Definition. An Steiner system is a - design (i.e. ). The values are the parameters of the Steiner system.
Since , a Steiner system is a simple design, and therefore we may interpret a block to be a set of points (), which we will do from now on.
Given parameters , there may be several non-isomorphic systems, or no systems at all.
Let be an system with point set and block set , and choose a point (often, the system is so symmetric that it makes no difference which point you choose). The choice uniquely induces an system with point set and block set consisting of for only those that contained . This works because for any with there was a unique that contained .
This recurses down all the way to (a partition of into blocks of ) and finally to (one arbitrary block of ). 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 (the first Steiner systems studied, by Kirkman, before Steiner) exist for and all or , and no other .
The reverse construction, turning an into an , need not be unique and may be impossible. Famously an and a have the Mathieu groups and as their automorphism groups, while , and are those of an , and , with connexions to the binary Golay code and the Leech lattice.
Remark. A Steiner system can be equivalently characterized as a -uniform hypergraph on vertices such that every set of vertices is contained in exactly one edge. Notice that any is just a -uniform linear space.
|Date of creation||2013-03-22 13:05:37|
|Last modified on||2013-03-22 13:05:37|
|Last modified by||mathcam (2727)|
|Defines||Steiner triple system|