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| [[image:Fano plane.svg|250px|right|thumbnail|The [[Fano plane]] is an S(2,3,7) Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line.]]
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| In [[Combinatorics|combinatorial]] [[mathematics]], a '''Steiner system''' (named after [[Jakob Steiner]]) is a type of [[block design]], specifically a [[Block design#Generalization:t-design|t-design]] with λ = 1 and ''t'' ≥ 2.
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| A Steiner system with parameters ''t'', ''k'', ''n'', written S(''t'',''k'',''n''), is an ''n''-element [[Set (mathematics)|set]] ''S'' together with a set of ''k''-element [[subset]]s of ''S'' (called '''blocks''') with the property that each ''t''-element subset of ''S'' is contained in exactly one block. In an alternate notation for block designs, an S(''t'',''k'',''n'') would be a ''t''-(''n'',''k'',1) design.
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| This definition is relatively modern, generalizing the ''classical'' definition of Steiner systems which in addition required that ''k'' = ''t'' + 1. An S(2,3,''n'') was (and still is) called a ''Steiner triple system'', while an S(3,4,''n'') was called a ''Steiner quadruple system'', and so on. With the generalization of the definition, this naming system is no longer strictly adhered to.
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| As of 2012, an outstanding problem in [[block design|design theory]] is if any nontrivial (<math>t < k < n</math>) Steiner systems have ''t'' ≥ 6. It is also unknown if infinitely many have ''t'' = 4 or 5.<ref>{{cite web|url=http://designtheory.org/library/encyc/tdes/g |title=Encyclopaedia of Design Theory: t-Designs |publisher=Designtheory.org |date=2004-10-04 |accessdate=2012-08-17}}</ref>
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| == Examples ==
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| === Finite projective planes ===
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| A finite [[projective plane]] of order ''q'', with the lines as blocks, is an S(2, ''q''+1, ''q''<sup>2</sup>+''q''+1), since it has ''q''<sup>2</sup>+''q''+1 points, each line passes through ''q''+1 points, and each pair of distinct points lies on exactly one line.
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| ===Finite affine planes===
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| A finite [[Affine plane (incidence geometry)|affine plane]] of order ''q'', with the lines as blocks, is an S(2, ''q'', ''q''<sup>2</sup>). An affine plane of order ''q'' can be obtained from a projective plane of the same order by removing one block and all of the points in that block from the projective plane. Choosing different blocks to remove in this way can lead to non-isomorphic affine planes.
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| ==Classical Steiner systems==
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| ===Steiner triple systems===
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| An S(2,3,''n'') is called a '''Steiner triple system''', and its blocks are called '''triples'''. It is common to see the abbreviation STS(''n'') for a Steiner triple system of order ''n''.
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| The number of triples is ''n''(''n''−1)/6. A necessary and sufficient condition for the existence of an S(2,3,''n'') is that ''n'' <math>\equiv</math> 1 or 3 (mod 6). The projective plane of order 2 (the [[Fano plane]]) is an STS(7) and the [[Affine plane (incidence geometry)|affine plane]] of order 3 is an STS(9).
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| Up to isomorphism, the STS(7) and STS(9) are unique, there are two STS(13)s, 80 STS(15)s, and 11,084,874,829 STS(19)s.<ref name=Handbook>{{harvnb|Colbourn|Dinitz|2007|loc=pg.60}}</ref>
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| We can define a multiplication on the set ''S'' using the Steiner triple system by setting ''aa'' = ''a'' for all ''a'' in ''S'', and ''ab'' = ''c'' if {''a'',''b'',''c''} is a triple. This makes ''S'' an [[idempotent]], [[commutative]] [[quasigroup]]. It has the additional property that "ab" = "c" implies "bc" = "a" and "ca" = "b".<ref>This property is equivalent to saying that (xy)y = x for all x and y in the idempotent commutative quasigroup.</ref> Conversely, any (finite) quasigroup with these properties arises from a Steiner triple system. Commutative idempotent quasigroups satisfying this additional property are called ''Steiner quasigroups''.<ref>{{harvnb|Colbourn|Dinitz|2007|loc=pg. 497, definition 28.12}}</ref>
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| ===Steiner quadruple systems===
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| An S(3,4,''n'') is called a '''Steiner quadruple system'''. A necessary and sufficient condition for the existence of an S(3,4,''n'') is that ''n'' <math>\equiv</math> 2 or 4 (mod 6). The abbreviation SQS(''n'') is often used for these systems.
