A theorem on "pseudo symmetric designs"

A theorem on "pseudo symmetric designs"

Citation for published version (APA):

Haemers, W. H. (1977). A theorem on "pseudo symmetric designs". (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7701). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1977 Document Version:

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics

Memorandum 1977-01 January 1977

A theorem on "pseudo symmetric designs"

Technological University Department of Mathematics PO Box 513, Eindhoven The Netherlands by Willem Haemers

•

A theorem on "pseudo symmetric designs" by

Willem Haemers

Let us denote a t-(V,k,A) design having S as the set oLall block

intersectionnumbers (Le. S ={x\x = \B

I n B21, Bl and B2 are distrinct

blocks}.) by S - t -(V.k~A). Here the subject is {p} - 0 - (v,k,v-l)

designs. The only known examples are obtained from symmetric 2-designs by rather simple modifications, cf[W],[BM], where these disigns are called linear square A-linked designs" and Itpseudo (v,k"A) designs", respectively. We'll treat a construction method, which makes it unlikely that these examples are the only possibilities.

As usual a t-(V,k,A) design is not allowed to have repeated blocks, and the parameters must satisfy 0 S t < k < v-t; i f t = 0 then A is the number of blocks, denoted by b. The matrixJ denotes the all-one matrix

of appropriate size, j is the all-one vector of length n, and I is the

""'1l n

idendity matrix of size n.

Theorem 1

A (O,I)-matrix A of size v x b is an incidencematrix of a {p} - 0 - (v,k,b) iff

t

A A

=

p J + (k -p) Ib

The proof of theorem 1 follows directly from the definition.

t

Since repeated blocks are not allowed, we have k ~ p, hence A A is not

singular, which implies b S v. A {p} - 0 -(v,k,v) with < k < v-I is the

same as a symmetric 2-(v,k,p) design, cf[W] prop.A; if k = or k = v-I

the matrix of a {p} - 0 - (v,k,v) is a permutationmatrix P or J-P. For

o

< k < v the number k(v-I)/v is never an integer, this means that

a {p} - 0 - (v,k,v-I) is never a I-design, however it has some nice regularity properties, cf[W] th.8.

The following examples are easely checked with the help of theorem I. Examples

[

' A

J. Let A -

~ ~l

be an incidence matrix of a {A} - 0 - (v ,k, v)

A J-A.·t

ot

then

[A~],

[ A 2

1

],

[~)

and

[~

) are incidence matrices of a

{A} - 0 - (v,k,v-l), {k-A} - 0 - (v,2(k-A), v-I),

-2-2. Let H =

[l

H, ] be a Hadamard matrix of size v, then ~ (~1 + J) represents

a {tv} - 0 - (v, !v, v-I).

For the next theorem a strongly resolveble 2-(v,k,A) design with C

blockclasses is used, cf[H],[HPJ; What we need here is that b+l=v+c holds, and that we may take its incidencematrix Al in a "decent" form,

such that AltA) = (k-p) Ib +

(~J +(p-~)Ic)

® J, for some integers

~

and p.

Theorem 2

Let Al be a "decent" incidence matrix of a strongly resolveble

{p, J.l}- 2 -(v,k,>.) with c classes. Let l2 be the incidence matrix of a

symmetric 2-(c,kl,k'+p_~) design. Let A

2

:=k

2 ® ~/c'

Then A,-

[::J

is the incidencematrix of a {k'+p} - 0 - (b+l,k+k',b).

Proof. We'll use theorem I. The matrix A has b columns and v + c = b+

t t t

rows, and satisfies A A

=

AlAI + A2A2

=

(k-p) Ib + (J.lJ +(p-J.l) Ic)0 J +

+ «k'+ p-~) J + (ll-p)I

c) ® J = (k-p) Ib + (k'+p) J ® J = (k-p) Ib + (k'+p) J

0

It is not too difficult to see that only if AI or J-A

I represents an

affine design (i.e. p=O), the construction can lead to one of the treated examples. It is much more difficult to find a suitable pair of such

designs. In fact we did not succeed, however the following corollary makes it unlikely that the construction never works.

Corollary

I f a regular Hadamard matrix of size 4n2 and a projective plane of order

2 . 4 2 4 2 4 2 4 2

4n - I eX1st, then a {4n -3n -n+l}-O-(16n -4n +1,80 -4n -n+1,16n -4n ) exis ts.

Proof. A regular Hadamard matrix of size 4n2 is equivalent to a symmetric

2-(4n2,n(2n-I).n(n-I» design. The Hadamard matrix also provides us with a

symmetric 2-(4n2-1,2n2-I,n2-1) design (Hadamard 2-design). According to

[SRJ or [H] this design together with an affine plane of order 4n2-1

guarantees the existence of a strongly resolveble

{(n2_1)(4n2_1),(2n2_1)2} -2_«4n2_])2,(4n2_])(2n2_]),n2(4n2-3» with 4n2

-3-Regular Hadamard matrices do often exist, for instance if the size ~s

4, 16, 36, 64 or 100, cf[GS]. However the existance of projectivl'> planeo

of order 4n2-1 is yet unsolved, except for n=l. From the parameters

it follows that only if n=i our construction can lead to one of the treated examples, and unfortunately it does.

References

[BM] Butson A.T.

&

O. Marrero, "Modular Hadamard Matrices and Related

Designs", J. Combinatorial Theory (A) ]5, (]973), 257-269.

[GS] Goethals J.M. & J.J. Seidel, "Strouly regular graphs derived

from combinatorial designs", Can. J. Math. 22,(1970), 597-614.

[H] Harris R., "On Automorphisms and Resolutions of Designs",

Ph. D. Thesis, University of London (1974).

[HP] Hughes D.R.

&

F.C. Piper, liOn Resolutions and Bose's Theorem",

Geometriae Dedicata 5, (1976), 129-]33.

[SR] Shrikhande S.S. & D. Raghavarao, itA Method of Construction of

Imcomplete Block Designs", Sankhya (A) 25, (J963), 399-402.

[W] Woodall D.R., IISquare A-Linked Designs", Proe.London Math.Soc. 20,

(1970), 669-687.