View source for Steiner system

A pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s0876701.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s0876702.png" /> is a finite set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s0876703.png" /> elements and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s0876704.png" /> is a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s0876705.png" />-subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s0876706.png" /> (called blocks) such that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s0876707.png" />-subset from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s0876708.png" /> is contained in exactly one block of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s0876709.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767010.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767011.png" /> is called the order of the Steiner system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767012.png" />. A Steiner system is a particular case of a [[block design]] and of a [[tactical configuration]]. A Steiner system for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767013.png" /> is a [[balanced incomplete block design]] (BIBD), and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767016.png" /> it is a finite [[projective plane]]. A necessary condition for the existence of a Steiner system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767017.png" /> is that the number

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767018.png" /></td> </tr></table>

is an integer for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767019.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767020.png" />. The sufficiency of this condition was proved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767024.png" /> (see [[#References|[3]]], [[#References|[4]]]).

In 1844 W. Woolhouse stated the existence problem for Steiner systems, and P. Kirkman solved it in 1847 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767025.png" /> (Steiner triple systems). In 1853 J. Steiner examined the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767026.png" />.

Problems usually considered for Steiner systems are: 1) the determination of the maximum number of mutually non-isomorphic Steiner systems of a given order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767027.png" />; 2) the existence of Steiner systems with a given group of automorphisms; 3) the imbedding of partial Steiner systems (not containing some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767028.png" />-subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767029.png" />) in a finite Steiner system; 4) the existence of resolvable Steiner systems (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767030.png" /> representable as a union of partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767031.png" />); 5) the maximal packing (minimal covering) of a complete set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767032.png" />-subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767033.png" /> by disjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767034.png" /> (using Steiner systems).

The majority of results on Steiner systems are obtained for small values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767036.png" /> (see [[#References|[2]]]–[[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Steiner,   "Combinatorische Aufgaben"  ''J. Reine Angew. Math.'' , '''45'''  (1853)  pp. 181–182</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Hall,   "Combinatorial theory" , Wiley  (1986)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C.C. Lindner,   A. Rosa,   "Steiner quadruple systems. A survey"  ''Discrete Math.'' , '''22'''  (1978)  pp. 147–181</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Hanani,   "Balanced incomplete block designs and related designs"  ''Discrete Math.'' , '''11'''  (1975)  pp. 255–369</TD></TR></table>



====Comments====
The name  "Steiner system"  is somewhat inappropriate, since the systems studied by Steiner in [[#References|[1]]] are not Steiner systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767037.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767038.png" />.

There is a rapidly growing literature on Steiner systems and related types of incidence structures, comprising thousands of papers. In what follows, only very few outstanding original papers are quoted; the reader is instead referred to the textbooks [[#References|[2]]], [[#References|[a1]]]–[[#References|[a2]]], the collections of papers [[#References|[a3]]]–[[#References|[a5]]] and the surveys [[#References|[a6]]]–[[#References|[a8]]] for a detailed discussion and complete references. Regarding the existence of small systems, the reader should consult the tables [[#References|[a9]]]–[[#References|[a10]]].

The main problems usually studied are as follows.

==Existence.==
The only pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767039.png" /> for which the existence problem is completely settled are still those listed in the original article above. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767040.png" /> a very strong asymptotic existence result is due to R.M. Wilson [[#References|[a11]]]–[[#References|[a13]]]: For any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767041.png" />, the necessary conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767042.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767044.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767045.png" /> are sufficient for all but finitely many values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767046.png" />. Unfortunately, the bounds on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767047.png" /> extractable from Wilson's work are exceedingly large. Progress has been made for some small values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767048.png" />; e.g., for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767049.png" /> there are only 89 values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767050.png" /> left for which the existence of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767051.png" /> is undecided, the largest of which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767052.png" />. Similar results hold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767053.png" />. Considerably less is known for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767054.png" />. The outstanding problem of the existence of non-trivial Steiner systems with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767055.png" /> is still open. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767056.png" />, existence is known only for the seven pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767062.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767063.png" />.