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| Up to isomorphism, SQS(8) and SQS(10) are unique, there are 4 SQS(14)s and 1,054,163 SQS(16)s.<ref>{{harvnb|Colbourn|Dinitz|2007|loc=pg.106}}</ref>
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| ===Steiner quintuple systems===
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| An S(4,5,''n'') is called a ''Steiner quintuple system''. A necessary condition for the existence of such a system is that ''n'' <math>\equiv</math> 3 or 5 (mod 6) which comes from considerations that apply to all the classical Steiner systems. An additional necessary condition is that ''n'' <math>\not\equiv</math> 4 (mod 5), which comes from the fact that the number of blocks must be an integer. Sufficient conditions are not known.
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| There is a unique Steiner quintuple system of order 11, but none of order 15 or order 17.<ref>{{harvnb|Östergard|Pottonen|2008}}</ref> Systems are known for orders 23, 35, 47, 71, 83, 107, 131, 167 and 243. The smallest order for which the existence is not known (as of 2011) is 21.
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| == Properties ==
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| It is clear from the definition of S(''t'',''k'',''n'') that <math>1 < t < k < n</math>. (Equalities, while technically possible, lead to trivial systems.)
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| If S(''t'',''k'',''n'') exists, then taking all blocks containing a specific element and discarding that element gives a ''derived system'' S(''t''−1,''k''−1,''n''−1). Therefore the existence of S(''t''−1,''k''−1,''n''−1) is a necessary condition for the existence of S(''t'',''k'',''n'').
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| The number of ''t''-element subsets in S is <math>\tbinom n t</math>, while the number of ''t''-element subsets in each block is <math>\tbinom k t</math>. Since every ''t''-element subset is contained in exactly one block, we have <math>\tbinom n t = b\tbinom k t</math>, or <math>b = \frac{\tbinom nt}{\tbinom kt}</math>, where ''b'' is the number of blocks. Similar reasoning about ''t''-element subsets containing a particular element gives us <math>\tbinom{n-1}{t-1}=r\tbinom{k-1}{t-1}</math>, or <math>r=\frac{\tbinom{n-1}{t-1}}{\tbinom{k-1}{t-1}}</math>, where ''r'' is the number of blocks containing any given element. From these definitions follows the equation <math>bk=rn</math>. It is a necessary condition for the existence of S(''t'',''k'',''n'') that ''b'' and ''r'' are integers. As with any block design, [[Fisher's inequality]] <math>b\ge n</math> is true in Steiner systems.
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| Given the parameters of a Steiner system S(t,k,n) and a subset of size <math>t' \leq t</math>, contained in at least one block, one can compute the number of blocks intersecting that subset in a fixed number of elements by constructing a [[Pascal triangle]].<ref>{{harvnb|Assmus|Key|1994|loc=pg. 8}}</ref> In particular, the number of blocks intersecting a fixed block in any number of elements is independent of the chosen block.
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| It can be shown that if there is a Steiner system S(2,''k'',''n''), where ''k'' is a prime power greater than 1, then ''n'' <math>\equiv</math> 1 or ''k'' (mod ''k''(''k''−1)). In particular, a Steiner triple system S(2,3,''n'') must have ''n'' = 6''m''+1 or 6''m''+3. It is known that this is the only restriction on Steiner triple systems, that is, for each [[natural number]] ''m'', systems S(2,3,6''m''+1) and S(2,3,6''m''+3) exist.
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| == History ==
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| Steiner triple systems were defined for the first time by [[Wesley S. B. Woolhouse|W.S.B. Woolhouse]] in 1844 in the Prize question #1733 of Lady's and Gentlemen's Diary.<ref>{{harvnb|Lindner|Rodger|1997|loc=pg.3}}</ref> The posed problem was solved by {{Harvs|first=Thomas|last=Kirkman|authorlink=Thomas Kirkman|year=1847|txt}}. In 1850 Kirkman posed a variation of the problem known as [[Kirkman's schoolgirl problem]], which asks for triple systems having an additional property (resolvability). Unaware of Kirkman's work, {{harvs|first=Jakob|last=Steiner|authorlink=Jakob Steiner|year=1853|txt}} reintroduced triple systems, and as this work was more widely known, the systems were named in his honor.
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| == Mathieu groups ==
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| Several examples of Steiner systems are closely related to [[group theory]]. In particular, the [[List of finite simple groups|finite simple groups]] called [[Mathieu group]]s arise as [[automorphism group]]s of Steiner systems:
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| * The [[Mathieu group M11|Mathieu group M<sub>11</sub>]] is the automorphism group of a S(4,5,11) Steiner system
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| * The [[Mathieu group M12|Mathieu group M<sub>12</sub>]] is the automorphism group of a S(5,6,12) Steiner system
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| * The [[Mathieu group M22|Mathieu group M<sub>22</sub>]] is the unique index 2 subgroup of the automorphism group of a S(3,6,22) Steiner system
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| * The [[Mathieu group M23| Mathieu group M<sub>23</sub>]] is the automorphism group of a S(4,7,23) Steiner system
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| * The [[Mathieu group M24|Mathieu group M<sub>24</sub>]] is the automorphism group of a S(5,8,24) Steiner system.
