Embedding Change

Log24

Saturday, January 29, 2022

California Dreamin’

Filed under: General — Tags: Bombogenesis, Embedding Change, Strange Change, The Drill Imperative — m759 @ 2:38 pm

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Bombogenesis

Filed under: General — Tags: Bombogenesis, Embedding Change, Strange Change — m759 @ 12:18 pm

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On the Diamond-Theorem Group* of Order 322,560

Filed under: General — Tags: Embedding Change, Kummerhenge, Octad, Overarching Group, Priority, Strange Change — m759 @ 2:06 am

Taormina and Wendland have often discussed this group, which they
call "overarching" within the context of their Mathieu-moonshine research.

This seems to be the first time they have attempted to explore its geometric
background as an affine group, apart from its role as "the octad group" in the
researches of R. T. Curtis and John Conway on the large Mathieu group M₂₄.

* See a Log24 post of June 1, 2013.

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Thursday, January 27, 2022

The Lexicographic Octad Generator

Filed under: General — Tags: Embedding Change, Octad, Octad Generator, Strange Change, The Guy Embedding — m759 @ 2:30 pm

A Lexicographic Basis for the Binary Golay Code:

Brouwer and Guven — "Long ago," in
"The generating rank of the space of short vectors
in the Leech lattice mod 2," by
Andries Brouwer & Cicek Guven,
https://www.win.tue.nl/~aeb/preprints/udim24a.pdf —

"One checks by computer" that this is a basis:

000000000000000011111111
000000000000111100001111
000000000011001100110011
000000000101010101010101
000000001001011001101001
000000110000001101010110
000001010000010101100011
000010010000011000111010
000100010001000101111000
001000010001001000011101
010000010001010001001110
100000010001011100100100

(Copied from the Brouwer-Guven paper)

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Adlam at Harvard —
"Constructing the Extended Binary Golay Code,"
by Ben Adlam, Harvard University, August 9, 2011,
https://fliphtml5.com/llqx/wppz/basic —

Adlam also asserts, citing a reference, that this same
set of twelve vectors is a basis:

(Copied from the Adlam paper)
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Sources —

Background for Log24 posts on 'Embedding Change'

"One checks by computer" —

At http://magma.maths.usyd.edu.au/calc/ —

> V24 := VectorSpace(FiniteField(2), 24);
> G := sub< V24 |
> [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1],
> [0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1],
> [0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1],
> [0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],
> [0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1],
> [0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,1,0,1,0,1,1,0],
> [0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,1,1,0,0,0,1,1],
> [0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,1,1,1,0,1,0],
> [0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,1,1,1,0,0,0],
> [0,0,1,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,1,1,1,0,1],
> [0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,1,1,1,0],
> [1,0,0,0,0,0,0,1,0,0,0,1,0,1,1,1,0,0,1,0,0,1,0,0]>;
> Dimension(G);
12

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