What is a Block Design?

Block designs are viewed in different ways by combinatorialists and statisticians. To a statistician, a block design is a set of ``plots'' or ``experimental units'' which carries a partition into ``blocks'', and a function from this set to the set of ``treatments''. A combinatorialist regards the set of treatments as basic (and usually calls them ``points''), and identifies each block with the multiset of treatments occurring on plots in that block; thus, a block design is a set of points together with a multiset of multisets of points.

A multiset is essentially the same thing as a sorted list which may contain repeated items. In this documentation, we represent a multiset as a list in square brackets $[\;]$. (The XML representation is a bit more complicated, as the above example shows.)

For the purpose of this specification, we have chosen to use the representation as a multiset of multisets.

Here is a small example. Suppose that we have six plots, numbered $1, 2, 3, 4, 5, 6$, with blocks $[1, 2, 3]$ and $[4, 5, 6]$. Suppose that treatment $A$ is applied to plots $1, 2, 4, 5$, and treatment $B$ to plots $3$ and $6$. Then we represent the block design as having point set $[A, B]$ and blocks $[[A, A, B], [A, A, B]]$. (Since the lists are sorted, we would represent the design in same way even if, say, treatment $B$ was applied to plots $1$ and $5$.) The names of the plots have disappeared, but the plots can be recovered as incident point-block pairs or ``flags''. We always represent block designs in this way.

In this example, blocks have the awkward property that they are multisets (rather than sets) of points. While this does occur in practice, we have decided to exclude such designs for the time being, for various reasons. A block design is called binary if no treatment occurs more than once in a block, that is, if the blocks are represented by sets (rather than general multisets) of points. All block designs in this document will be binary.

Here is an example of a binary design. It is the Fano plane from the Introduction, viewed in a slightly different way. There are 21 plots, partitioned into seven blocks of three; there are seven treatments, numbered from 0 to 6, as shown in the following table (whose columns represent the blocks):

0	0	0	1	1	2	2
1	3	5	3	4	3	4
2	4	6	5	6	6	5

Further details can be found in the items on block designs in the http://designtheory.org/library/encycEncyclopaedia of Design Theory, or in our survey paper [1].