Log24

Thursday, August 27, 2020

The Complete Extended Binary Golay Code

Filed under: General — Tags: , , , , , — m759 @ 12:21 pm

All 4096 vectors in the code are at . . .

http://neilsloane.com/oadir/oa.4096.12.2.7.txt.

Sloane’s list* contains the 12 generating vectors
listed in 2011 by Adlam —

As noted by Conway in Sphere Packings, Lattices and Groups ,
these 4096 vectors, constructed lexicographically, are exactly
the same vectors produced by using the Conway-Sloane version
of the Curtis Miracle Octad Generator (MOG). Conway says this
lexico-MOG equivalence was first discovered by M. J. T. Guy.

(Of course, any  permutation of the 24 columns above produces
a version of the code qua  code. But because the lexicographic and
the MOG constructions yield the same result, that result is in
some sense canonical.)

See my post of July 13, 2020 —

The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.

For some related results, Google the twelfth generator:

* Sloane’s list is of the codewords as the rows of  an orthogonal array

See also http://neilsloane.com/oadir/.

Monday, November 27, 2017

The Golay Code via Witt’s Construction

Filed under: General,Geometry — Tags: — m759 @ 2:08 pm

Reference

Hansen, Robert Peter, "Construction and Simplicity of
the Large Mathieu Groups" (2011). Master's Theses. 4053. 
http://scholarworks.sjsu.edu/etd_theses/4053.

See also The Matrix Meets the Grid (Log24, Nov. 24).
More generally, see SPLAG in this journal.

Tuesday, January 16, 2024

Saturday, January 13, 2024

Tuesday, January 9, 2024

“Eternal Sunshine” Snark

Filed under: General — Tags: — m759 @ 10:01 pm

From Monday morning's post in memory of actor Tom Wilkinson —

Related material — Plato's "form of the Good" and . . .

http://m759.net/wordpress/?s=Schwartz+Golay.

Tuesday, December 19, 2023

Copilot Report

Filed under: General — Tags: , , — m759 @ 12:16 am
 
From Microsoft Copilot (with "Precise" setting)
on Dec. 18, 2023 —
 
How is the 1910 work of G. M. Conwell on PG(3,2) related to the Miracle Octad Generator of R. T. Curtis?

Searching for: Relation between G. M. Conwell work on PG(3,2) and Miracle Octad Generator of R. T. Curtis

Searching for: Relation between G. M. Conwell work on PG(3,2) and Miracle Octad Generator of R. T. Curtis explained

Generating answers for you…

The work of George M. Conwell on the finite projective 3-space PG(3,2) and the Miracle Octad Generator (MOG) of R. T. Curtis are both significant contributions to the field of combinatorial mathematics12.

Conwell’s work on PG(3,2) provided a detailed study of the structure of this finite projective space1On the other hand, the MOG is a tool for exploring the binary Golay code and its automorphism group M242.

The connection between these two works lies in the underlying combinatorial structures they explore. The Rosenhain and Göpel tetrads, which are key structures in the finite geometry PG(3,2), play a significant role in the Curtis Miracle Octad Generator3These structures provide a link between the finite geometric structures studied by Conwell and the combinatorial structures generated by the MOG3.

For a more detailed understanding of this connection, I recommend checking out the resources mentioned in the references123. If you have any specific questions about the concepts in these works, feel free to ask! I’ll do my best to help.

Learn more

5 archive.org
_________________________________________________________

__________________________________________________________


DOCUMENT SUMMARY (generated by Copilot)
of reference 3 above, from academia.edu —
"Rosenhain and Göpel Tetrads in PG(3,2),"
by Steven H. Cullinane on March 17, 2013

Here is a summary of the document:

Reference:
 
Copilot suggestions following document summary —
 



Correction:

"Subsets of four points in a projective 3-space" above is a Copilot error, and not
from the document being summarized. It should be "subsets of four points in an
affine 4-space."

Saturday, October 21, 2023

Chapter 11 Continues: A Larger Box

Filed under: General — Tags: , , — m759 @ 12:00 pm

"… really we should use larger boxes." — Ursula K. Le Guin

"The Steiner system S (5, 8, 24) is a block design
made up of 24 points and 759 blocks, each of size 8,
with the property that every 5 points lie in exactly one block.
This design is naturally associated with the Golay code, and
its automorphism group is the simple Mathieu group M24;
see [3, Ch. 11].

3.  J.H. Conway and N.J.A. Sloane, Sphere Packings,
     Lattices and Groups
, Springer-Verlag, New York, 1988."

New Zealand Journal of Mathematics,
Volume 25 (1996), 133-139.
"Markings of the Golay Code," by
Marston Conder and John McKay.
(Received July 1995.)

