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1. 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

2. Classification of doubly resolvabledesigns and orthogonal resolutions Svetlana Topalova, Stela Zhelezova Institute of Mathematics and Informatics, BAS, Bulgaria

3. 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

4. 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.

5. 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.

6. 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)

7. 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

8. 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)

9. 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.

10. 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.

11. 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

12. 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

13. 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.

14. 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

15. 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

16. 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

17. Classification of of RORs and DRDs

18. 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.

19. Classification of doubly resolvable designs and orthogonal resolutions Classification of the sets of mutually orthogonal resolutions (m-MORs)

20. 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.

21. 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.

22. 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.

23. 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

24. 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

25. 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

26. Classification of doubly resolvable designs and orthogonal resolutions Lower bounds on the number of m-MORs