Editing Steiner system (section)

== The Steiner system S(5, 8, 24) ==
The Steiner system S(5, 8, 24), also known as the '''Witt design''' or '''Witt geometry''', 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]]. 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")

=== Direct lexicographic generation ===
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.

The list of octads for the elements 01, 02, 03, ..., 22, 23, 24 is then:

::    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.

=== Construction from the binary Golay code ===
The 4096 codewords of the 24-bit [[binary Golay code]] are generated, and the 759 codewords with a [[Hamming weight]] of 8 correspond to the S(5,8,24) system.

The Golay code can be constructed by many methods, such as generating all 24-bit binary strings in lexicographic order and discarding those that [[Hamming distance|differ from some earlier one in fewer than 8 positions]]. The result looks like this:
<pre>
    000000000000000000000000
    000000000000000011111111
    000000000000111100001111
    .
    . (next 4090 24-bit strings omitted)
    .
    111111111111000011110000
    111111111111111100000000
    111111111111111111111111
</pre>
The codewords form a [[group (mathematics)|group]] under the [[XOR]] operation.

=== Projective line construction ===
This construction is due to Carmichael (1931).<ref>{{harvnb|Carmichael|1931}}</ref>

Add a new element, call it {{mvar|∞}}, to the 23 elements of the [[finite field]] {{mvar|'''F'''}}<sub>23</sub> (that is, the integers mod 23). This set, {{mvar|''S''}}, of 24 elements can be formally identified with the points of the [[projective line]] over {{mvar|'''F'''}}<sub>23</sub>. Call the following specific subset of size 8,
:<math>\{\infty,0,1,3,12,15,21,22\}, </math>
a "block".  (We can take any octad of the extended [[binary Golay code]], seen as a quadratic residue code.) From this block, we obtain the other blocks of the {{mvar|S}}(5,8,24) system by repeatedly applying the [[linear fractional transformation]]s:
:<math>z' = f(z) = \frac{az + b}{cz + d},</math>
where {{mvar|a,b,c,d}} are in {{mvar|'''F'''}}<sub>23</sub> and {{math|1= ''ad &minus; bc'' = 1}}.
With the usual conventions of defining {{math|1= ''f'' (&minus;''d''/''c'') = ∞}} and {{math|1= ''f'' (∞) = ''a''/''c''}}, these functions map the set {{mvar|''S''}} onto itself. In geometric language, they are [[Projectivity|projectivities]] of the projective line. They form a [[group (mathematics)|group]] under composition which is the [[projective special linear group]] {{mvar|PSL}}(2,23) of order 6072. There are exactly 8 elements of this group that leave the initial block fixed setwise. So there will be 6072/8 = 759 images of that block.  These form the octads of S(5,8,24).

=== Construction from the [[Miracle Octad Generator]] ===
The [[Miracle Octad Generator]] (MOG) is a tool to generate octads, such as those containing specified subsets. It consists of a 4x6 array with certain weights assigned to the rows. In particular, an 8-subset should obey three rules in order to be an octad of S(5,8,24). First, each of the 6 columns should have the same [[Parity (mathematics)|parity]], that is, they should all have an odd number of cells or they should all have an even number of cells. Second, the top row should have the same parity as each of the columns. Third, the rows are respectively multiplied by the weights 0, 1, 2, and 3 over the [[Finite field#Field with four elements|finite field of order 4]], and column sums are calculated for the 6 columns, with multiplication and addition using the [[finite field arithmetic]] definitions. The resulting column sums should form a valid ''[[hexacode]]word'' of the form {{nowrap|(''a'', ''b'', ''c'', ''a'' + ''b'' + ''c'', ''3a'' + ''2b'' + ''c'', ''2a'' + ''3b'' + ''c'')}} where ''a, b, c'' are also from the finite field of order 4. If the column sums' parities don't match the row sum parity, or each other, or if there do not exist ''a, b, c'' such that the column sums form a valid hexacodeword, then that subset of 8 is not an octad of S(5,8,24).

The MOG is based on creating a [[bijection]] (Conwell 1910, "The three-space PG(3,2) and its group") between the 35 ways to partition an 8-set into two different 4-sets, and the 35 lines of the [[Fano plane#Fano three-space|Fano 3-space]] PG(3,2). It is also geometrically related (Cullinane, "Symmetry Invariance in a Diamond Ring", Notices of the AMS, pp A193-194, Feb 1979) to the 35 different ways to partition a 4x4 array into 4 different groups of 4 cells each, such that if the 4x4 array represents a four-dimensional finite [[affine space]], then the groups form a set of parallel subspaces.