See also the Harlan Kane Special from Broomsday 2023.
That post relates properties of the 4×4 box (Cullinane, 1979)
to those of the 4×6 box (Conway and Sloane, 1988, without
mention of Cullinane 1979).

Thursday, September 29, 2022

The 4×6 Problem*

Filed under: General — Tags: , — m759 @ 4:03 pm

The exercise posted here on Sept. 11, 2022, suggests a 
more precisely stated problem . . .

The 24 coordinate-positions of the 4096 length-24 words of the 
extended binary Golay code G24 can be arranged in a 4×6 array
in, of course, 24! ways.

Some of these ways are more geometrically natural than others.
See, for instance, the Miracle Octad Generator of R. T. Curtis.
What is the size of the largest subcode C of G24 that can be 
arranged in a 4×6 array in such a way that the set  of words of C 
is invariant under the symmetry group of the rectangle itself, i.e. the
four-group of the identity along with horizontal and vertical reflections
and 180-degree rotation.

Recent Log24 posts tagged Bitspace describe the structure of
an 8-dimensional (256-word) code in a 4×6 array that has such
symmetry, but it is not yet clear whether that "cube-motif" code
is a Golay subcode. (Its octads are Golay, but possibly not all its
dodecads; the octads do not quite generate the entire code.) 
Magma may have an answer, but I have had little experience in
its use.

* Footnote of 30 September 2022.  The 4×6 problem is a
special case of a more general symmetric embedding problem.
Given a linear code C and a mapping of C to parts of a geometric
object A with symmetry group G, what is the largest subcode of C
invariant under G? What is the largest such subcode under all
such mappings from C to A?

Wednesday, September 28, 2022

Bitspace Note

Filed under: General — Tags: , — m759 @ 5:28 pm

Update of 5:20 AM ET on Sept. 29. 2022 —

The octads of the [24, 8, 8] cube-motif code
can be transformed by the permutation below
into octads recognizable, thanks to the Miracle
Octad Generator (MOG) of R. T. Curtis, as
belonging to the Golay code.

Tuesday, September 20, 2022

Motif-Space Updates

Filed under: General — Tags: — m759 @ 3:55 am

A linear code of length 24, dimension 8, and minimum weight 8
(a "[24, 8, 8] code") that was discussed in recent posts tagged
Bitspace might, viewed as a vector space, be called "motif space."

Yesterday evening's post "From a Literature Search for Binary [24, 8, 8] Codes
has been updated.  A reference from that update —

Computer Science > Information Theory

arXiv:cs/0607074 (cs)

[Submitted on 14 Jul 2006]

On Construction of the (24,12,8) Golay Codes

Xiao-Hong PengPaddy Farrell

Download PDF

Two product array codes are used to construct the (24, 12, 8) binary Golay code through the direct sum operation. This construction provides a systematic way to find proper (8, 4, 4) linear block component codes for generating the Golay code, and it generates and extends previously existing methods that use a similar construction framework. The code constructed is simple to decode.

Comments: To appear in IEEE Trans. on Information Theory Vol. 24 No. 8
Subjects: Information Theory (cs.IT)
Cite as: arXiv:cs/0607074 [cs.IT]

From Peng and Farrell, 2006 —

Monday, September 19, 2022

From a Literature Search for Binary [24, 8, 8] Codes

Filed under: General — Tags: , — m759 @ 7:34 pm

For one example of a binary [24, 8, 8] code, see other bitspace posts.

It is not clear whether that example is a subcode of the Golay code.

See also 

http://www.codetables.de/BKLC/
Tables.php?q=2&n0=1&n1=256&k0=1&k1=256

and  

http://www.codetables.de/BKLC/BKLC.php?q=2&n=12&k=8 .

Update of 3:22 AM ET on 20 September 2022 —

Update of  3:44 AM ET 20 September 2022 —

Another relevant document:

Friday, September 16, 2022

Symmetric Generation

Filed under: General — Tags: , — m759 @ 2:20 pm

Symmetric Generation of a Linear Code

The above is about a subspace of the
24-dimensional vector space over GF(2) 
. . . "An entire world of just 24 squares,"
to adapt a phrase from other Log24
posts tagged "Promises."

 

Update of 1:45 AM ET Sept. 18, 2022 —

It seems* from a Magma calculation that
the resemblance of the above extended
cube-motif code to the Golay code is only
superficial.

 

Without  the highly symmetric generating codewords that were added
to extend its dimension from 8 to 12, the cube-motifs code apparently
does , like the Golay code, have nonzero weights of only 8, 12, 16, and 24 —

Perhaps someone can prove there is no  way that adding more generating
codewords can turn the cube-motif code into the Golay code.

