Editing Steiner system (section)

=== 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).