Transcript Document 7377133

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 ={P1,P2,…,Pv} is finite set of v elements (points) and
• B={B1,B2,…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-(4m-1,2m-1,m-1) design is called a
Hadamard design.
1. Introduction
_____________________________________________________________________________________________________________________________________________
Isomorphisms and Automorphisms
•
Isomorphic designs (D1~D2).
D1
•
•
•
P1
φ
P2
B1
φ
B2
D2
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.
D1
•
•
– can be partitioned into two 2-(v,k,
D2
A reducible quasidouble is called a double.
We denote the double design which is reducible to the
designs D1 and D2 by [D1||D2].
)
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
D1
D1
D2
D2
• The design is completely determined by its incidence matrix. The
incidence matrices of two isomorphic designs are equivalent.
D1
D2
D1
D2
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 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 H5 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
5
1
15!
n 1
Aut ( H n )
 31 524 292 800.
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 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
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 H1||Hi , i=1,2,3,4,5 is greater
than 4,7.1012.
H1
•
Hi
Our purpose is to construct exactly one representative of
each isomorphism class. That is why it is very important
to show which permutations S15 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 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).
7. Classification results
______________________________________________________________________________________________________________________________________________
•
|Aut(D)|
Table 1. Order of the full automorphism group of H1||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
4
1
1
1
1
1566454
Des.
|Aut(D)|
Des.
•
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.