Steiner system

Definition. An $S(\tau ,\kappa ,\nu )$ Steiner system is a $\tau$-$(\nu ,\kappa ,1)$ design (i.e. $\lambda =1$). The values $\tau ,\kappa ,\nu$ are the parameters of the Steiner system.

Since $\lambda =1$, a Steiner system is a simple design, and therefore we may interpret a block to be a set of points ($B={𝒫}_B$), which we will do from now on.

Given parameters $\tau ,\kappa ,\nu$, there may be several non-isomorphic systems, or no systems at all.

Let $𝒮$ be an $S(\tau ,\kappa ,\nu )$ system with point set $𝒫$ and block set $ℬ$, and choose a point $P\in 𝒫$ (often, the system is so symmetric that it makes no difference which point you choose). The choice uniquely induces an $S(\tau -1,\kappa -1,\nu -1)$ system ${𝒮}_1$ with point set ${𝒫}_1=𝒫\setminus \{P\}$ and block set ${ℬ}_1$ consisting of $B\setminus \{P\}$ for only those $B\in ℬ$ that contained $P$. This works because for any $T_1\subseteq {𝒫}_1$ with $|T_1|=\tau -1$ there was a unique $B\in ℬ$ that contained $T=T_1\cup \{P\}$.

This recurses down all the way to $\tau =1$ (a partition of $\nu -\tau +1$ into blocks of $\kappa -\tau +1$) and finally to $\tau =0$ (one arbitrary block of $\kappa -\tau$). If any of the divisibility conditions (see the entry design (http://planetmath.org/Design) for more detail) on the way there do not hold, there cannot exist a Steiner system with the original parameters either.

For instance, Steiner triple systems $S(2,3,\nu )$ (the first Steiner systems studied, by Kirkman, before Steiner) exist for $\nu =0$ and all $\nu \equiv 1$ or $3(mod6)$, and no other $\nu$.

The reverse construction, turning an $S(\tau ,\kappa ,\nu )$ into an $S(\tau +1,\kappa +1,\nu +1)$, need not be unique and may be impossible. Famously an $S(4,5,11)$ and a $S(5,6,12)$ have the Mathieu groups $M_{11}$ and $M_{12}$ as their automorphism groups, while $M_{22}$, $M_{23}$ and $M_{24}$ are those of an $S(3,6,22)$, $S(4,7,23)$ and $S(5,8,24)$,with connexions to the binary Golay code and the Leech lattice.

Remark. A Steiner system $S(t,k,n)$ can be equivalently characterized as a $k$-uniform hypergraph on $n$ vertices such that every set of $t$ vertices is contained in exactly one edge. Notice that any $S(2,k,n)$ is just a $k$-uniform linear space.