Partially balanced incomplete block design

PBIBD

A (symmetric) association scheme with $m$ classes on the $v$ symbols $\{1,...,v\}$ satisfies:

Two distinct symbols $x$ and $y$ are termed $i$th associates for exactly one $i\in\{1,...,m\}$; each symbol has exactly $n_i$ $i$th associates; and when two distinct symbols $x$ and $y$ are $i$th associates, the number of other symbols that are $j$th associates of $x$ and also $k$th associates of $y$ is $p^i_jk$, independent of the choice of the $i$th associates $x$ and $y$. The matrices $A_0,...,A_m$ of an $m$-class association scheme are defined as $A_0=I$, and for $1\leq \leq m$, $A_i$ is a $(0,1)$-matrix whose entry $(x,y)$ is $1$ exactly when $x$ and $y$ are $i$th associates.

Let $X$ be a $v$-set with a symmetric $m$-class association scheme defined on it. A partially balanced incomplete block design with $m$ associate classes (or $\mathrm{PBIBD}(m)$) is a block design based on $X$ with $b$ sets (the blocks), each of size $k$, and with each symbol appearing in $r$ blocks. Any two symbols that are $i$th associates appear together in $\lambda_i$ blocks of $\mathrm{PBIBD}(m)$. The numbers $v,b,r,k,\lambda_i$ ($1\leq i\leq m$) are the parameters of $\mathrm{PBIBD}(m)$. The notation $\mathrm{PBIBD}(v,k,\lambda_i)$ is also used. $N$ is used for the $v\times b$ $(0,1)$ incidence matrix of $\mathrm{PBIBD}(m)$.

Let $A_0,...,A_m$ be the matrices of an association scheme corresponding to a $\mathrm{PBIBD}(m)$. Then $NN^T=rI+\sum_i\lambda_iA_i$ and $JN=kJ$. Conversely, if $N$ is a $(0,1)$-matrix which satisfies these conditions and the $A_i$ are the matrices of an association scheme, then $N$ is the incidence matrix of a $\mathrm{PBIBD}(m)$.

It is easily verified that $vr=bk$, that $\sum^m_{i=1}n_i=v-1$, and that $\sum^m_{i=1}\lambda_in_i=r(k-1)$. A $\mathrm{PBIBD}(1)$ is a balanced incomplete block design (a BIBD; cf. Block design); also, a $\mathrm{PBIBD}(2)$ in which $\lambda_1=\lambda_2$ is a BIBD.

There are six types of $\mathrm{PBIBD}(2)$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 $v$-set $X$ into $m$ groups each of size $n$. 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 $NN^T$ are $rk$, $r-\lambda_1$ and $rk-v\lambda_2$, with multiplicities $1$, $m(n-1)$, and $m-1$, respectively. A group-divisible partially balanced incomplete block design is singular if $r=\lambda_1$; semi-regular if $r>\lambda_1$, $rk-v\lambda_2=0$; and regular if $r>\lambda_1$ and $rk>v\lambda_2$.

Let $v=n(n-1)/2$, $n\geq 5$, and arrange the $v$ elements of $X$ in a symmetrical $(n\times n)$-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 $\mathrm{PBIBD}(2)$s are the residual designs of symmetric BIBDs with $\lambda=2$. Triangular schemes and generalized triangular schemes are also known as Johnson schemes.

Let $v=n^2$ and arrange the $v$ symbols in an $n\times n$ array. Superimpose on this array a set of $i-2$ mutually orthogonal Latin squares (see [a1] and also Latin square) of order $n$. 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 $L_i$-type association scheme. If $i=n$, then the scheme is group divisible; if $i=n+1$, then all the symbols are first associates of each other.

Let $\{1,...,v-1\}=D\cup E$. A non-group divisible association scheme defined on $Z_v$ is cyclic if $D=-D$ and if the set of $n_1(n_1-1)$ differences of distinct elements of $D$ has each element of $D$ $p^1_11$ times and each element of $E$ $p^2_11$ times. The first associates of $i$ are $i+D$.

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