Sets of mutually orthogonal resolutions of BIBDs Svetlana Topalova, Stela Zhelezova Institute of Mathematics and Informatics, BAS, Bulgaria

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.

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

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.

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.

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

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

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

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

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

Sets of mutually orthogonal resolutions of BIBDs qvk brDRDsRORs2-MORs3-MORs4-MORsNo / /11/ / ≥485 ≥485 / ≥ / / 11 / / 170 / 6060 / / / 75 / / / 11 / / ≥7180 / ≥27≥27 / ≥ / / 55 / / / 75 / / / 3341 / / 3211 / 21 / / 11 / 1-44 m-MORs construction and classification

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 m-MORs of multiple designs

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 (1,2,3)-conjugate(2,1,3)-conjugate E1E1 E2E2

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

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

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

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

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 R1R R2R R3R m-MORs of multiple designs with v/k = 2 Example: 4 equal parallel classes of 3 mutually orthogonal resolutions, v = 2k Latin rectangle  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)

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

Sets of mutually orthogonal resolutions of BIBDs m-MORs of true m-fold multiple designs with v/k > R1R R2R R3R R4R Example: true 4-fold multiple, 4 equal parallel classes of 4 mutually orthogonal resolutions, v = 3k M = M (1,2,3,4 )   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.

Sets of mutually orthogonal resolutions of BIBDs 1234 R1R R2R R3R R4R m-MORs of true m-fold multiple designs with v/k > 2 automorphism  transforming first blocks into second blocks and vice versa 1234 R1R R2R R3R R4R relation to M (1,3,2,4) - the (1, 3, 2, 4) conjugate of M M = M (1,2,3,4 ) R1R R2R R3R R4R Example: 4 equal parallel classes of 4 mutually orthogonal resolutions, v = 3k M = M (1,2,3,4 )  

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

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

Sets of mutually orthogonal resolutions of BIBDs vk NrROR sDRDs2-MORs3-MORs4-MORsmsmsm /≥ s m /≥ s m /≥ s m /≥ s m /≥ s m /≥8≥8/≥ /≥ s m /≥ s m /≥ s m /≥ s m ≥ /≥ /≥95≥ ≥5 /≥ s m ≥1 /≥ s m 512 Lower bounds on the number of m-MORs