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 > 2.

3. Introduction 2-(v,k,λ) design (BIBD);  V  V – finite set of v points  Bb blocksk V  B – finite collection of b blocks : k-element subsets of V  D = (V, B ) VλB  D = (V, B ) – 2-(v,k,λ) design if any 2-subset of V is in λ blocks of B Sets of mutually orthogonal resolutions of BIBDs

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

5. Sets of mutually orthogonal resolutions of BIBDs Introduction  Equal blocks  Equal blocks – incident with the same set of points.  Equal designs  Equal designs – if each block of the first design is equal to a block of the second one.  Equal parallel classes  Equal parallel classes – if each block of the first parallel class is equal to a block of the second one.  2-(v,k,m )m-fold multiple m m  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  True m-fold multiple of 2-(v,k, ) design – if the 2-(v,k, ) designs are equal.

6. Sets of mutually orthogonal resolutions of BIBDs  Isomorphic resolutions  Isomorphic resolutions - exists an automorphism of the design transforming each parallel class of the first resolution into a parallel class of the second one. 2- (qk; k; )(r; qk; r- ) q (qk-1)/(k-1),  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 > 1 (Semakov and Zinoviev) orthogonal  Parallel class, orthogonal to a resolution – intersects each parallel class of the resolution in at most one block. Introduction

7. Sets of mutually orthogonal resolutions of BIBDs  Orthogonal resolutions  Orthogonal resolutions – all classes of the first resolution are orthogonal to the parallel classes of the second one. DRD  Doubly resolvable design (DRD) – has at least two orthogonal resolutions  ROR  ROR – resolution, orthogonal to at least one other resolution. Introduction

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

9. Sets of mutually orthogonal resolutions of BIBDs  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 History

10. Sets of mutually orthogonal resolutions of BIBDs DRD  Start with a DRD.  Block by blockm  Block by block construction of the m resolutions. lexicographically greater and orthogonal  Construction of a resolution R m – lexicographically greater and orthogonal to the resolutions R 1, R 2, …, R m-1.  Isomorphism test new m-MORmaximal  Output a new m-MOR – if it is maximal. m-MORs construction and classification

11. Sets of mutually orthogonal resolutions of BIBDs qvk brDRDsRORs2-MORs3-MORs4-MORsNo 26384020111/1--236 263126030110/11/1-596 263168040110 / ≥485 ≥485 / ≥4851078 28462814111 / 1--101 28494221110 / 11 / 1-278 284125628447 / 170 / 6060 /60524 2105167236555 / 5--891 21052410854662 / 75 / 5-- 2126104422111 / 1--319 2126156633110 / 11 / 1-743 2126208844546 691 / ≥7180 / ≥27≥27 / ≥27- 2168146030555 / 5--618 2168219045550 / 55 / 5-- 22010187638333 / 3--1007 39333612352 / 75 / 5-66 393448163883388 / 495333 / 3341 / 1145 4123244112070319 / 3211 / 21 / 155 416424010110 / 11 / 1-44 m-MORs construction and classification

12. Sets of mutually orthogonal resolutions of BIBDs  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. 1423 2314 4132 3241 m-MORs of multiple designs 1423 2314 4132

13. Sets of mutually orthogonal resolutions of BIBDs  L – latin square of order n: E 1, E 2, E 3 – sets of n elements.   = {(x 1,x 2,x 3 ) : L(x 1,x 2 ) = x 3 }{a,b,c} = {1,2,3}  (a,b,c)-conjugate of L – rows indexed by E a, columns by E b and symbols by E c, L (a,b,c) (x a,x b ) = x c for each (x 1,x 2,x 3 )   E 2 E 1 m-MORs of multiple designs 1423 2314 4132 3241 1243 4312 2134 3421 (1,2,3)-conjugate(2,1,3)-conjugate E1E1 E2E2

14. Sets of mutually orthogonal resolutions of BIBDs  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. m-MORs of multiple designs

15. Sets of mutually orthogonal resolutions of BIBDs  L 1 = (a ij ), S 1 ; L 2 = (b ij ), S 2 ; i,j=1,2,…,n; n – order of latin squares.  Orthogonal latin squares – every element in S 1 x S 2 occurs exactly once among the pairs (a ij, b ij ).  Mutually orthogonal set of latin squares (set of MOLS) – each pair of latin squares in the set is orthogonal. m-MORs of multiple designs

16. Sets of mutually orthogonal resolutions of BIBDs  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. m-MORs of multiple designs

17. Sets of mutually orthogonal resolutions of BIBDs  M – set of m MOLS: E 1, E 2, E 3, …, E m+2 – n-sets.   = {(x 1,x 2, …, x m+2 ) : L(x 1,x 2 ) = x i+2, i=1,2, …, m}  {a 1,a 2,…, a m+2 } = {1,2,…, m+2}  M (a 1,a 2,…, a m+2 ) – contains the Latin squares L i : L i (a 1, a 2 ) = a i+2, i = 1, 2, …,m for each (x 1, x 2, …, x m+2 )  . m-MORs of multiple designs

