(To be extended)

This is the area of greatest interest to combinatorialists.

Let be natural numbers with and . A - design is a block design with the properties

- there are points;
- each block contains exactly points;
- any points are contained in exactly blocks.

A -design is a block design which is a - design for some .

If our design is a -design for some , we record
in the element `t_design_properties` the
attributes
. Here and have
their usual meaning, and are the replication number
and block size, and and have the properties of
the definition. We do not guarantee that the design is not a
-design for some . (On the other hand, a
-design is also an -design for any .)

We also record some properties of the -design. At present, we have the following:

`square`

True if the numbers of points and blocks are equal.`projective_plane`

True if the design is a projective plane.`affine_plane`

True if the design is an affine plane.`steiner_system`

True if the design is a design for some . We also record the relevant value of (which may not be the same as the attribute called`t`).`steiner_triple_system`

True if the design is a design.

For example, the *t*-design properties of the Fano plane are as follows:

<t_design_properties> <parameters b="7" k="3" lambda="1" r="3" t="2" v="7"> </parameters> <square flag="true"> </square> <projective_plane flag="true"> </projective_plane> <affine_plane flag="false"> </affine_plane> <steiner_system flag="true" t="2"> </steiner_system> <steiner_triple_system flag="true"> </steiner_triple_system> </t_design_properties>

More properties will be included here. Among others, these will include
different specific types of *t*-designs, and intersection triangles
for Steiner systems.