Steiner system

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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.

In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design.

A Steiner system with parameters l, m, n, written S(l,m,n), is an n-element set S together with a set of m-element subsets of S (called blocks) with the property that each l-element subset of S is contained in exactly one block. A Steiner system with parameters l, m, n is often called simply "an S(l,m,n)".

An S(2,3,n) is called a Steiner triple system, and its blocks are called triples. The number of triples is n(n-1)/6. We can define a multiplication on a 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 into an idempotent commutative quasigroup.

An S(3,4,n) is sometimes called a Steiner quadruple system. Systems with higher values of m are not usually called by special names.

[edit] Examples

[edit] Finite projective planes

A finite projective plane of order q, with the lines as blocks, is an S(2, q+1, q²+q+1), since it has q²+q+1 points, each line passes through q+1 points, and each pair of distinct points lies on exactly one line.

[edit] Properties

It can be shown that if there is a Steiner system S(2,m,n), where m is a prime power greater than 1, then n = 1 or m (mod m(m−1)). In particular, a Steiner triple system S(2,3,n) must have n = 6k+1 or 6k+3. It is known that this is the only restriction on Steiner triple systems, that is, for each natural number k, systems S(2,3,6k+1) and S(2,3,6k+3) exist.

[edit] History

This section requires expansion.

Steiner systems were introduced in (Kirkman 1847), in the form of Kirkman's schoolgirl problem, which treats triple systems. Triple systems were introduced and further studied independently in (Steiner 1853), whence the name.

[edit] Mathieu groups

Several examples of Steiner systems are closely related to group theory. In particular, the finite simple groups called Mathieu groups arise as automorphism groups of Steiner systems:

• The Mathieu group M₁₁ is the automorphism group of a S(4,5,11) Steiner system
• The Mathieu group M₁₂ is the automorphism group of a S(5,6,12) Steiner system
• The Mathieu group M₂₂ is the automorphism group of a S(3,6,22) Steiner system
• The Mathieu group M₂₃ is the automorphism group of a S(4,7,23) Steiner system
• The Mathieu group M₂₄ is the automorphism group of a S(5,8,24) Steiner system.

[edit] The Steiner system S(5, 6, 12)

There is a unique S(5,6,12) Steiner system; its automorphism group is the Mathieu group M₁₂, and in that context it is denoted by W₁₂.

[edit] Constructions

To construct it, take a 12-point set and think of it as the projective line over F₁₁ — in other words, the integers mod 11 together with a point called infinity. Among the integers mod 11, six are perfect squares:

Call this set a "block". From this, we may obtain other blocks by applying fractional linear transformations:

These blocks then form a (5,6,12) Steiner system.

W₁₂ can also constructed from the affine geometry on the vector space F₃xF₃, an S(2,3,9) system.

An alternative construction of W₁₂ is the 'Kitten' of R.T. Curtis.^[1]

[edit] The Steiner system S(5, 8, 24)

Particularly remarkable is the Steiner system S(5, 8, 24), also known as the Witt Design. It was described in a 1931 paper by R.D. Carmichael and rediscovered by Ernst Witt, who published a paper on the subject in 1938: (Witt 1938). This system is connected with many of the sporadic simple groups and with the exceptional 24-dimensional lattice known as the Leech lattice.

The automorphism group of S(5, 8, 24) is the Mathieu group M₂₄, and in that context it is denoted W₂₄ ("W" for "Witt")

[edit] Constructions

There are many ways to construct the S(5,8,24). The method given here is perhaps the simplest to describe, and is easy to convert into a computer program. It uses sequences of 24 bits. The idea is to write these down in lexicographic order, missing out any one which differs from some earlier one in fewer than 8 positions. The result looks like this:

   000000000000000000000000
   000000000000000011111111
   000000000000111100001111
   000000000000111111110000
   000000000011001100110011
   000000000011001111001100
   000000000011110000111100
   000000000011110011000011
   000000000101010101010101
   000000000101010110101010
   .
   . (next 4083 omitted)
   .
   111111111111000011110000
   111111111111111100000000
   111111111111111111111111

The list contains 4096 items, which are each code words of the extended binary Golay code. They form a 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 octads) are given by the patterns of 1's in the code words with eight 1-bits.

A still more direct approach is taken by running through all 8-element subsets of a 24-element set and skipping any such subset which differs from some octad already found in fewer than four positions.

Departing from 01, 02, 03, ..., 22, 23, 24 one obtains

   01 02 03 04 05 06 07 08
   01 02 03 04 09 10 11 12
   01 02 03 04 13 14 15 16
   . 
   . (next 753 octads omitted)
   .
   13 14 15 16 17 18 19 20
   13 14 15 16 21 22 23 24
   17 18 19 20 21 22 23 24

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.

[edit] See also

• Kirkman's schoolgirl problem

[edit] References

[edit] External links

• Implementation of S(5,8,24) by Dr. Alberto Delgado, Gabe Hart, and Michael Kolkebeck
• S(5, 8, 24) Software and Listing by Johan E. Mebius
• The Witt Design computed by Ashay Dharwadker

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