one of its conjugates.

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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of true m-fold multiple designs with v/k gt
2
Example 4 equal parallel classes of 4 mutually
orthogonal resolutions, v 3k
1 2 3 4
R1 111213 212223 313233 414243
R2 112233 211243 314213 413223
R3 113243 214233 311223 412213
R4 114223 213213 312243 411233
M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4)
1 2 3 4 1 2 3 4
2 1 4 3 3 4 1 2
3 4 1 2 4 3 2 1
4 3 2 1 2 1 4 3
?
automorphism ? transforming first blocks into
second blocks and vice versa
1 2 3 4
R1 121113 222123 323133 424143
R2 122133 221143 324113 423123
R3 123143 224133 321123 422113
R4 124123 223113 322143 421133
relation to M (1,3,2,4) - the (1, 3, 2, 4)
conjugate of M
1 2 3 4
R1 111213 212223 313233 414243
R2 112243 211233 314223 413213
R3 113223 214213 311243 412233
R4 114233 213243 312213 411223
M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4) M M(1,2,3,4)
1 2 3 4 1 2 3 4
2 1 4 3 4 3 2 1
3 4 1 2 2 1 4 3
4 3 2 1 3 4 1 2
?
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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of multiple designs

• Proposition 2 Let lq-1,m be the number of main
  class inequivalent sets of q - 1 MOLS of side m.
  Let q v/k and mq. Let the 2-(v,k,m?) design D
  be a true m-fold multiple of a resolvable 2-(v,k,
  ?) design d. If lq-1,m gt 0, then D is doubly
• resolvable and has at least
  m-MORs.

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Sets of mutually orthogonal resolutions of BIBDs
m-MORs of multiple designs

• Corollary 3 Let lm be the number of main class
  inequivalent Latin squares of side m. Let v/k 2
  and m 2. Let the 2-(v,k,m?) design D be a true
  m-fold multiple of a resolvable 2-(v,k,?) design
  d. Then D is doubly resolvable and has at least
• m-MORs, no maximal
  i-MORs for i lt m, and if d
• is not doubly resolvable, no i-MORs for i gt
  m.

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Sets of mutually orthogonal resolutions of BIBDs
Lower bounds on the number of m-MORs
v k ? Nr ROR s DRDs 2-MORs 3-MORs 4-MORs m sm
6 3 20 1 1 1 0/ sm 0/ sm 0/ sm 5 11
6 3 24 1 1 1 0/ sm 0/ sm 0/ sm 6 352 716
6 3 28 1 1 1 0/ sm 0/ sm 0/ sm 7 2.1015
6 3 32 1 1 1 0/ sm 0/ sm 0/ sm 8 3.1042
6 3 36 1 1 1 0/ sm 0/ sm 0/ sm 9 2.1096
8 4 15 82 4 4 /8 8 /8 5 8
8 4 18 240 13 13 / sm / sm / sm 6 31824
8 4 21 650 16 16 / sm / sm / sm 7 33.1010
8 4 24 1803 44 44 / sm / sm / sm 8 29.1033
8 4 27 4763 70 70 / sm / sm / sm 9 19.1067
10 5 32 27.106 27.106 /13.106 /95 95 - -
10 5 40 5 5 5 / sm / sm / sm 5 95
12 6 25 1 1 1 / sm / sm / sm 5 12