t-design properties

(To be extended)

This is the area of greatest interest to combinatorialists.

Let $t,v,k,\lambda$ be natural numbers with $t \leq k \leq v$ and $\lambda > 0$. A $t$-$(v,k,\lambda)$ design is a block design with the properties

• there are $v$ points;

• each block contains exactly $k$ points;

• any $t$ points are contained in exactly $\lambda$ blocks.

A $t$-design is a block design which is a $t$-$(v,k,\lambda)$ design for some $v,k,\lambda$.

If our design is a $t$-design for some $t > 1$, we record in the element t_design_properties the attributes $t,v,b,r,k,\lambda$. Here $v$ and $b$ have their usual meaning, $r$ and $k$ are the replication number and block size, and $t$ and $\lambda$ have the properties of the definition. We do not guarantee that the design is not a $t'$-design for some $t' > t$. (On the other hand, a $t$-design is also an $s$-design for any $s < t$.)

We also record some properties of the $t$-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 $t-(v,k,1)$ design for some $t,v,k$. We also record the relevant value of $t$ (which may not be the same as the attribute called t).

steiner_triple_system

True if the design is a $2-(v,3,1)$ 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.