* The "seems" is because I have not yet encountered any of these
relatively rare (42 out of 4096) purported weight-4 codewords. Their
apparent existence may be due to an error in my typing of 0's and 1's.

Tuesday, September 13, 2022

A Helpful Survey of the Literature

Filed under: General — Tags: — m759 @ 9:39 pm

Some background for the exercise of 9/11

Vera Pless, "More on the uniqueness of the Golay codes,"
Discrete Mathematics 106/107 (1992) 391-398 —

"Several people [1-2,6] have shown that
any set of 212 binary vectors of length 24,
distance ≥ 8, containing 0, must be the
unique (up to equivalence) [24,12,8] Golay code." 

[1] P. Delsarte and J.M. Goethals, "Unrestricted codes
with the Golay parameters are unique
,"
Discrete Math. 12 (1975) 211-224.

[2] A. Neumeier, private communication, 1990.

[6] S.L. Snover, "The uniqueness of the
Nordstrom-Robinson and the Golay binary codes
,"
Ph.D. Thesis, Dept. of Mathematics, 
Michigan State Univ., 1973.

Related images —

"Before time began, there was the Cube."

              — Optimus Prime in 2007

      

"Remember, remember the fifth of November"

  — Hugo Weaving in 2005

Monday, September 5, 2022

2001: A Time Odyssey

Filed under: General — m759 @ 4:15 pm

See as well a search for intelligent life at Santa Cruz —

Related art —

Friday, August 19, 2022

The Guy Embedding

Filed under: General — Tags: , , — m759 @ 3:43 pm

M. J. T. Guy discovered that the lexicographic  version
of the Golay code contains, embedded within it, the
Miracle Octad Generator  (MOG)  of R. T. Curtis.

For 12 basis vectors of the lexicographic version, see below.

Basis vectors for the lexicographic version of the binary Golay code

For some context, click the embedded guy.

For a closely related, but simpler, mathematical
structure, see posts tagged The Omega Matrix.

Thursday, February 3, 2022

Through the Asian Looking Glass

Filed under: General — Tags: , , , — m759 @ 10:59 pm

The Miracle Octad Generator (MOG) —
The Conway-Sloane version of 1988:

Embedding Change, Illustrated

See also the 1976 R. T. Curtis version, of which the Conway-Sloane version
is a mirror reflection —

“There is a correspondence between the two systems
of 35 groups, which is illustrated in Fig. 4 (the MOG or
Miracle Octad Generator).”
—R.T. Curtis, “A New Combinatorial Approach to M24,” 
Mathematical Proceedings of the Cambridge Philosophical
Society
 (1976), 79: 25-42

http://www.log24.com/log/pix10A/100514-Curtis1976MOG.jpg

Curtis’s 1976 Fig. 4. (The MOG.)

The Guy Embedding (named for M.J.T., not Richard K., Guy) states that
the MOG is naturally embedded in the codewords of the extended binary
Golay code, if those codewords are generated in lexicographic order.

MOG in LOG embedding

The above reading order for the MOG 4×6 array —
down the columns, from left to right — yields the Conway-Sloane MOG.

Since that is a mirror image of the original Curtis MOG, the reading order
yielding that  MOG is down the columns, from right to left.

"Traditionally, ChineseJapaneseVietnamese and Korean are written vertically
in columns going from top to bottom and ordered from right to left, with each
new column starting to the left of the preceding one." — Wikipedia

The Asian reading order has certain artistic advantages:

Tuesday, February 1, 2022

Bit Space

Filed under: General — Tags: , — m759 @ 10:54 am

Twelve significant bit-sequences —

Twelve basis vectors, in lexicographic order, for the binary Golay-code space

http://www.log24.com/log/pix11/110412-IconicArt.jpg

Thursday, January 27, 2022

The Lexicographic Octad Generator

Filed under: General — Tags: , , , , — 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)

_________________________________________________

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:

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

(Copied from the Adlam paper)
__________________________________________________

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

Wednesday, January 26, 2022

The Embedding

Filed under: General — Tags: — m759 @ 12:05 pm

Continues.

Twelve basis vectors, in lexicographic order, for the binary Golay-code space

For further details,
click on the monolith.

Thursday, December 30, 2021

Antidote to Chaos?

Filed under: General — Tags: , , , , — m759 @ 3:57 pm

Some formal symmetry —

"… each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding  a 1×8 array
from Turyn's 1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again  yields
the 2x2x2 eightfold cube ."

— Steven H. Cullinane on April 19, 2016 — The Folding.

Related art-historical remarks:

The Shape of Time  (Kubler, Yale U.P., 1962).

