design

design

A

τ

-

(

ν

,

κ

,

λ

)

design

, aka

τ

-design

or

block design

, is an

incidence structure

(

𝒫

,

ℬ

,

ℐ

)

with

•

|

𝒫

|

=

ν

points in all,

•

|

𝒫

B

|

=

κ

points in each block

B

, and such that

•

any set

T

⊆

𝒫

of

|

T

|

=

τ

points occurs as
subset

T

⊆

𝒫

B

in exactly

λ

blocks.

The numbers

τ

,

ν

,

κ

,

l

⁢

a

⁢

m

⁢

b

⁢

d

⁢

a

are called the

parameters

of a design. They are often called

t

,

v

,

k

,

λ

(in mixed Latin and Greek alphabets) by some authors.

Given parameters

τ

,

ν

,

κ

,

l

⁢

a

⁢

m

⁢

b

⁢

d

⁢

a

, there may be several non-isomorphic designs, or no designs at all.

Designs need not be simple (they can have

repeated blocks

), but they usually are (and don’t) in which case

B

can again be used as synonym for

𝒫

B

.

∘

0-designs (

τ

=

0

) are allowed.

∘

1-designs (

τ

=

1

) are known as

tactical configurations

.

∘

2-designs are called

balanced incomplete block designs

or

BIBD

.

∘

3, 4, 5… -designs have all been studied.

Being a

τ

-

(

ν

,

κ

,

λ

)

design implies also being an

ι

-

(

ν

,

κ

,

λ

ι

)

design for every

0

≤

ι

≤

τ

(on the same

ν

points and with the same block size

κ

), with

λ

ι

given by

λ

τ

=

λ

and recursively

λ

ι

=

ν

-

ι

κ

-

ι

⁢

λ

ι

+

1

from which we get the number of blocks as

λ

0

=

ν

!

/

(

ν

-

τ

)

!

κ

!

/

(

κ

-

τ

)

!

=

(

ν

τ

)

/

(

κ

τ

)

Being a 0-design says nothing more than all blocks having the same size. As soon as we have

τ

≥

1

however we also have a 1-design, so the number

λ

1

=

|

ℬ

P

|

of blocks per point

P

is

constant

throughout the

structure

. Note now

λ

0

⁢

κ

=

λ

1

⁢

ν

which is also evident from their

interpretation

.

As an example: designs (simple designs) with

κ

=

2

are

multigraphs

(simple

graphs

), now

∘

τ

=

0

implies no more than that,

∘

τ

=

1

gives

regular graphs

, and

∘

τ

=

2

gives

complete graphs

.

A more elaborate “

lambda calculus

” (pun intended) can be introduced as follows. Let

I

⊆

P

and

O

⊆

P

with

|

I

|

=

ι

and

|

O

|

=

o

. The number of blocks

B

such that all the points of

I

are inside

B

and all the points of

O

are outside

B

is

independent

of the choice of

I

and

O

, only depending on

ι

and

o

, provided

ι

+

o

≤

τ

. Call this number

λ

ι

o

. It

satisfies

a kind of reverse

Pascal triangle

like

recursion

λ

ι

o

=

λ

ι

+

1

o

+

λ

ι

o

+

1

that starts off for

o

=

0

with

λ

ι

0

=

λ

ι

. An important quantity (for designs with

τ

≥

2

) is the

order

λ

1

1

=

λ

1

0

-

λ

2

0

=

λ

1

-

λ

2

.

Finally, the dual of a design can be a design but need not be.

•

A

square design

aka

symmetric design

is one where

τ

=

2

and

|

𝒫

|

=

|

ℬ

|

, now also

|

𝒫

B

|

=

|

ℬ

P

|

. Here the dual is also a
square design.

Note that for

τ

≥

3

no designs exist with

|

𝒫

|

=

|

ℬ

|

other than

trivial

ones (where any

κ

=

ν

-

1

points form a block).

Title

design

Canonical

name

Design

Date of creation

2013-03-22 19:14:09

Last modified on

2013-03-22 19:14:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62K10

Classification

msc 51E30

Classification

msc 51E05

Classification

msc 05B25

Classification

msc 05B07

Classification

msc 05B05

Synonym

block design

Synonym

tau-design

Synonym

τ

-design

Synonym

BIBD

Defines

block

Defines

simple design

Defines

square design

Defines

symmetric design

Defines

tactical configuration

Defines

balanced incomplete block design

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