I have been reading about the Mathieu finite simple groups, which has taken me on an interesting detour through coding theory, and the Miracle Octad Generator (MOG). There are a lot of instructions online on how the MOG works, but much less on why it works, and it's the why I'd like some pointers on.
In terms of the how, I understand that you've got 24 points laid out onto a grid, and obeying some rules, you can extend any 5 of them to an octad of the steiner system S(5,8,24). The rules are based around parity of the rows and columns, and scoring the columns to give hexacode words, which are all words from a 6-dimensional codebook over the finite field $F_4$. Although a tad fiddly, this is all well and good.
In order to try and find out why it works, I have read the reference everyone gives: chapter 11 of Conway/Sloane's Sphere Packing, Lattices and Groups. But (at least it seemed to me) this source only described how to use the MOG, not why it worked. Another source that looked promising was a master's thesis by Sitt Chee Keen (http://eprints.usm.my/6567/1/ON_STEINER_SYSTEM_S%285%2C_8%2C_24%29_AND_THE_MIRACLE_OCTAD_GENERATOR_%28MOG%29.pdf) but I could only find the first half online.
Like everything Conway was interested in, it still just feels a bit magic!
Could anyone give me any intuition as to what is going on behind the scenes in the MOG to make it work? Why the hexacode? How might you prove it? Any good references? Any help much appreciated, thank you.
If you are just looking for a proof that the MOG works, the discussion in Sphere Packing, Lattices and Groups, Chapter 11, up to section 6, shows that any octad can be uniquely completed from 5 points of the MOG. So it is a proof that the MOG gives a Steiner system.
As for the intuition behind why we would construct such a thing, well, I don't have a good answer. I have the same feeling that Conway had a way of distilling a lot of ideas to something simple and oftentimes magical-looking.
To some extent, though, I think the 5,8,24 Steiner system inherently has to be a bit magical, however it is constructed, because it is an exceptional structure. It is specific to the number 24, and doesn't generalize.