See yesterday's post The Thing 

Sunday, July 25, 2021

Math Rights

Filed under: General — Tags: — m759 @ 7:00 pm

"The Algebra Project  was born.

The project was a five-step philosophy of teaching
that can be applied to any concept, he wrote, 
including physical experience, pictorial representation,
people talk (explain it in your own words), feature talk
(put it into proper English) and symbolic representation."

The New York Times  today

"He wrote"

See pages 120-122 in . . .

Thursday, February 25, 2021

Compare and Contrast

Filed under: General — Tags: , , , — m759 @ 12:31 pm

“… What is your dream—your ideal?  What is your News from Nowhere,
or, rather, What is the result of the little shake your hand has given to
the old pasteboard toy with a dozen bits of colored glass for contents?
And, most important of all, can you present it in a narrative or romance
which will enable me to pass an idle hour not disagreeably? How, for instance,
does it compare in this respect with other prophetic books on the shelf?”

— Hudson, W. H.. A Crystal Age  (p. 2). Open Road Media. Kindle Edition.

See as well . . .

The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.

Friday, November 27, 2020

The Sun-Ra Code

Filed under: General — Tags: , — m759 @ 3:24 pm

Saturday, October 24, 2020

Switchin’ the Positions*

Filed under: General — Tags: , — m759 @ 11:27 am

“C24 is the list of codewords of the extended
binary Golay code C24.  Each codeword is expressed
by a subset of the set M  of the positions  [1, . . . , 24]
of MOG.”

— From Shimada’s notes on computational data at
http://www.math.sci.hiroshima-u.ac.jp/~shimada/
preprints/Edge/PaperEdge/compdataEdge.pdf
.

*  Related material — A new Ariana Grande video and . . .

a recent digital artwork, “Code Girl,” with accompanying story —

Friday, August 21, 2020

Logic and Efficiency

Filed under: General — m759 @ 11:14 am

From remarks in  this  journal on Aug. 7 —

“You’ve got to pick up every stitch.” — Donovan

“Looking carefully at Golay’s code
is like staring into the sun.”

— Richard Evan Schwartz

Friday, August 7, 2020

Primary Color

Filed under: General — Tags: , — m759 @ 12:25 pm

From a Log24 search for Schwartz + “The Sun”

“Looking carefully at Golay’s code
is like staring into the sun.”

— Richard Evan Schwartz

Monday, July 13, 2020

The Lexicographic Octad Generator (LOG)*

The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.

By Steven H. Cullinane, July 13, 2020

Background —


The Miracle Octad Generator (MOG)
of R. T. Curtis (Conway-Sloane version) —

Embedding Change, Illustrated

A basis for the Golay code, excerpted from a version of
the code generated in lexicographic order, in

"Constructing the Extended Binary Golay Code"
Ben Adlam
Harvard University
August 9, 2011:

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

Below, each vector above has been reordered within
a 4×6 array, by Steven H. Cullinane, to form twelve
independent Miracle Octad Generator  vectors
(as in the Conway-Sloane SPLAG version above, in
which Curtis's earlier heavy bricks are reflected in
their vertical axes) —

01 02 03 04 05 . . . 20 21 22 23 24 -->

01 05 09 13 17 21
02 06 10 14 18 22
03 07 11 15 19 23
04 08 12 16 20 24

0000 0000 0000 0000 1111 1111 -->

0000 11
0000 11
0000 11
0000 11 as in the MOG.

0000 0000 0000 1111 0000 1111 -->

0001 01
0001 01
0001 01
0001 01 as in the MOG.

0000 0000 0011 0011 0011 0011 -->

0000 00
0000 00
0011 11
0011 11 as in the MOG.

0000 0000 0101 0101 0101 0101 -->

0000 00
0011 11
0000 00
0011 11 as in the MOG.

0000 0000 1001 0110 0110 1001 -->

0010 01
0001 10
0001 10
0010 01 as in the MOG.

0000 0011 0000 0011 0101 0110 -->

0000 00
0000 11
0101 01
0101 10 as in the MOG.

0000 0101 0000 0101 0110 0011 -->

0000 00
0101 10
0000 11
0101 01 as in the MOG.

0000 1001 0000 0110 0011 1010 -->

0100 01
0001 00
0001 11
0100 10 as in the MOG.

0001 0001 0001 0001 0111 1000 -->

0000 01
0000 10
0000 10
1111 10 as in the MOG.

0010 0001 0001 0010 0001 1101 -->

0000 01
0000 01
1001 00
0110 11 as in the MOG.

0100 0001 0001 0100 0100 1110 -->

0000 01
1001 11
0000 01
0110 00 as in the MOG.