18. Sets of mutually orthogonal resolutions of BIBDs  v / k = 2  If one block of a parallel class is known, the point set of the second one is known too;  R 1 – B 1, B 2  R 2 – B 1, B 2 ’; B 2, B 1 ’  ?R3   block  ≥ 2 equal blocks 1234 R1R1 11121112 21222122 31323132 41424142 R2R2 11221122 21122112 31423142 41324132 R3R3 11321132 21422142 31123112 41224122 m-MORs of multiple designs with v/k = 2 Example: 4 equal parallel classes of 3 mutually orthogonal resolutions, v = 2k Latin rectangle 1234 2143 3412  B 2 = B 2 ’, B 1 = B 1 ’   block  ≥ 1 equal block  n1n1 n2n2 …nana Partition of parallel classes into n a subcollections, each contains equal parallel classes. min n i = m 2-(6,3,16) 2-(8,4,12) 2-(10,5,32) 2-(12,6,20) 2-(16,8,28)

19. Sets of mutually orthogonal resolutions of BIBDs  Proposition 1:D2-(v,k, )v = 2k  Proposition 1: Let D be a 2-(v,k, ) design and v = 2k. k at least two blocks 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. at least one setk m0moremD at least one maximal m-MOR 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 > m and no maximal i-MORs for i < m. m-MORs of multiple designs with v/k = 2

20. Sets of mutually orthogonal resolutions of BIBDs m-MORs of true m-fold multiple designs with v/k > 2 1234 R1R1 111213111213 212223212223 313233313233 414243414243 R2R2 112233112233 211243211243 314213314213 413223413223 R3R3 113243113243 214233214233 311223311223 412213412213 R4R4 114223114223 213213213213 312243312243 411233411233 Example: true 4-fold multiple, 4 equal parallel classes of 4 mutually orthogonal resolutions, v = 3k M = M (1,2,3,4 ) 12341234 21433412 34124321 43212143   Permutation of resolution classes, numbers of equal classes, resolutions of the m-MOR -> columns, symbols and rows of all the latin squares in M.  A nontrivial automorphism of the design transforms M into one of its conjugates.

21. Sets of mutually orthogonal resolutions of BIBDs 1234 R1R1 121113121113 222123222123 323133323133 424143424143 R2R2 122133122133 221143221143 324113324113 423123423123 R3R3 123143123143 224133224133 321123321123 422113422113 R4R4 124123124123 223113223113 322143322143 421133421133 m-MORs of true m-fold multiple designs with v/k > 2 automorphism  transforming first blocks into second blocks and vice versa 1234 R1R1 111213111213 212223212223 313233313233 414243414243 R2R2 112243112243 211233211233 314223314223 413213413213 R3R3 113223113223 214213214213 311243311243 412233412233 R4R4 114233114233 213243213243 312213312213 411223411223 relation to M (1,3,2,4) - the (1, 3, 2, 4) conjugate of M M = M (1,2,3,4 ) 12341234 21434321 34122143 43213412 1234 R1R1 111213111213 212223212223 313233313233 414243414243 R2R2 112233112233 211243211243 314213314213 413223413223 R3R3 113243113243 214233214233 311223311223 412213412213 R4R4 114223114223 213213213213 312243312243 411233411233 Example: 4 equal parallel classes of 4 mutually orthogonal resolutions, v = 3k M = M (1,2,3,4 ) 12341234 21433412 34124321 43212143  

22. Sets of mutually orthogonal resolutions of BIBDs  Proposition 2: l q-1,m q - 1 MOLSmq = v/k m≥q2-(v,k,m )Dtrue m-fold resolvable 2-(v,k, )dl q-1,m > 0D  Proposition 2: Let l q-1,m be the number of main class inequivalent sets of q - 1 MOLS of side m. Let q = v/k and m≥q. Let the 2-(v,k,m ) design D be a true m-fold multiple of a resolvable 2-(v,k, ) design d. If l q-1,m > 0, then D is doubly resolvable and has at least m-MORs. m-MORs of multiple designs

23. Sets of mutually orthogonal resolutions of BIBDs  Corollary 3:l m main class inequivalent mv/k = 2m ≥ 22- (v,k,m )Dtrue m-fold multipleresolvable dD  Corollary 3: Let l m 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-MORsd m-MORs, no maximal i-MORs for i < m, and if d not doubly is not doubly resolvable, no i-MORs for i > m. m-MORs of multiple designs

24. Sets of mutually orthogonal resolutions of BIBDs vk NrROR sDRDs2-MORs3-MORs4-MORsmsmsm 63201110/≥ s m 511 63241110/≥ s m 6352 716 63281110/≥ s m 72.10 15 63321110/≥ s m 83.10 42 63361110/≥ s m 92.10 96 84158244/≥8≥8/≥858 841824013 /≥ s m 631824 842165016 /≥ s m 733.10 10 8424180344 /≥ s m 829.10 33 8427476370 /≥ s m 919.10 67 10532≥27.10 6 /≥13.10 6 /≥95≥95-- 10540≥5 /≥ s m 595 12625≥1 /≥ s m 512 Lower bounds on the number of m-MORs