Doubles of Hadamard 2-(15,7,3) Designs

1. Doubles of Hadamard 2-(15,7,3) Designs Zlatka Mateva,Department of Mathematics, Technical University, Varna, Bulgaria

2. Doubles of Hadamard 2-(15,7,3) Designs 1. Introduction 2. Hadamard2-(15,7,3) designs 3. Thepresent work 4. Doublesof 2-(15,7,3) 5. Construction algorithm 6. Isomorphism test 7. Classification resulrs.

3. 1. Introduction_____________________________________________________________________________________________________________________________________________2-(v,k,) design • 2-(v,k,) designis a pair(P,B)where • P ={P1,P2,…,Pv} isfiniteset of velements (points) and • B={B1,B2,…Bb}is a finite collectionof k–element subsets of V called blocks, such that • each 2-subset of P occurs in exactly blocks of B. Any point of P occurs in exactly r blocks of B. • If v=bthe design issymmetricandr=ktoo. • A symmetric2-(4m-1,2m-1,m-1) design is called aHadamard design.

4. 1. Introduction_____________________________________________________________________________________________________________________________________________Isomorphisms and Automorphisms • Isomorphicdesigns(D1~D2). • An automorphismis an isomorphismof the design to itself. • The set of all automorphisms of a design D form a group called itsfull group of automorphisms. We denote this group byAut(D). • Each subgroup of this group is agroup of automorphismsof the design. P1 φ P2 D1 D2 - B1 φ B2

5. Introduction_________________________________________________________________________________________________________________________________DoublesIntroduction_________________________________________________________________________________________________________________________________Doubles • Each 2-(v,k,)design determines the existenceof2-(v,k,2)designs. which are called quasidoublesof 2-(v,k,). • Reducible2-(v,k,2)–can be partitioned into two2-(v,k,)designs. • A reducible quasidouble is called adouble. • We denote the double design which is reducible to the designsD1and D2by [D1||D2]. 2-(v,k,) 2-(v,k,2) D1 D2

6. Incidence matrix of a designwithvpoints andbblocks is a(0,1) matrix with v rows and b columns, where the element of the i-th (i=1,2,…,v) row and j-th (j=1,2,…b) column is 1 if the i-th point of P occurs in the j-th block ofBand 0 otherwise. The design is completely determined by its incidence matrix. The incidence matrices of two isomorphic designs are equivalent. 1. Introduction_____________________________________________________________________________________________________________________________________________Incidence matrix of a design D1 D1 D(vxk)dij = 1iff Pi Bj , dij = 0iff Pi Bj D2 D2 D1 D2 D1 D2

7. Let the incidence matrix of the designDbeD. Define standard lexicographic order on the rows and columns of D. We denote byDsortthecolumn-sorted matrix obtained from D by sorting the columns in decreasing order. Define a standard lexicographic order on the matrices considering each matrix as an ordered v-tupleof the v rows. LetDmax=max{(D)sort :Sv}(corresponds to the notation romim introduced from A.Proskurovski about the incidence matrix of a graph). Dmaxis a canonical form of the incidence matrix D. 1. Introduction_____________________________________________________________________________________________________________________________________________A canonical form of the incidence matrix of a design

8. 2. Hadamard2-(15,7,3) Designs_______________________________________________________________________________________________________________ • There exist 5 nonisomorphic 2-(15,7,3) Hadamarddesigns. We denote them by H1, H2, H3, H4 and H5 such that for i=1,2,3,4 Himax> Hi+1max. • The full automorphism groups of H1, H2, H3, H4 and H5areof orders 20160, 576, 96, 168 and 168 respectively. We use automorphisms and point orbits of these groups to decrease the number of constructed isomorphic designs. • The number of isomorphic but distinguished 2-(15,7,3) designs is

9. 3.Thepresent work______________________________________________________________________________________________________ • Subject of the present work are 2-(15,7,6) designs. • R.Mathon, A.Rosa-Handbook of Combinatorial Designs (2007)-there exist at least 57810 nonisomorphic 2-(15,7,6) designs. • This lower bound is improved in two works of S.Topalova and Z.Mateva (2006, 2007) where all 2-(15,7,6) designs with automorphisms of prime odd orders were constructed. Their number was determined to be 92 323 and 12 786 of them were found to be reducible • The results of the present work coincide with those in the previous works and the lower bound is improved to 1 566 454 reducible 2-(15,7,6) designs.