1000 0001 0001 0111 0010 0100 -->

10 00 00
00 01 01
00 01 10
01 11 00 as in the MOG (heavy brick at center).

Update at 7:41 PM ET the same day —
A check of SPLAG shows that the above result is not new:
MOG in LOG embedding

And at 7:59 PM ET the same day —
Conway seems to be saying that at some unspecified point in the past,
M.J.T. Guy, examining the lexicographic Golay code,  found (as I just did)
that weight-8 lexicographic Golay codewords, when arranged naturally
in 4×6 arrays, yield certain intriguing visual patterns. If the MOG existed
at the time of his discovery, he would have identified these patterns as
those of the MOG.  (Lexicographic codes have apparently been
known since 1960, the MOG since the early 1970s.)

* Addendum at 4 AM ET  the next day —
See also Logline  (Walpurgisnacht 2013).

Monday, May 25, 2020

The Shimada Documents

Filed under: General — Tags: — m759 @ 12:24 am

(For Harlan Kane)

From Shimada’s notes on computational data at
http://www.math.sci.hiroshima-u.ac.jp/~shimada/
preprints/Edge/PaperEdge/compdataEdge.pdf

“C24 is the list of codewords of the extended
binary Golay code C24.  Each codeword is expressed
by a subset of the set M  of the positions [1, . . . , 24]
of MOG.”

Saturday, May 2, 2020

Turyn’s Octad Theorem: The Next Level*

Filed under: General — Tags: , , — m759 @ 11:24 am

From  the obituary of a game inventor  who reportedly died
on Monday, February 25, 2013 —

” ‘He was hired because of the game,’ Richard Turyn,
a mathematician who worked at Sylvania, told
the Washington Post in a 2004 feature on Diplomacy.”

* For the theorem, see Wolfram Neutsch,  Coordinates .
(Published by de Gruyter, 1996. See pp. 761-766.)

Having defined (pp. 751-752) the Miracle Octad Generator (MOG)
as a 4×6 array to be used with Conway’s “hexacode,” Neutsch says . . .

“Apart from the three constructions of the Golay codes
discussed at length in this book (lexicographic and via
the MOG or the projective line), there are literally
dozens of alternatives. For lack of space, we have to
restrict our attention to a single example. It has been
discovered by Turyn and can be connected in a very
beautiful way with the Miracle Octad Generator….

To this end, we consider the natural splitting of the MOG into
three disjoint octads L, M, R (‘left’, ‘middle’, and ‘right’ octad)….”

— From page 761

“The theorem of Turyn”  is on page 764

Saturday, March 7, 2020

The “Octad Group” as Symmetries of the 4×4 Square

From "Mathieu Moonshine and Symmetry Surfing" —

(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)

(1)  Institute for Theoretical Physics, ETH Zurich
(2)  Department of Mathematics, ETH Zurich

https://arxiv.org/abs/1609.09302v2 —

"This presentation of the symmetry groups Gi  is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the so-called octad group

G = (Z2)4  A8 .

It can be described as a maximal subgroup of M24
obtained by the setwise stabilizer of a particular
'reference octad' in the Golay code, which we take
to be O= {3,5,6,9,15,19,23,24} ∈ 𝒢24. The octad
subgroup is of order 322560, and its index in M24
is 759, which is precisely the number of
different reference octads one can choose."

This "octad group" is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, "Symmetry invariance in a
diamond ring
," by Steven H. Cullinane in Notices of the American Mathematical
Society
, February 1979, pages A-193, 194.

* The Galois tesseract .

Update of March 15, 2020 —

Conway and Sloane on the "octad group" in 1993 —

Monday, December 2, 2019

Gray Space

Filed under: General — Tags: , — m759 @ 3:00 pm

See as well a search for Gray Space in this journal.

Related material:  The Schwartz Omega .

“Looking carefully at Golay’s code
  is like staring into the sun.”

— Richard Evan Schwartz

Saturday, May 4, 2019

The Chinese Jars of Shing-Tung Yau

Filed under: General — Tags: , , , , — m759 @ 11:00 am

The title refers to Calabi-Yau spaces.

T. S. Eliot —

Four Quartets

. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.

A less "cosmic" but still noteworthy code — The Golay code.

This resides in a 12-dimensional space over GF(2).

Related material from Plato and R. T. Curtis

Counting symmetries with the orbit-stabilizer theorem

A related Calabi-Yau "Chinese jar" first described in detail in 1905

Illustration of K3 surface related to Mathieu moonshine

A figure that may or may not be related to the 4x4x4 cube that
holds the classical  Chinese "cosmic code" — the I Ching

ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3-pix.pdf

Friday, January 18, 2019

The Woke Grids …

Filed under: General — Tags: , , , — m759 @ 10:45 am

… as opposed to The Dreaming Jewels .

