Partially balanced incomplete block design

PBIBD

A (symmetric) association scheme with classes on the symbols satisfies:

Two distinct symbols and are termed th associates for exactly one ;

each symbol has exactly th associates; and

when two distinct symbols and are th associates, the number of other symbols that are th associates of and also th associates of is , independent of the choice of the th associates and . The matrices of an -class association scheme are defined as , and for , is a -matrix whose entry is exactly when and are th associates.

Let be a -set with a symmetric -class association scheme defined on it. A partially balanced incomplete block design with associate classes (or ) is a block design based on with sets (the blocks), each of size , and with each symbol appearing in blocks. Any two symbols that are th associates appear together in blocks of . The numbers () are the parameters of . The notation is also used. is used for the incidence matrix of .

Let be the matrices of an association scheme corresponding to a . Then and . Conversely, if is a -matrix which satisfies these conditions and the are the matrices of an association scheme, then is the incidence matrix of a .

It is easily verified that , that , and that . A is a balanced incomplete block design (a BIBD; cf. Block design); also, a in which is a BIBD.

There are six types of s, [a3], based on the underlying types of association schemes:

1) group divisible;

2) triangular;

3) Latin-square-type;

4) cyclic;

5) partial-geometry-type; and

6) miscellaneous.

Partition the -set into groups each of size . In a group-divisible association scheme the first associates are the symbols in the same group and the second associates are all the other symbols. The eigenvalues of are , and , with multiplicities , , and , respectively. A group-divisible partially balanced incomplete block design is singular if ; semi-regular if , ; and regular if and .

Let , , and arrange the elements of in a symmetrical -array with the diagonal entries blank. In the triangular association scheme, the first associates of a symbol are those in the same row or column of the array; all other symbols are second associates. The duals of triangular s are the residual designs of symmetric BIBDs with . Triangular schemes and generalized triangular schemes are also known as Johnson schemes.

Let and arrange the symbols in an array. Superimpose on this array a set of mutually orthogonal Latin squares (see [a1] and also Latin square) of order . Let the first associates of any symbol be those in the same row or column of the array or be associated with the same symbols in one of the Latin squares. This is an -type association scheme. If , then the scheme is group divisible; if , then all the symbols are first associates of each other.

Let . A non-group divisible association scheme defined on is cyclic if and if the set of differences of distinct elements of has each element of times and each element of times. The first associates of are .

In a partial-geometry-type association scheme, two symbols are first associates if they are incident with a line of the geometry and second associates if they are not incident with a line of the geometry.

See [a2], [a4], [a5] for further information.

References

[a1]	R.J.R. Abel, A.E. Brouwer, C.J. Colbourn, J.H. Dinitz, "Mutually orthogonal latin squares" C.J. Colbourn (ed.) J.H. Dinitz (ed.) , CRC Handbook of Combinatorial Designs , CRC (1996) pp. 111–141
[a2]	R.A. Bailey, "Partially balanced designs" N.L. Johnson (ed.) S. Kotz (ed.) C. Read (ed.) , Encycl. Stat. Sci. , 6 , Wiley (1985) pp. 593–610
[a3]	W.H. Clatworthy, "Tables of two-associate-class partially balanced designs" , Applied Math. Ser. , 63 , Nat. Bureau of Standards (US) (1973)
[a4]	D. Raghavarao, "Constructions and combinatorial problems in design of experiments" , Wiley (1971)
[a5]	D.J. Street, A.P. Street, "Partially balanced incomplete block designs" C.J. Colbourn (ed.) J.H. Dinitz (ed.) , CRC Handbook of Combinatorial Designs , CRC (1996) pp. 419–423