# 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 ($B={\mathcal{P}}_{B}$), 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 $\mathcal{S}$ be an $S(\tau ,\kappa ,\nu )$ system with point set $\mathcal{P}$ and block set $\mathcal{B}$, and choose a point $P\in \mathcal{P}$ (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 ${\mathcal{S}}_{1}$ with point set ${\mathcal{P}}_{1}=\mathcal{P}\setminus \{P\}$ and block set ${\mathcal{B}}_{1}$ consisting of $B\setminus \{P\}$ for only those $B\in \mathcal{B}$ that contained $P$. This works because for any ${T}_{1}\subseteq {\mathcal{P}}_{1}$ with $|{T}_{1}|=\tau -1$ there was a unique $B\in \mathcal{B}$ 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 $3\phantom{\rule{veryverythickmathspace}{0ex}}(mod6)$, 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^{}.

Title | Steiner system |
---|---|

Canonical name | SteinerSystem |

Date of creation | 2013-03-22 13:05:37 |

Last modified on | 2013-03-22 13:05:37 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 9 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 51E10 |

Classification | msc 05C65 |

Related topic | Hypergraph |

Related topic | IncidenceStructures |

Defines | Steiner triple system |