A July 2014 Amsterdam master's thesis on the Golay code
and Mathieu group —

"The properties of G24 and M24 are visualized by
four geometric objects:  the icosahedron, dodecahedron,
dodecadodecahedron, and the cubicuboctahedron."

Some "geometric objects"  — rectangular, square, and cubic arrays —
are even more fundamental than the above polyhedra.

A related image from a post of Dec. 1, 2018

Thursday, January 17, 2019

The Dreaming Jewels’ Nightmares

Filed under: General — Tags: — m759 @ 11:00 pm

Some Log24 posts related to Theodore Sturgeon's 1950 tale
of The Dreaming Jewels  have been tagged with that title.

For a purely mathematical approach to Sturgeon's concept see . . .

July 6, 2014, Amsterdam master's thesis on geometric models of the Golay code and Mathieu group

For some related nightmares, see July 2014 in this journal.

Tuesday, December 25, 2018

Simply

Filed under: General — Tags: — m759 @ 11:42 am
 

"So to obtain the isomorphism from L2(7) onto L3(2) we simply
multiply any given permutation of L2(7) by the affine translation
that restores to its rightful place."

— Sphere Packings, Lattices and Groups ,
by John H. Conway and N. J. A. Sloane.
First edition, 1988, published by Springer-Verlag New York, Inc.
Chapter 11 (by J. H. Conway), "The Golay Codes and the Mathieu Groups," 
Section 12, "The trio group 26:(S3×L2(7))"

Compare and contrast —

Why PSL(2,7) is isomorphic to GL(3.2)

This post was suggested by a New York Times  headline today —

Friday, December 14, 2018

Small Space Odyssey

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 11:00 am

References in recent posts to physical space and 
to mathematical space suggest a comparison.

Physical space is well known, at least in the world
of mass entertainment.

Mathematical space, such as the 12-dimensional
finite space of the Golay code, is less well known.

A figure from each space —

The source of the Conway-Sloane brick —

Quote from a mathematics writer —

“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

The former practice yields reflections like those of Conway and Sloane.
The latter practice is not recommended.

Sunday, December 2, 2018

Symmetry at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: , , , , — m759 @ 6:43 am

A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018

http://www.math.sci.hiroshima-u.ac.jp/
branched/files/2018/abstract/Aitchison.txt

 

Iain AITCHISON

Title:

Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II

Abstract:

Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness.

Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein's quartic curve, respectively), and Bring's genus 4 curve arises in Klein's description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare's homology 3-sphere, and Kummer's surface in real dimension 4 also play special roles.

In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay's binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois' exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol'd Trinities'.

Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set.

Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered.

Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato's concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective.

Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones' genus 70 Riemann surface previously proposed as a completion of an Arnol'd Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston's highly symmetric 6- and 8-component links, the latter related by Thurston to Klein's quartic curve.

See also yesterday morning's post, "Character."

Update: For a followup, see the next  Log24 post.

Tuesday, March 28, 2017

Bit by Bit

Filed under: General,Geometry — Tags: , , — m759 @ 11:45 am

From Log24, "Cube Bricks 1984" —

An Approach to Symmetric Generation of the Simple Group of Order 168

Also on March 9, 2017 —

For those who prefer graphic  art —

Broken Symmetries  in  Diamond Space  

Tuesday, March 7, 2017

Signature Backdrop

Filed under: General — Tags: , — m759 @ 9:00 pm

"The Bitter End’s signature stage backdrop —
a bare 150-year-old brick wall — helped distinguish it from
other popular bohemian hangouts like the Village Gate  
and the Village Vanguard. It appeared on the cover of
Peter, Paul and Mary’s first album."

The New York Times  this evening on a Sunday death

Commentary

“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

See also Schwartz in "The Omega Matrix," a post of 5 PM ET Sunday:

Monday, November 28, 2016

Interpenetration

Filed under: General — Tags: , — m759 @ 3:00 pm

Or:  A Candle for Sunrise  

(Continued)

Commentary —

“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

Monday, September 12, 2016

The Kummer Lattice

The previous post quoted Tom Wolfe on Chomsky's use of
the word "array." 

An example of particular interest is the 4×4  array
(whether of dots or of unit squares) —

      .

Some context for the 4×4 array —

The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .

Further background on the Kummer lattice:

Alice Garbagnati and Alessandra Sarti, 
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action." 
To appear in Rocky Mountain J. Math.

The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite  geometry, see the website
Finite Geometry of the Square and Cube.