==Enumeration.==
Once existence is settled, one is interested in the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767064.png" /> of isomorphism classes of Steiner systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767065.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767066.png" /> grows very rapidly with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767067.png" /> (for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767068.png" />); e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767070.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767071.png" />. Wilson [[#References|[a11]]]–[[#References|[a13]]] proved <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767072.png" /> for all admissible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767073.png" /> which are sufficiently large (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767074.png" /> is a constant depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767075.png" />). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767076.png" />, one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767077.png" /></td> </tr></table>

The only known similar result for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767078.png" /> concerns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767079.png" />. For some small parameter triples, a complete enumeration is possible. E.g., the projective and affine planes are unique up to order 8, and so are the Witt designs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767081.png" /> [[#References|[a14]]].

==Automorphism groups.==
Much effort is spent in constructing Steiner systems with  "nice"  (usually this means large, at least point-transitive) automorphism groups. It appears that most Steiner systems are rigid (i.e., have no non-trivial automorphism), but this has only been demonstrated for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767082.png" /> (where it is known that the proportion of rigid Steiner triple systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767083.png" /> among all Steiner triple systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767084.png" /> approaches 1 as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767085.png" />). The particularly interesting existence problem for cyclic Steiner systems (those with a point-transitive cyclic group of automorphisms) has also been settled completely only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767086.png" />; the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767087.png" /> is at present studied extensively.

==Characterization and classification.==
In view of the results on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767088.png" /> mentioned above, the complete classification of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767089.png" /> is only feasible for small values of the parameters. One therefore is interested in characterizing the  "classical"  series of examples (in particular, the Steiner systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767091.png" /> formed by the points and lines in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767092.png" />-dimensional affine, respectively projective, space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767093.png" /> and the Hermitian unitals formed by the absolute points and the non-absolute lines of a unitary polarity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767094.png" />) and in classifying Steiner systems with interesting automorphism groups. E.g., all systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767095.png" /> with a group which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767096.png" />-homogeneous on points have been classified; except for the three classical series just mentioned, there are only two further series and a few sporadic examples. At present one works on the classification of systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767097.png" /> with a flag-transitive or a point-primitive group.

==Resolvable Steiner systems.==
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767098.png" /> is a union of parallel classes (i.e., partitions of the point set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s08767099.png" />). A resolvable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670100.png" /> is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670101.png" /> and also called a Kirkman system (since the seminal case of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670102.png" /> is equivalent to a solution of Kirkman's famous schoolgirl problem posed in 1850). One clearly has the further necessary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670103.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670104.png" /> for the existence of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670105.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670106.png" />, the necessary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670107.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670108.png" /> is known to be sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670110.png" />; it is also asymptotically sufficient for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670111.png" />. The cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670113.png" /> are almost settled (with about 100 undecided values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670114.png" /> in either case). Much less is known for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670115.png" />; only the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670116.png" /> is almost settled.

==Subsystems.==
Another problem of interest is the existence of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670117.png" /> with a subsystem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670118.png" />. Up till now (1990) this has only been considered for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670119.png" />. Here one obtains the additional necessary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670120.png" />, which is known to be sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670121.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670122.png" />. Similarly, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670123.png" /> with a sub-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670124.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670125.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670127.png" />.

==Imbeddings.==
One question concerns the possibility of imbedding a  "partial"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670128.png" /> in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670129.png" />. E.g., a partial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670130.png" /> can be imbedded in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670131.png" /> for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670132.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670133.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670134.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670135.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670136.png" />. Another question which has been studied in many special instances concerns imbeddings of partial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670137.png" />'s in finite projective planes. Here the outstanding problem is the conjecture that any partial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670138.png" /> can be imbedded in a finite projective plane (possibly even in a Desarguesian plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670139.png" />).

==Large sets of Steiner systems and packings.==
A large set is a partition of the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670140.png" />-subsets of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670141.png" />-set into Steiner systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670142.png" />. The outstanding result here is the existence of a large set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670143.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670144.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670145.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670146.png" /> (with 6 possible exceptions) proved by Lu Jia-Xi in a series of papers [[#References|[a15]]]. One also knows the existence of a large set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670147.png" />'s. More generally, one is interested in (bounds on) the maximum number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670148.png" /> of pairwise disjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670149.png" />'s (i.e. packings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670150.png" />'s).

