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Sets of mutually orthogonal resolutions of BIBDs

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

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Sets of mutually orthogonal resolutions of BIBDs

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  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. Sets of mutually orthogonal resolutions of BIBDs Introduction 2-(v,k,λ) design (BIBD); • V– finite set of vpoints • B – finite collection of bblocks : k-element subsets of V • D = (V, B) – 2-(v,k,λ) design if any 2-subset of V is in λ blocks of B

  4. Sets of mutually orthogonal resolutions of BIBDs Introduction • Isomorphicdesigns – 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.

  5. 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 m2-(v,k,) designs. • 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 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- )qequidistant codes, r =(qk-1)/(k-1),q > 1 (Semakov and Zinoviev) • Parallel class, orthogonal to a resolution – intersects each parallel class of the resolution in at most one block.

  7. 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.

  8. 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.

  9. 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 anddoubly near resolvable BIBDS with block size 3, Discrete Mathematics 308, 2008 • Colbourn C.J. and Dinitz J.H. (Eds.), The CRC Handbook ofCombinatorial 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

  10. Sets of mutually orthogonal resolutions of BIBDs m-MORs construction and classification • Start with a DRD. • Block by blockconstruction of the m resolutions. • Construction of a resolution Rm–lexicographicallygreater and orthogonal to the resolutions R1, R2, …, Rm-1. • Isomorphism test • Output a new m-MOR – if it is maximal.

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

  12. 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.

  13. 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) = xcfor each (x1,x2,x3)   E2 E1 E1 E2 (1,2,3)-conjugate (2,1,3)-conjugate

  14. Sets of mutually orthogonal resolutions of BIBDs m-MORs of multiple designs • Equivalent latin squares – threebijections from the rows, columns and symbols of the first square to the rows, columns andsymbols, 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.

  15. Sets of mutually orthogonal resolutions of BIBDs m-MORs of multiple designs • L1 = (aij), S1; L2 = (bij), S2; i,j=1,2,…,n; n – order of latin squares. • Orthogonal latin squares – every element in S1x S2occurs 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.

  16. 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 oneis 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-(16,8,28) 

  17. Sets of mutually orthogonal resolutions of BIBDs m-MORs of multiple designswith 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 leastone maximal m-MOR, no i-MORs for i > m and no maximal i-MORs for i < m.

  18. Sets of mutually orthogonal resolutions of BIBDs m-MORs of true m-fold multiple designs with v/k > 2 Example: true 4-fold multiple, 4 equal parallel classes of 4 mutually orthogonal resolutions, v = 3k  • 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.

  19. 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 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 lq-1,m > 0, then D is doubly resolvable and has at least m-MORs.

  20. 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 < m, and if d is not doubly resolvable, no i-MORs for i > m.

  21. Sets of mutually orthogonal resolutions of BIBDs Lower bounds on the number of m-MORs

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