points in all,
points in each block , and such that
any set of points occurs as subset in exactly blocks.
The numbers are called the parameters of a design. They are often called , , , (in mixed Latin and Greek alphabets) by some authors.
Given parameters , there may be several non-isomorphic designs, or no designs at all.
Designs need not be simple (they can have repeated blocks), but they usually are (and don’t) in which case can again be used as synonym for .
0-designs () are allowed.
1-designs () are known as tactical configurations.
2-designs are called balanced incomplete block designs or BIBD.
3, 4, 5… -designs have all been studied.
Being a - design implies also being an - design for every (on the same points and with the same block size ), with given by and recursively
from which we get the number of blocks as
Being a 0-design says nothing more than all blocks having the same size. As soon as we have however we also have a 1-design, so the number of blocks per point is constant throughout the structure. Note now
which is also evident from their interpretation.
As an example: designs (simple designs) with are multigraphs (simple graphs), now
A more elaborate “lambda calculus” (pun intended) can be introduced as follows. Let and with and . The number of blocks such that all the points of are inside and all the points of are outside is independent of the choice of and , only depending on and , provided . Call this number . It satisfies a kind of reverse Pascal triangle like recursion
that starts off for with . An important quantity (for designs with ) is the order .
Finally, the dual of a design can be a design but need not be.
A square design aka symmetric design is one where and , now also . Here the dual is also a square design.
Note that for no designs exist with other than trivial ones (where any points form a block).
|Date of creation||2013-03-22 19:14:09|
|Last modified on||2013-03-22 19:14:09|
|Last modified by||CWoo (3771)|
|Defines||balanced incomplete block design|