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:
“Let us consider that we are all partially insane. It will explain us to each other; it will unriddle many riddles; it will make clear and simple many things which are involved in haunting and harassing difficulties and obscurities now.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“... all the cares and anxieties, the trials and disappointments of my whole life, are light, when balanced with my sufferings in childhood and youth from the theological dogmas which I sincerely believed, and the gloom connected with everything associated with the name of religion, the church, the parsonage, the graveyard, and the solemn, tolling bell.”
—Elizabeth Cady Stanton (1815–1902)
“I have no designs on society, or nature, or God. I am simply what I am, or I begin to be that. I live in the present. I only remember the past, and anticipate the future. I love to live.”
—Henry David Thoreau (1817–1862)