# A generalization of PBIBDs

A PBIBD is an incidence structure together with an underlying symmetric association scheme. One can relax the symmetry requirement, and ask for incidence structures with an underlying (not necessarily symmetric) association scheme. Here are the definitions:

An association scheme with $m$ associate classes is a finite set $X$ together with $m+1$ binary relations $R_i\subseteq X^2$, $0\leq i\leq m$, such that:

• $R_0=\{(x,x)\mid x\in X\}$,
• for each relation $R_i$, the inverse relation $R_i^{-1}=\{(y,x)\mid (x,y)\in R_i\}$ is also an associate class,
• $\bigcup\limits_{i=0}^m R_i=X^2$, and $R_i\cap R_j=\emptyset$ whenever $i\neq j$,
• for any pair $(x,z)\in R_k$, the number of elements $y\in X$ such that $(x,y)\in R_i$ and $(y,z)\in R_j$ depends only on $i,j,k$. This number is denoted by $p_{ij}^k$.

The numbers in the last item above are called the intersection numbers of the association scheme.

A symmetric association scheme is one in which all the binary relations are symmetric.

A partially balanced incomplete block design with $m$ associate classes, denoted $\mathrm{PBIBD}(m)$, is a block design $D=(P,B)$ with $|P|=v$ points, $|B|=b$ blocks, where each block has size $k$, each point lies on $r$ blocks, and such that there is a symmetric association scheme defined on $P$ where, if $(x,y)$ are $j$-th associates then $x$ and $y$ occur together in precisely $\lambda_j$ blocks for $1\leq j\leq m$. The numbers $v,b,r,k,\lambda_j (1\leq j\leq m)$ are the parameters of the $\mathrm{PBIBD}(m)$.

Our definition of a generalized PBIBD is verbatim the definition above, dropping the word symmetric. It follows that if $R_j$ is the inverse relation of $R_i$ then $\lambda_j=\lambda_i$.

It seems that such structures were considered by Nair in a 1964 paper, published in Journal of the American Statistical Association, titled "A new class of designs", but I was not able to find any other reference to such objects. Moreover, the review from MR, by Shrikhande, looks skeptic about the accuracy and usefulness of the objects constructed by Nair.

My questions are:

1. Are you aware of this generalization of PBIBDs?
2. Is it interesting/useful for statisticians or others?