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Wednesday, July 6, 2011

Nordstrom-Robinson Automorphisms

Filed under: General,Geometry — Tags: , — m759 @ 1:01 AM

A 2008 statement on the order of the automorphism group of the Nordstrom-Robinson code—

"The Nordstrom-Robinson code has an unusually large group of automorphisms (of order 8! = 40,320) and is optimal in many respects. It can be found inside the binary Golay code."

— Jürgen Bierbrauer and Jessica Fridrich, preprint of "Constructing Good Covering Codes for Applications in Steganography," Transactions on Data Hiding and Multimedia Security III, Springer Lecture Notes in Computer Science, 2008, Volume 4920/2008, 1-22

A statement by Bierbrauer from 2004 has an error that doubles the above figure—

The automorphism group of the binary Golay code G is the simple Mathieu group M24 of order |M24| = 24 × 23 × 22 × 21 × 20 × 48 in its 5-transitive action on the 24 coordinates. As M24 is transitive on octads, the stabilizer of an octad has order |M24|/759 [=322,560]. The stabilizer of NR has index 8 in this group. It follows that NR admits an automorphism group of order |M24| / (759 × 8 ) = [?] 16 × 7! [=80,640]. This is a huge symmetry group. Its structure can be inferred from the embedding in G as well. The automorphism group of NR is a semidirect product of an elementary abelian group of order 16 and the alternating group A7.

— Jürgen Bierbrauer, "Nordstrom-Robinson Code and A7-Geometry," preprint dated April 14, 2004, published in Finite Fields and Their Applications , Volume 13, Issue 1, January 2007, Pages 158-170

The error is corrected (though not detected) later in the same 2004 paper—

In fact the symmetry group of the octacode is a semidirect product of an elementary abelian group of order 16 and the simple group GL(3, 2) of order 168. This constitutes a large automorphism group (of order 2688), but the automorphism group of NR is larger yet as we saw earlier (order 40,320).

For some background, see a well-known construction of the code from the Miracle Octad Generator of R.T. Curtis—

Click to enlarge:

IMAGE - The 112 hexads of the Nordstrom-Robinson code

For some context, see the group of order 322,560 in Geometry of the 4×4 Square.

Friday, March 29, 2019

Front-Row Seed

Filed under: General — Tags: — m759 @ 4:17 PM

"This outer automorphism can be regarded as
the seed from which grow about half of the
sporadic simple groups…." — Noam Elkies

Closely related material —

The Kummer 16_6 Configuration and the Nordstrom-Robinson Code

The top two cells of the Curtis "heavy brick" are also
the key to the diamond-theorem correlation.

Thursday, March 28, 2019

Culture

Filed under: General,Geometry — Tags: , — m759 @ 9:35 PM

The previous post, "Dream of Plenitude," suggests . . .

The Kummer 16_6 Configuration and the Nordstrom-Robinson Code

"So here's to you, Nordstrom-Robinson . . . ."

Wednesday, December 12, 2018

Kummerhenge Continues.

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 7:24 PM

Those pleased by what Ross Douthat today called
"The Return of Paganism" are free to devise rituals
involving what might be called "the sacred geometry
of the Kummer 166  configuration."

As noted previously in this journal, 

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

Geometric incarnation and the Kummer configuration

See also earlier posts also tagged "Kummerhenge" and 
another property of the remarkable Kummer 166 

The Kummer 16_6 Configuration and the Nordstrom-Robinson Code

For some related literary remarks, see "Transposed" in  this journal.

Some background from 2001 —

Tuesday, August 19, 2014

Lux

Filed under: General — m759 @ 7:00 PM

(As opposed to an alleged bearer  of lux  .)

IMAGE- T. Lux Feininger on 'Gestaltung'

(Here “eidolon” should instead be “eidos.”)

An example of Gestaltung :

This journal on the day of Feininger’s death, and the day before —

The 256 Code (Thursday, July 7, 2011) and Nordstrom-Robinson Automorphisms.

Thursday, July 7, 2011

The 256 Code:

Filed under: General — m759 @ 2:45 AM

Earlier Determinations —

IMAGE- Robert L. Griess Jr. on the automorphism group of the 256-word Nordstrom-Robinson code

"I could tell you a lot, but you got
 to be true to your code…." —Sinatra

Update of 4:28 PM ET 7/7—

The above remark by Griess is from a preprint of an article
published in Journal of Combinatorial Theory Series A
Volume 115 Issue 7, October 2008. (A copy dated
April 28, 2006, was placed in the arXiv on May 7, 2006.)

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