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. Hadamard 2-(15,7,3) designs 3. The present work 4. Doubles of 2-(15,7,3) 5. Construction algorithm 6. I somorphism test 7. C lassification resulrs.

3. 1. Introduction _____________________________________________________________________________________________________________________________________________ 2-(v,k,  ) design 2-(v,k,  ) designis a pair ( P, B ) where P ={P 1,P 2,…,P v } is finite set of v elements (points) and B ={B 1,B 2,…B b } is a finite collection of 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=b the design is symmetric and r=k too. A symmetric 2-(4m-1,2m-1,m-1) design is called a Hadamard design.

4. Isomorphic designs (D 1 ~D 2 ). An automorphism is an isomorphism of the design to itself. The set of all automorphisms of a design D form a group called its full group of automorphisms. We denote this group by Aut(D). Each subgroup of this group is a group of automorphisms of the design. P1P1 P2P2 D1D1 D2D2 1. Introduction _____________________________________________________________________________________________________________________________________________ Isomorphisms and Automorphisms B1B1 B2B2 φ φ

5. 1.Introduction _________________________________________________________________________________________________________________________________ Doubles Each 2-(v,k,  ) design determines the existence of 2-(v,k,2  ) designs. which are called quasidoubles of 2-(v,k,  ). Reducible 2-(v,k,2  ) – can be partitioned into two 2-(v,k,  ) designs. A reducible quasidouble is called a double. We denote the double design which is reducible to the designs D 1 and D 2 by [D 1 ||D 2 ].  2-(v, k, )  2-(v, k,2 ) D1D1D1D1 D2D2D2D2

6. Incidence matrix of a design with v points and b blocks 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 of B and 0 otherwise. The design is completely determined by its incidence matrix. The incidence matrices of two isomorphic designs are equivalent. D (v x k) d ij = 1 iff P i  B j, d ij = 0 iff P i  B j D1D1 D1D1 1. Introduction _____________________________________________________________________________________________________________________________________________ Incidence matrix of a design D2D2 D2D2 D1D1 D1D1 D2D2 D2D2

7. Let the incidence matrix of the design D be D. Define standard lexicographic order on the rows and columns of D. We denote by D sort the column-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-tuple of the v rows. Let D max =max{(  D) sort :  S v } (corresponds to the notation romim introduced from A.Proskurovski about the incidence matrix of a graph). D max is a canonical form of the incidence matrix D. 1. Introduction _____________________________________________________________________________________________________________________________________________ A canonical form of the incidence matrix of a design

8. 2. Hadamard 2-(15,7,3) Designs _______________________________________________________________________________________________________________ There exist 5 nonisomorphic 2-(15,7,3) Hadamard designs. We denote them by H 1, H 2, H 3, H 4 and H 5 such that for  i=1,2,3,4 H i max > H i+1 max. The full automorphism groups of H 1, H 2, H 3, H 4 and H 5 are of 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. The present 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 H 1 and  H i,  S v is presented. Their block collection is obtained as a union of the block collections of H 1 and  H i, i=1,2,3,4,5 and  S v. The action of Aut(H 1 ) and Aut(H i ) is considered and doubles are not constructed for part of the permutations of S v because it is shown that they lead to isomorphic doubles. The classification of the obtained designs is made by the help of D max.

11. 4. D oubles 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 H 1 ||  H i, i=1,2,3,4,5 is greater than 4,7.10 12. Our purpose is to construct exactly one representative of each isomorphism class. That is why it is very important to show which permutations  S 15 applied to H i lead to isomorphic designs and skip them. H1H1 Hi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  S v 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 N n ={1,2,…,n}. We look for the greatest m  N v-1  {0} such that if i  N m then  i =  i and  m+1 <  m+1,  m+1  N v \{  1,  2,…,  m }. the number  m+1 is taken from the set N m // that contains a unique representative of each of the orbits of the permutation group Aut(D // )  1,  2,…,  m. If j  N m and  j >  m+1 then points P j / and P / m+1 should not be in one orbit with respect to the stabilizer Aut(D / ) 1,2,…,,j-1.

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

15. 7. Classification results ______________________________________________________________________________________________________________________________________________ The number of nonismorphic reducible 2-(15,7,6) designs from the five cases H 1 ||  H i, i  N 5 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. Classification results ______________________________________________________________________________________________________________________________________________ Table 1. Order of the full automorphism group of H 1 ||  H i, i=1,2,3,4,5. Only H 1 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. |Aut(D)| 12346789101214 Des. 1 559 0075 0129901731191586014324 |Aut(D)| 1618212432364248566496 Des. 61154861214363 |Aut(D)| 1201681922883363845762048268820160 All des. Des. 12411411111566454