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This is in regards to Chapter 11 of SPLAG. The tetracode construction of M12 is based on col-col, col+tet, tet-tet, col+col-tet, which are 6 + 36 + 36 + 54 = 132. (Unsigned hexads in the C12 code, of the Ternary Golay Code). Now I noticed the coincidence, that the cols are the inverse of the tets in the C12 code, and also S3 in C4 X S3 is the inverse of D4 in D4 X C3, in terms of generators, that is, S3 applied to a set will have the inverse effect of D4 applied there. (And cols=3 elements while tets=4 elements). Perhaps just a coincidence---

Anyway, 12P5 = 95040 and this is also the order of M12, the stabilizer of a S(5,6,12) Steiner system, with the tetracode construction being one possible construction (with various labellings possible). Now I know that it is also a stabilizer of the C12 code, and that M12 is quintuply sharply transitive on these sets. It sends blocks to blocks in S(5,6,12), My question being, how a single g (member of) M12 accomplishes this, does each permutation of 12P5, (which sends every pentad to every possible pentad) correspond to a g, so that there are 95040 elements, acting regularly transitive on the Steiner system, and/or C12..

The best example would be the hexads (264) of the Ternary Golay Code, constructed with the Tetracode, how does this stabilize C12?

Thanks

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col = ?, tet = ?, 12P5 = ? –  Ricky Demer Apr 18 '11 at 6:00
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I studied how M12 is built up from M9, in stages, and also got proficient with the tetracodeword construction. An interesting fact is that the Ternary Golay Code and the tetracodeword construction in SPLAG for S(5,6,12) (doubly = 264) are not so easily related as the Binary Golay Code and the hexacodeword construction for S(5,8,24), which is more direct. This of course relates to the MiniMOG and the MOG, respectively.

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