nasrin soltankhah department of mathematical sciencesdepartment of mathematical sciences alzahra...
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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.
•
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
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