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| == The Steiner system S(5, 6, 12) ==
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| There is a unique S(5,6,12) Steiner system; its automorphism group is the [[Mathieu group]] M<sub>12</sub>, and in that context it is denoted by W<sub>12</sub>.
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| === Constructions ===
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| To construct it, take a 12-point set and think of it as the [[projective line]] over '''F'''<sub>11</sub> — in other words, the integers mod 11 together with a point called infinity. Among the integers mod 11, six are perfect squares:
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| :<math>\{0,1,3,4,5,9\}.\ </math>
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| Call this set a "block". From this, we may obtain other blocks by applying fractional linear transformations:
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| :<math>z \mapsto \frac{az + b}{cz + d}.</math>
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| These blocks then form a (5,6,12) Steiner system.
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| W<sub>12</sub> can also constructed from the [[affine geometry]] on the [[vector space]] F<sub>3</sub>xF<sub>3</sub>, an S(2,3,9) system.
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| An alternative construction of W<sub>12</sub> is obtained by use of the 'kitten' of R.T. Curtis.<ref>{{Harvnb|Curtis|1984}}</ref>
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| == The Steiner system S(5, 8, 24) ==
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| Particularly remarkable is the Steiner system S(5, 8, 24), also known as the '''Witt design''' or '''Witt geometry'''. It was first described by {{Harvs|txt|authorlink=Robert Daniel Carmichael|last=Carmichael|year=1931}} and rediscovered by {{Harvs|txt|last=Witt|year=1938|authorlink=Ernst Witt}}. This system is connected with many of the [[sporadic simple group]]s and with the [[exceptional object|exceptional]] 24-dimensional [[lattice (group)|lattice]] known as the [[Leech lattice]].
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| The automorphism group of S(5, 8, 24) is the [[Mathieu group M24|Mathieu group M<sub>24</sub>]], and in that context the design is denoted W<sub>24</sub> ("W" for "Witt")
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| === Constructions ===
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| There are many ways to construct the S(5,8,24). Two methods are described here:
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| * Method based on 8-combinations of 24 elements: All 8-element subsets of a 24-element set are generated in lexicographic order, and any such subset which differs from some subset already found in fewer than four positions is discarded.
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| The list of octads for the elements 01, 02, 03, ..., 22, 23, 24 is then:
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| :: 01 02 03 04 05 06 07 08
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| :: 01 02 03 04 09 10 11 12
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| :: 01 02 03 04 13 14 15 16
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| :: .
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| :: . (next 753 octads omitted)
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| :: .
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| :: 13 14 15 16 17 18 19 20
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| :: 13 14 15 16 21 22 23 24
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| :: 17 18 19 20 21 22 23 24
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| Each single element occurs 253 times somewhere in some octad. Each pair occurs 77 times. Each triple occurs 21 times. Each quadruple (tetrad) occurs 5 times. Each quintuple (pentad) occurs once. Not every hexad, heptad or octad occurs.
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| * Method based on 24-bit binary strings: All 24-bit binary strings are generated in lexicographic order, and any such string that [[Hamming distance|differs from some earlier one in fewer than 8 positions]] is discarded. The result looks like this:
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| <pre>
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| 000000000000000000000000
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| 000000000000000011111111
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| 000000000000111100001111
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| 000000000000111111110000
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| 000000000011001100110011
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| 000000000011001111001100
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| 000000000011110000111100
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| 000000000011110011000011
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| 000000000101010101010101
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| 000000000101010110101010
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| .
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| . (next 4083 omitted)
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| .
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| 111111111111000011110000
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| 111111111111111100000000
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| 111111111111111111111111
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| </pre>
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| The list contains 4096 items, which are each code words of the [[binary Golay code|extended binary Golay code]]. They form a [[group (mathematics)|group]] under the XOR operation. One of them has zero 1-bits, 759 of them have eight 1-bits, 2576 of them have twelve 1-bits, 759 of them have sixteen 1-bits, and one has twenty-four 1-bits. The 759 8-element blocks of the S(5,8,24) (called [http://igor.gold.ac.uk/~mas01rwb/octad.html octads]) are given by the patterns of 1's in the code words with eight 1-bits.
