Partially Balanced Designs (PBIBDs)
An n-class association scheme consists of a set X of size v together with a partition S of X × X into n + 1 binary relations, R0, R1, ..., Rn. A pair of elements in relation Ri are said to be ith–associates. Each element of X has ni ith associates. Furthermore:
- and is called the Identity relation.
- Defining, if R in S, then R* in S
- If, the number of such that and is a constant depending on i, j, k but not on the particular choice of x and y.
An association scheme is commutative if for all i, j and k. Most authors assume this property.
A partially balanced incomplete block design with n associate classes (PBIBD(n)) is a block design based on a v-set X with b blocks each of size k and with each element appearing in r blocks, such that there is an association scheme with n classes defined on X where, if elements x and y are ith associates, 1 ≤ i ≤ n, then they are together in precisely λi blocks.
A PBIBD(n) determines an association scheme but the converse is false.
Read more about this topic: Symmetric Design
Famous quotes containing the words partially, balanced and/or designs:
“There was an Old Man who supposed,
That the street door was partially closed;”
—Edward Lear (1812–1888)
“With a balanced combination of the two principal energies from mother and father, a girl can both be in touch with her womanly strengths and be a powerful force in the world—strong and nurturing, decisive and caring, goal- oriented and aware of the needs of others. She has the courage to voice what she thinks and feels and the strength to follow her destiny.”
—Jeanne Elium (20th century)
“Our remedies oft in ourselves do lie,
Which we ascribe to heaven. The fated sky
Gives us free scope, only doth backward pull
Our slow designs when we ourselves are dull.”
—William Shakespeare (1564–1616)