Constructing $M_{24}$ using Steiner Systems. I'm trying to (re)construct the Mathieu Group $M_{24}$ using Steiner systems.
I'm not so familiar with t-designs and Steiner systems. I have just the definition of t-designs, Steiner systems and the automorphism group of a t-design.
According to different texts I found:


*

*I have to construct the Steiner system $S(5,8,24)$.

*simplest way to construct this Steiner system is to use binary lexicodes.

*$M_{24}$ is the automorphism group of $S(5,8,24)$.

*I have to prove $M_{24}$ is 5-transitive.
How can I construct $S(5,8,24)$?
 A: Typing $$\rm Steiner\ system\ octads$$ into Google will lead you to many references, among them http://en.wikipedia.org/wiki/Binary_Golay_code and http://www.math.umn.edu/~webb/oldteaching/Year2010-11/LeechTriangleHandout.pdf
A: This paper by Robin Chapman (Wayback Machine) gives the proof of equivalence of $S(5,8,24)$ systems with binary Golay codes, followed by several constructions of binary Golay codes.  
Beware that even though the binary lexicode construction is one of the "simplest" to describe, it is one of the most difficult to prove that it has the properties you want.  In particular, it is not at all trivial to prove that the binary lexicode is linear - the only proof (Wayback Machine) I know involves a detour through the (beautiful!) theory of impartial combinatorial games.  It's also very hard to use to do explicit calculations (such as determining whether a particular word is in the code).
By contrast, the construction via the Miracle Octad Generator is more complicated, but it's much easier to prove that it has the right properties, and to do calculations with (see Chapman's paper, or Chapter 11 of Conway and Sloane's book Sphere Packings, Lattices, and Groups).