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| == See also ==
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| * [[Constant weight code]]
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| * [[Kirkman's schoolgirl problem]]
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| * [[Sylvester–Gallai configuration]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation|first1=E. F., Jr.|last1=Assmus|first2=J. D.|last2=Key|title=Designs and Their Codes|publisher=[[Cambridge University Press]]|isbn=0-521-45839-0|contribution = 8. Steiner Systems|pages=295–316|year=1994}}.
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| *{{citation|first1=Thomas|last1=Beth|first2=Dieter|last2=Jungnickel|first3=Hanfried|last3=Lenz|title=Design Theory|publisher=[[Cambridge University Press]]|location=Cambridge|year=1986}}. 2nd ed. (1999) ISBN 978-0-521-44432-3.
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| *{{citation|last=Carmichael|first=Robert|authorlink=Robert Daniel Carmichael|title=Tactical Configurations of Rank Two|journal=American Journal of Mathematics|volume=53|pages=217–240|year=1931|url=http://www.jstor.org/stable/10.2307/2370885}}
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| * {{citation | last1=Colbourn|first1=Charles J. | last2=Dinitz|first2=Jeffrey H. | title=Handbook of Combinatorial Designs | publisher=Chapman & Hall/ CRC | location=Boca Raton | zbl=0836.00010 | year=1996 | isbn=0-8493-8948-8 }}
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| * {{citation | last1=Colbourn|first1=Charles J. | last2=Dinitz|first2=Jeffrey H. | title=Handbook of Combinatorial Designs | year=2007 | publisher=Chapman & Hall/ CRC | location=Boca Raton | isbn=1-58488-506-8 | edition=2nd | zbl=1101.05001}}
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| * {{citation | last=Curtis|first=R.T.|contribution=The Steiner system S(5,6,12), the Mathieu group M<sub>12</sub> and the "kitten"|title=Computational group theory (Durham, 1982)|publisher=Academic Press|place=London|year=1984|isbn=0-12-066270-1|mr=0760669|ed=Michael D. Atkinson|pages=353–358}}
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| *{{citation|first1=D. R.|last1=Hughes|first2=F. C.|last2=Piper|title=Design Theory|publisher=Cambridge University Press|isbn=0-521-35872-8|pages=173–176|year=1985}}.
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| * {{Citation |last= Kirkman |first=Thomas P. |authorlink= Thomas Kirkman |title= On a Problem in Combinations |journal= The Cambridge and Dublin Mathematical Journal |volume= II |pages= 191–204 | publisher = Macmillan, Barclay, and Macmillan |year= 1847 |postscript= .}}
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| * {{citation|last=Lindner|first=C.C.|last2=Rodger|first2=C.A.|title=Design Theory|year=1997|publisher=CRC Press|location=Boca Raton|isbn=0-8493-3986-3}}
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| * {{citation|last=Östergard|first=Patric R.J.|last2=Pottonen |first2=Olli|title=There exists no Steiner system S(4,5,17)|journal=Journal of Combinatorial Theory Series A|year=2008|volume=115|issue=8|pages=1570–1573|doi=10.1016/j.jcta.2008.04.005}}
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| *{{citation|first=J.|last=Steiner|authorlink=Jakob Steiner|title=Combinatorische Aufgabe|journal=[[Crelle's Journal|Journal für die Reine und Angewandte Mathematik]]|volume=45|year=1853|pages=181–182}}.
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| *{{citation|doi=10.1007/BF02948947|last=Witt|first=Ernst|authorlink=Ernst Witt|title=Die 5-Fach transitiven Gruppen von Mathieu|journal=Abh. Math. Sem. Univ. Hamburg|volume=12|pages=256–264|year=1938}}
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| == External links ==
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| *{{MathWorld|title=Steiner System|urlname=SteinerSystem|author=Rowland, Todd and Weisstein, Eric W.}}
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| *{{springer|title=Steiner system|id=Steiner_system|last=Rumov|first=B.T.}}
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| * [http://www.win.tue.nl/~aeb/graphs/S.html Steiner systems] by Andries E. Brouwer
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| * [http://hilltop.bradley.edu/~delgado/gandg/SteinerApplet.html Implementation of S(5,8,24)] by Dr. Alberto Delgado, Gabe Hart, and Michael Kolkebeck
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| * [http://www.xs4all.nl/~jemebius/Steiner5824.htm S(5, 8, 24) Software and Listing] by Johan E. Mebius
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| * [http://www.dharwadker.org/witt.html The Witt Design] computed by Ashay Dharwadker
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| [[Category:Design theory]]
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| [[Category:Design of experiments]]
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| [[Category:Set families]]
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| [[de:Steiner-Tripel-System]]
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