Some further context

"To our knowledge, the relation of the Golay code
to the Kummer lattice is a new observation."

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 
"

As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface.  The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.

* Update of Sept. 14: "Uncoordinatized," but parametrized  by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.

Monday, May 23, 2016

23

Filed under: General,Geometry — m759 @ 8:25 am

IMAGE- R. D. Carmichael's 1931 construction of the Steiner system S(5, 8, 24)

Keywords: Golay, Mathieu, Steiner system, Carmichael, Harvard

Tuesday, April 19, 2016

The Folding

Filed under: General,Geometry — Tags: , , , — m759 @ 2:00 pm

(Continued

A recent post about the eightfold cube  suggests a review of two
April 8, 2015, posts on what Northrop Frye called the ogdoad :

As noted on April 8, each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding  a 1×8 array from Turyn's
1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again  yields the 2x2x2 eightfold cube .

Those who prefer an entertainment  approach to concepts of space
may enjoy a video (embedded yesterday in a story on theverge.com) —
"Ghost in the Shell: Identity in Space." 

Thursday, December 10, 2015

Art Brut?

Filed under: General — m759 @ 12:00 pm

Two images from December 2, 2015

From Log24 and from
the American Mathematical Society

Monday, September 28, 2015

Cracker Jack Prize

Filed under: General,Geometry — Tags: — m759 @ 11:00 pm

From a post of July 24, 2011

Mira Sorvino in 'The Last Templar'

A review —

“The story, involving the Knights Templar, the Vatican, sunken treasure,
the fate of Christianity and a decoding device that looks as if it came out of 
a really big box of medieval Cracker Jack, is the latest attempt to combine
Indiana Jones derring-do with ‘Da Vinci Code’ mysticism.”

— The New York Times

A feeble attempt at a purely mathematical "decoding device"
from this journal earlier this month

Image that may or may not be related to the extended binary Golay code and the large Witt design

For some background, see a question by John Baez at Math Overflow
on Aug. 20, 2015.

The nonexistence of a 24-cycle in the large Mathieu group
might discourage anyone hoping for deep new insights from
the above figure.

See Marston Conder's "Symmetric Genus of the Mathieu Groups" —

Saturday, September 19, 2015

Geometry of the 24-Point Circle

Filed under: General,Geometry — Tags: , , — m759 @ 1:13 am

The latest Visual Insight  post at the American Mathematical
Society website discusses group actions on the McGee graph,
pictured as 24 points arranged in a circle that are connected
by 36 symmetrically arranged edges.

Wikipedia remarks that

"The automorphism group of the McGee graph
is of order 32 and doesn't act transitively upon
its vertices: there are two vertex orbits of lengths
8 and 16."

The partition into 8 and 16 points suggests, for those familiar
with the Miracle Octad Generator and the Mathieu group M24,
the following exercise:

Arrange the 24 points of the projective line
over GF(23) in a circle in the natural cyclic order
, 1, 2, 3,  , 22, 0 ).  Can the McGee graph be
modeled by constructing edges in any natural way?

Image that may or may not be related to the extended binary Golay code and the large Witt design

In other words, if the above set of edges has no
"natural" connection with the 24 points of the
projective line over GF(23), does some other 
set of edges in an isomorphic McGee graph
have such a connection?

Update of 9:20 PM ET Sept. 20, 2015:

Backstory: A related question by John Baez
at Math Overflow on August 20.

Tuesday, March 24, 2015

Brouwer on the Galois Tesseract

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 pm

Yesterday's post suggests a review of the following —

Andries Brouwer, preprint, 1982:

"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)

Pages 8-9:

Substructures of S(5, 8, 24)

An octad is a block of S(5, 8, 24).

Theorem 5.1

Let B0 be a fixed octad. The 30 octads disjoint from B0
form a self-complementary 3-(16,8,3) design, namely 

the design of the points and affine hyperplanes in AG(4, 2),
the 4-dimensional affine space over F2.

Proof….

… (iv) We have AG(4, 2).

(Proof: invoke your favorite characterization of AG(4, 2) 
or PG(3, 2), say 
Dembowski-Wagner or Veblen & Young. 

An explicit construction of the vector space is also easy….)

Related material:  Posts tagged Priority.

Thursday, January 8, 2015

Gitterkrieg

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 am

(Continued)

From the abstract of a talk, "Extremal Lattices," at TU Graz
on Friday, Jan. 11, 2013, by Prof. Dr. Gabriele Nebe
(RWTH Aachen) —

"I will give a construction of the extremal even
unimodular lattice Γ of dimension 72  I discovered
in summer 2010. The existence of such a lattice
was a longstanding open problem. The
construction that allows to obtain the
minimum by computer is similar to the one of the
Leech lattice from E8 and of the Golay code from
the Hamming code (Turyn 1967)."

