Block Design - Partially Balanced Designs (PBIBDs)

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.

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