Doubles of Hadamard 2 15 7 3 Designs

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

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. Isomorphism test 7. Classification resulrs.

1. Introduction _______________________________________________________________________ 2 -(v, k, ) design • 2 -(v, k, ) design is a pair (P, B) where • P ={P 1, P 2, …, Pv} is finite set of v elements (points) and • B={B 1, B 2, …Bb} 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 -(4 m-1, 2 m-1, m-1) design is called a Hadamard design.

1. Introduction _______________________________________________________________________ Isomorphisms and Automorphisms • Isomorphic designs (D 1~D 2). D 1 • • • P 1 φ P 2 B 1 φ B 2 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.

1. Introduction _________________________________________________________________ Doubles • Each 2 -(v, k, ) design determines the existence of 2 -(v, k, 2 ) designs. 2 -(v, k, ) 2 -(v, k, 2 ) which are called quasidoubles of 2 -(v, k, ). • Reducible 2 -(v, k, 2 ) designs. D 1 • • – can be partitioned into two 2 -(v, k, ) D 2 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].

1. Introduction _______________________________________________________________________ Incidence matrix of a design • 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. D(vxk) dij = 1 iff Pi Bj , dij = 0 iff Pi Bj D 1 D 2 • The design is completely determined by its incidence matrix. The incidence matrices of two isomorphic designs are equivalent. D 1 D 2

1. Introduction _______________________________________________________________________ A canonical form of the incidence matrix of a design • Let the incidence matrix of the design D be D. • Define standard lexicographic order on the rows and columns of D. We denote by Dsort 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 Dmax=max{( D)sort : Sv } (corresponds to the notation romim introduced from A. Proskurovski about the incidence matrix of a graph). • Dmax is a canonical form of the incidence matrix D.

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 Himax> Hi+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

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.

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

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 H 1|| Hi , i=1, 2, 3, 4, 5 is greater than 4, 7. 1012. H 1 • Hi 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 Hi lead to isomorphic designs and skip them.

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.

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.

6. Isomorphism test _______________________________________________________________________ • 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.

7. Classification results _______________________________________________________________________ • The number of nonismorphic reducible 2 -(15, 7, 6) designs from the five cases H 1|| Hi, 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).

7. Classification results _______________________________________________________________________ • |Aut(D)| Table 1. Order of the full automorphism group of H 1|| Hi, i=1, 2, 3, 4, 5. 1 2 3 4 6 7 8 9 10 12 14 1 559 007 5 012 990 173 119 15 860 1 4 32 4 |Aut(D)| 16 18 21 24 32 36 42 48 56 64 96 Des. 61 1 5 48 6 1 2 14 3 6 3 120 168 192 288 336 384 576 2048 2688 20160 All des. 1 2 4 1 1 1 1 1566454 Des. |Aut(D)| Des. • 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.