A GENERALIZATION OF TRADES

Nasrin Soltankhah

Department of Mathematical Sciences

Alzahra University Tehran, I.R. Iran

Given a set of v treatments V. Let k and t be two positive integers such that t<k<v.

 

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In a (v,k,t) trade both collections of blocks must cover the same set of elements. This set of elements is called the foundation of the trade and is denoted by found(T).

ExampleA (6,4,1) trade of volume 2

   

xy12xy34

xy13xy24

   

x12x34y13y24z14z23

x13x24y14y23z12z34

A (7,3,2) trade of volume 6

1. Hedayat introduced the concept of trade [1] in the 1960s.

2. Hedayat and Li applied the method of trade-off and trades for building BIBDs by repeated blocks (1979-1980).

3. Milici and Quattrocchi introduced the steiner trade named it DMB (1984).

4. Hwang (1986), Mahmoodian and Soltankhah [1992 ] and Asgari and Soltankhah [ 2009] deal with the existence and non-existence of (v,k,t) trades.

Some Known results

 

 

   

   

  

x

x

x

x

 

 

 

 

 

 

minimal

Mimimal (v,k,t) trade has unique structure

 

If found(T)=k+t+1

 

 

There exists (v,k,t) trade of volume m for

 

 

 

Combinatorial trade1. Trade in other block designs2. Trade in Latin squares (Latin trade)3. G-trade in graphs (Decomposition H)

trade

Latin trade

-(v,k,t) Latin trade

-(v,k,t) trade

A Generalization of combinatorial trade

 

1 2 3 4

3 4 2 1

4 3 1 2

2 1

4 3

 

Definition:

 

 

Example:

 

 

Definition

 

 

Example:

x12 x34 y13 y24 z14 z23

x14 x23 y12 y34 z13 z24

x13 x24 y14 y23 z12 z34

     

3-way (7,3,2) trade

xy12

xy34

xy13

xy24

xy14

xy23

     

3-way (6,4,1) trade

123 147 158 248 267 357368456

124 138 157 237 268 467458356

127 135 148 246 238 367457568

     

3-way (8,3,2) trade

Application of Trade

1. Intersection problem2. Defining set

Trade off

BIBD

Balanced incomplete block designsLet v, k, and λ be positive integers such that v > k ≥ 2. A (v, k, λ)-balanced incomplete block design ((v, k, λ)-BIBD) is a pair (X,A) such that the following properties are satisfied:

1. |X| = v,

2. each block contains exactly k points, and

3. every pair of distinct points is contained in exactly λ blocks.

A Steiner triple system of order v, or STS(v), is a (v, 3, 1)-BIBD.

x12 x34 y13 y24 z14 z23

x14 x23 y12 y34 z13 z24

x13 x24Y14y23z12 z34

STS(7):

3-(7,3,2)TRADE

     

x12 x34 y13 y24 z14 z23 xyz

INTERSECTION

x12 x34 y13 y24 z14 z23 xyz

x14 x23 y12 y34 z13 z24 xyz

x13 x24 y14 y23 z12 z34 xyz

3-(7,3,2) trade     

Defining Set

• Given parameters k, t. For which volume

does there exist a µ – way (v, k, t) trade ?

What is the volume spectrum ?

µ = 3 

 

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

of volume m

Construction 1

     

1234

1324

1423

3-way (4,2,1) trade

Example:

x12x34y13y24z14z23

x13x24y14y23z12z34

x14x23y12y34z13z24

3-way (7,3,2) trade

     

Unique structure

 

     

     

Construction 2

Example:

     

1234

1324

1423

3-way (4,2,1) tradeof volume 2

 

     

3-way (8,4,3) trade of volume 12

   

     

     

 

 

 

   

   

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

Construction 2

 

t m Construction

1 2

2 6 1

3 12 2

4 36 1

5 72 2⁞ ⁞ ⁞

Question 1

Does there exist a 3-way (v,k,t) trade of volume less than  

Conjecture:The minimum volume is  

For t=2

For t=3

 

For t=2 and k=3

For t=3 and k=4

Question 2

 

•  

•  

k = t+1

•