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Classification of doubly resolvable designs and orthogonal resolutions

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## Classification of doubly resolvable designs and orthogonal resolutions

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**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 1 2 … … … 12.m m**Classification of doubly resolvabledesigns and orthogonal**resolutions Svetlana Topalova, Stela Zhelezova Institute 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**Classification of doubly resolvable designs and orthogonal**resolutions Definitions 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 • 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.**Classification of doubly resolvable designs and orthogonal**resolutions Design resolutions and equidistant constant weight codes • Resolvability – at least one resolution. • Resolution – partition of the blocks into parallel classes - each point is in exactly one block of each parallel class. • Isomorphic 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 Design resolutions and equidistant constant weight codes • q-ary code Cq(n,M,d) of length n with M words and minimal distance d is a set of M elements of Znq (words), any two words are at Hamming distance at least d. • any two words are at the same distance d-> equidistant code • all codewords have the same weight -> constant weight code. • One-to-one correspondence between resolutions of 2-(qk; k;) designs and the maximal distance (r; qk; r-)qequidistant codes, r =(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)**Classification of doubly resolvable designs and orthogonal**resolutions Design resolutions and equidistant constant weight codes Equidistant ternary constant weight (4,9,3)code Incidence matrix of2-(9,3,1) points V = 9 parallel class blocksb = 12 words**Classification of doubly resolvable designs and orthogonal**resolutions Design resolutions and equidistant constant weight codes 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)**Classification of doubly resolvable designs and orthogonal**resolutions Mutually orthogonal resolutions Definitions • Parallel class, orthogonal to a resolution – intersects each parallel class of the resolution in at most one block (blocks are labelled). • 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.**Classification of doubly resolvable designs and orthogonal**resolutions Mutually orthogonal resolutions Definitions • 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.**Classification of doubly resolvable designs and orthogonal**resolutions Mutually orthogonal resolutions 2-MOR of 2-(9,3,3) One resolution has r=12 parallel classes All 2r=24 parallel classes**Classification of doubly resolvable designs and orthogonal**resolutions Design resolutions and equidistant constant weight codes -History • Colbourn C.J. and Dinitz J.H. (Eds.), The CRC Handbook ofCombinatorial Designs, 2007 • Kaski P.and Östergärd P., Classification algorithms for codes and designs, 2006. • KaskiP. and Östergärd P., Classification of resolvable balanced incomplete blockdesigns - the unitals on 28 points, 2008. • Kaski P., Östergärd P., Enumeration of 2-(9,3,) designs and theirresolutions, 2002 • Kaski P., Morales L., Östergärd P., Rosenblueth D., Velarde C., Classification of resolvable2-(14,7,12) and 3-(14,7,5) designs, 2003 • Morales L., Velarde C., Acomplete classification of (12,4,3)-RBIBDs, 2001 • Morales L., Velarde C., Enumeration of resolvable2-(10,5,16) and 3-(10,5,6) designs, 2005 • Östergärd P., Enumeration of2-(12,3,2) designs, 2000**Classification of doubly resolvable designs and orthogonal**resolutions Mutually orthogonal resolutions History • Abel R.J.R., Lamken E.R., Wang J., A few more Kirkman squares anddoubly 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.**Classification of doubly resolvable designs and orthogonal**resolutions Mutually orthogonal resolutions History • Colbourn C.J., Lamken E., Ling A., Mills W., Theexistence 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 • KaskiP., ÖstergrädP., TopalovaS., ZlatarskiR., 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**Classification of doubly resolvable designs and orthogonal**resolutions Classificationof doubly resolvable designs Incident matrix of2-(9,3,1) Corresponding ternary equidistant code – (4,9,3) • backtrack search word by word; • E test; • ORE test; points V= 9 parallel class blocksb = 12 words**Classification of doubly resolvable designs and orthogonal**resolutions Classificationof doubly resolvable designs partial solutionR n empty block ROR– each of two orthogonal resolutions (different partitions of the blocks with at most one common block) RORn – partial solution of ROR words < n e - E test after each one words > n e- ORE test followed by E test words > n o– ORE test n e≈ 10, V/2 ≤n o≤ 2V/3 Vn V resolution R**Classification of doubly resolvable designs and orthogonal**resolutions Classification of the sets of mutually orthogonal resolutions (m-MORs) • 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.**Classification of doubly resolvable designs and orthogonal**resolutions Classification of the sets of mutually orthogonal resolutions (m-MORs)**Classification of doubly resolvable designs and orthogonal**resolutions Correctness of the results for the numbers of RORs and DRDs • 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.**Classification of doubly resolvable designs and orthogonal**resolutions Correctness of the results for the numbers of RORs and DRDs • An affine design has a unique resolution andcan 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 multipledesigns with v = 2kinanother way. • For some parameters we obtained in parallel the maximum m for which m-MORs exist.**Classification of doubly resolvable designs and orthogonal**resolutions Bounds for higher parameters - Definitions • 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. • 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.**Classification of doubly resolvable designs and orthogonal**resolutions Bounds for higher parameters • 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. m-MORs of multiple designs with v/k = 2**Classification of doubly resolvable designs and orthogonal**resolutions Bounds for higher parameters • 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. m-MORs of true m-fold multiple designs with v/k > 2**Classification of doubly resolvable designs and orthogonal**resolutions Bounds for higher parameters • 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. m-MORs of true m-fold multiple designs with v/k > 2**Classification of doubly resolvable designs and orthogonal**resolutions Lower bounds on the number of m-MORs