This is very cool.

This is very cool.

I suspect it may influence some thinking I've been doing lately about the Klein Quartic.

Originally shared by Gerard Westendorp

I figured out a way to understand Golay codes, and their relation to the Leech lattice, M24, and the Klein Quartic. I just completed this website: http://westy31.home.xs4all.nl/Golay/GolayCodeAndSymmetry.html
Golay codes are used in space communication to as a way to send data with built-in error correction.
You can construct the Golay code by imagining 12 data bits on the respective 12 pentagonal faces of the great dodecahedron. Then on each of the 12 vertices, put a parity bit that is the parity sum over all face bits that are not connected to the vertex. That's it! Note that there are 7 of these faces, and each of the 12 faces has 7 parity bits related to it, so there is th 24-7 connection to the Klein Quartic.

Comments

  1. John BaezSeptember 2, 2015 at 12:47 AM

    Yes, let's think about this thing I read in a page Gerard linked to:

    The Mathieu group M24 is also constucted geometrically, but in terms of a rather complicated figure (an icosatetrahedron) having 24 heptagonal faces, 56 vertices and 84 edges. If we consider just the vertices and edges, we have a non-planar graph that can be drawn (without crossing lines) on a surface of genus 3. Since this group and the extended binary Golay code are intimately related, it seems natural (in view of our construction) to consider whether there are natural geometric generators for M24 that can be viewed via the dodecahedron.

    This "icosatetrahedron" is basically one of our favorite abstract regular polytopes associated to Klein's quartic curve.  In what sense does M24 arise from this?  The statement is too vague for me to understand it!

    http://giam.southernct.edu/DecodingGolay/conclusions.html

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  2. John BaezSeptember 2, 2015 at 12:50 AM

    I guess it's supposed to be explained here:

    R. T. Curtis, Geometric interpretations of the 'natural' generators of the Mathieu groups, Math. Proc. Camb. Phil. Soc. 107 (1990), 19-26.

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  3. Roice NelsonSeptember 2, 2015 at 8:02 AM

    John Baez, have you seen this page yet?

    http://homepages.wmich.edu/~drichter/mathieu.htm

    Gerard Westendorp's page linked to it, and it walks through constructing M_24 from the quartic abstract polyhedron.  (I'm guessing the fancy term "icosatetrahedron" simply designates this polyhedron has 24 faces.  Not sure if it has any more meaning than that.)

    Another thing that struck me about Gerard's page was his statement that the Golay code has legal words of 0, 8, 12, 16, or 24 ones.  These specific numbers have been showing up in the 350-cell we've been looking at, as connection counts between some of the natural cell partitions.  For example, in the partition into two 175 cell halves, a cell on the surface of one half connects to 8 cells on the other half and 16 cells on the same half.  So I'm wondering if something deeper might be lurking there.

    ReplyDelete
  4. John BaezSeptember 2, 2015 at 9:48 AM

    I haven't seen that page.  I'll try to look at it tomorrow (it's bedtime here) and start thinking about that stuff!  Sounds cool!

    ReplyDelete
  5. John BaezSeptember 2, 2015 at 4:29 PM

    Roice Nelson - actually I'd seen this page once before, but only paying attention to the claim that the small cubicuboctahedron "is" the Klein quartic.  I was somewhat disappointed to figure out that while they're both genus-3 surfaces, they are geometrically different, with different symmetry groups.  But now I see Richter is somehow taking advantage of this difference to find two sets of permutations of the 24 vertices, which together generate the group M24.  The first set is just the symmetry group of the Klein quartic, GL(3,2)=PSL(2,7). But the second set, I don't understand very well.  Alas it doesn't seem to be the symmetry group of the cubicuboctahedron!

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