==Generalizations.==
Steiner systems have been generalized in many ways, and problems analogous to the ones discussed here are studied for these generalizations. The most obvious generalization is that to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670151.png" />'s, where one requires that any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670152.png" />-set of points is contained in exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670153.png" /> common blocks. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670154.png" /> is called simple if there are no two blocks which are incident with the same set of points. L. Teirlinck [[#References|[a16]]] proved that there exist non-trivial simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670155.png" />'s for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670156.png" />, thus settling an outstanding problem. Another generalization is allowance of different block sizes; this has been mainly studied for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670157.png" />; the resulting structures are called pairwise balanced designs and are a fundamental tool for the existence theory of Steiner systems with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670158.png" />. One also considers packing and covering of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670159.png" />-subsets of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670160.png" />-set by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670161.png" />-subsets. Finally, sets of mutually [[Orthogonal Latin squares|orthogonal Latin squares]] can be considered as a generalization of affine planes, i.e. of systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670162.png" />.

==Connections to other areas.==
Finally, the study of Steiner systems and their generalizations is closely connected to several other areas of mathematics, such as geometry, group theory and coding theory. The best-known example is provided by the Witt designs mentioned above, which are essentially equivalent to the Mathieu groups and the Golay codes, see [[#References|[a1]]], [[#References|[a17]]], [[#References|[a18]]]. On the other hand, there are also important applications of combinatorial designs in general, and Steiner systems in particular, e.g. in computer science and cryptography [[#References|[a19]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Beth,   D. Jungnickel,   H. Lenz,   "Design theory" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.R. Hughes,   F.C. Piper,   "Design theory" , Cambridge Univ. Press  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.C. Lindner (ed.)  A. Rosa (ed.) , ''Topics on Steiner systems'' , North-Holland  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C.J. Colbourn (ed.)  R.A. Mathon (ed.) , ''Combinatorial design theory'' , North-Holland  (1987)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Hartman (ed.) , ''Combinatorial designs'' , North-Holland  (1989)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Doyen,   A. Rosa,   "An extended bibliography and survey of Steiner systems"  ''Congressus Numerantium'' , '''20'''  (1978)  pp. 297–361</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. Doyen,   A. Rosa,   "An extended bibliography of Steiner systems"  ''Ann. Discr. Math.'' , '''7'''  (1980)  pp. 317–349</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  D. Jungnickel,   "Design theory: an update"  ''Ars Comb.'' , '''28'''  (1989)  pp. 129–199</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R.A. Mathon,   A. Rosa,   "Tables of parameters of BIBD's with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670163.png" /> including existence, enumeration and resolvability results: an update"  ''Ars Comb.''  (To appear)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  Y.M. Chee,   C.J. Colbourn,   D.L. Kreher,   "Simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670164.png" />-designs with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670165.png" />"  ''Ars Comb.'' , '''29'''  (1990)  pp. 193–258</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  R.M. Wilson,   "An existence theory for pairwise balanced designs I"  ''J. Comb. Th. (A)'' , '''13'''  (1972)  pp. 220–245</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  R.M. Wilson,   "An existence theory for pairwise balanced designs II"  ''J. Comb. Th. (A)'' , '''13'''  (1972)  pp. 246–273</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  R.M. Wilson,   "An existence theory for pairwise balanced designs III"  ''J. Comb. Th. (A)'' , '''18'''  (1975)  pp. 71–79</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  E. Witt,   "Über Steinersche Systeme"  ''Abh. Math. Sem. Hamburg'' , '''12'''  (1938)  pp. 265–275</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  Jia-Xi Lu,   "On large sets of disjoint Steiner triple systems, VI"  ''J. Comb. Th. (A)'' , '''37'''  (1984)  pp. 189–192</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  L. Teirlinck,   "Non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670166.png" />-designs without repeated blocks exist for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087670/s087670167.png" />"  ''Discr. Math.'' , '''65'''  (1987)  pp. 301–311</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  F.J. MacWilliams,   N.J.A. Sloane,   "The theory of error-correcting codes" , '''I-II''' , North-Holland  (1978)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  H. Lüneburg,   "Transitive Erweiterungen endlicher Permutationsgruppen" , ''Lect. notes in math.'' , '''84''' , Springer  (1969)</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  C.J. Colbourn,   P.C. van Oorschot,   "Applications of combinatorial designs in computer science"  ''ACM Comp. Surveys'' , '''21'''  (1989)  pp. 223–250</TD></TR></table>