10. 3.The present work______________________________________________________________________________________________________ • Here a classification of all 2-(15,7,6) designs reducible into two Hadamard designs H1 and Hi, Sv is presented. Their block collection is obtained as a union of the block collections of H1 and Hi, i=1,2,3,4,5 and Sv. • The action of Aut(H1) and Aut(Hi) is considered and doubles are not constructed for part of the permutations of Svbecause it is shown that they lead to isomorphic doubles. • The classification of the obtained designs is made by the help of Dmax.

11. 4. Doubles of 2-(15,7,3) Designs______________________________________________________________________________________________________ • Let a 2-(15,7,6) design D is reducible into designs D/ and D//. D=D/||D//. • The number of doubles H1||Hi , i=1,2,3,4,5 is greater than 4,7.1012. • Our purpose is to construct exactly one representative of each isomorphism class. That is why it is very important to show which permutations S15applied to Hi lead to isomorphic designs and skip them. H1 Hi

12. 5. Construction algorithm______________________________________________________________________________________________________________________________________________ • The construction algorithm is based on the next simple proposition. Prorosition 1.Let D/and D//be two 2-(v,k,) designs and let / and //be automorphisms of D/ and D// respectively. Then for all permutations Sv the double designs [D/||D//], [D/||//D//] and [D/||/D//] are isomorphic. • Proof. • /Aut(D/)  [D/||/D//] ~ /-1[D/||/D//]=[D/||D//] and // Aut(D//)  [D/|| //D//]=[D/||D//]. • Corolary 1.If the double design [D/||D//] is already constructed, then all permutations in the set • Aut(D/)..Aut(D//) \{} • can be omited.

13. 5. Construction algorithm______________________________________________________________________________________________________________________________________________ • We implement a back-track search algorithm. • Let the last considered permutation be =(1,2,…,v). The next lexicographically greater than it permutation =(1,2,…,v) is formed in the following way: • Let Nn={1,2,…,n}. We look for the greatest mNv-1{0} such that • if i Nm then i=i and m+1< m+1,  m+1Nv\{1, 2,…, m}. • the number  m+1 is taken from the set Nm// that contains a unique representative of each of the orbits of the permutation group Aut(D//) 1, 2,…, m. • If jNm and j> m+1 then points Pj/ and P/m+1 should not be in one orbit with respect to the stabilizer Aut(D/)1,2,…,,j-1.

14. 6. Isomorphismtest______________________________________________________________________________________________________________________________________________ • The isomorphism test is applied when a new double D is constructed by the help of the canonical Dmax form of its incidence matrix. • The algorithm finding Dmax gives as additional effect the full automorphism group of D.

15. 7. Classificationresults______________________________________________________________________________________________________________________________________________ • The number of nonismorphic reducible 2-(15,7,6) designs from the five cases H1||Hi, iN5 is 1566454. • Their classification with respect to the order of the automorphism group is presented in Table 1. • A double design can have automorphisms of order 2 and automorphisms which preserve the two constituent designs (see for instance V. Fack, S. Topalova, J. Winne, R. Zlatarski, Enimeration of the doubles of the projective plane of order 4, Discrete Mathematics 306 (2006) 2141-2151).

16. 7. Classificationresults______________________________________________________________________________________________________________________________________________ • Table 1.Order of the full automorphism group of H1||Hi, i=1,2,3,4,5. • Only H1 has automorphisms of order 5. Therefore among the constructed designs should be all doubles with automorphisms of order 5. • Their number is the same as the one Mateva and Topalova obtained constructing 2-(15,7,6) designs with automorphisms of order 5.