Steiner system

Definition. An $S(\\tau,\\kappa,\u)$ Steiner system is a $\\tau$ - $(\u,\\kappa,1)$ design (i.e. $\\lambda=1$ ). The values $\\tau,\\kappa,\u$ 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 $\\tau,\\kappa,\u$ , there may be several non-isomorphic systems, or no systems at all.

Let $\\mathcal{S}$ be an $S(\\tau,\\kappa,\u)$ 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 $S(\\tau-1,\\kappa-1,\u-1)$ 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 $\u-\\tau+1$ 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 $S(2,3,\u)$ (the first Steiner systems studied, by Kirkman, before Steiner) exist for $\u=0$ and all $\u\\equiv 1$ or $3\\pmod{6}$ , and no other $\u$ .

The reverse construction, turning an $S(\\tau,\\kappa,\u)$ into an $S(\\tau+1,\\kappa+1,\u+1)$ , 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.

单词	SteinerSystem
释义	Steiner system Definition. An $S(\\tau,\\kappa,\u)$ Steiner system is a $\\tau$ - $(\u,\\kappa,1)$ design (i.e. $\\lambda=1$ ). The values $\\tau,\\kappa,\u$ 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 $\\tau,\\kappa,\u$ , there may be several non-isomorphic systems, or no systems at all. Let $\\mathcal{S}$ be an $S(\\tau,\\kappa,\u)$ 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 $S(\\tau-1,\\kappa-1,\u-1)$ 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 $\u-\\tau+1$ 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 $S(2,3,\u)$ (the first Steiner systems studied, by Kirkman, before Steiner) exist for $\u=0$ and all $\u\\equiv 1$ or $3\\pmod{6}$ , and no other $\u$ . The reverse construction, turning an $S(\\tau,\\kappa,\u)$ into an $S(\\tau+1,\\kappa+1,\u+1)$ , 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.
随便看	HartmanGrobmanTheorem1 HartogsExtensionTheorem Hartogs extension theorem HartogsFunctions Hartogs functions HartogsNumber Hartogs number HartogssTheoremOnSeparateAnalyticity HartogsTheorem HartogsTriangle Hartogs triangle Hartogs’s theorem on separate analyticity Hartogs’ theorem HasARank has a rank Hashing hashing Haskell Haskell HasseDiagram Hasse diagram HasseMinkowskiTheorem Hasse-Minkowski theorem HassePrinciple Hasse principle