On an earlier talk by Nebe at Oberwolfach in 2011 —

"Exciting new developments were presented by
Gabriele Nebe (Extremal lattices and codes ) who
sketched the construction of her recently found
extremal lattice in 72 dimensions…."

Nebe's Oberwolfach slides include one on 
"The history of Turyn's construction" —

Nebe's list omits the year 1976. This was the year of
publication for "A New Combinatorial Approach to M24"
by R. T. Curtis, the paper that defined Curtis's 
"Miracle Octad Generator."

Turyn's 1967 construction, uncredited by Curtis, may have
been the basis for Curtis's octad-generator construction.

See Turyn in this journal.

Thursday, August 7, 2014

The Omega Mystery

Filed under: General — Tags: — m759 @ 11:00 pm

See a post,  The Omega Matrix, from the date of her death.

Related material:

"When Death tells a story, you really have to listen."
— Cover of The Book Thief

A scene from the film of the above book —

“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

Some context — "Mathematics, Magic, and Mystery" —
See posts tagged April Awareness 2014.

Friday, March 28, 2014

Blazing Thule

Filed under: General — Tags: , , — m759 @ 10:20 am

The title is suggested by a new novel (see cover below),
and by an unwritten book by Nabokov —

Siri Hustvedt, 'The Blazing World'.

Related material:

Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: General,Geometry — Tags: , , , , — m759 @ 12:24 pm

IMAGE- Two constructions, by Turyn/Curtis, and by Conway, of the Miracle Octad Generator

See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.

Related material on the Turyn-Curtis construction
from the University of Cambridge —

— Slide by "Dr. Parker" — Apparently Richard A. Parker —
Lecture 4, "Discovering M24," in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on "Sporadic and Related Groups."
See also the Parker lectures of 2012-2013 on the same topic.

A third construction of Curtis's 35  4×6  1976 MOG arrays would use
Cullinane's analysis of the 4×4 subarrays' affine and projective structure,
and point out the fact that Conwell's 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis's 1976 MOG.

See The Diamond Theorem,  Finite RelativityGalois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 23 —

Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1's, all other squares as 0's) of the 35  4×6 arrays of the 1976
Curtis MOG would then reveal*  the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35  MOG arrays.  For more
details of this "by-hand" construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction (such as Turyn's), not  by hand, of the
extended binary Golay code. See the Brouwer preprint quoted above.

* "Then a miracle occurs," as in the classic 1977 Sidney Harris cartoon.

Illustration of array addition from March 23 —

IMAGE- Discovery of the S_3 action on bricks in the Conwell-Cullinane 'by-hand' approach to octad-building

Thursday, March 13, 2014

Story

Filed under: General — m759 @ 12:00 pm

A Presbyterian meditation —

A scene from the film of the above book —

“Looking carefully at Golay’s code is like staring into the sun.”

Richard Evan Schwartz

For more of the story, see Golay in this journal.

Saturday, December 14, 2013

Beautiful Mathematics

Filed under: General,Geometry — Tags: , , , , — m759 @ 7:59 pm

The title, which I dislike, is taken from a 2011 publication
of the MAA, also sold by Cambridge University Press.

Some material relevant to the title adjective:

"For those who have learned something of higher mathematics, nothing could be more natural than to use the word 'beautiful' in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic."— Jim Holt, opening of a book review in the Dec. 5, 2013, issue of The New York Review of Books

Some relevant links—

The above list was updated on Jan. 31, 2014, to include the
"Strangeness" and "Hidden quirks" links.  See also a post of
​Jan. 31, 2014.

Update of March 9, 2014 —

The link "Simply defined abstractions" is to the construction of the Steiner
system S(5, 8, 24) described by R. T. Curtis in his 1976 paper defining the
Miracle Octad Generator. It should be noted that this construction is due
to Richard J. Turyn, in a 1967 Sylvania research report. (See Emily Jennings's
talk of 1 Nov. 2012.) Compare  the Curtis construction, written in 1974,
with the Turyn construction of 1967 as described in Sphere Packings, Lattices
and Groups , by J. H. Conway and N. J. A. Sloane (first published in 1988).

Monday, August 5, 2013

Wikipedia Updates

Filed under: General,Geometry — Tags: , — m759 @ 12:30 pm

I added links today in the following Wikipedia articles:

The links will probably soon be deleted,
but it seemed worth a try.

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — Tags: , , — m759 @ 4:30 am

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Tuesday, January 24, 2012

The Infinity Point

Filed under: General,Geometry — Tags: , — m759 @ 2:20 pm

From Labyrinth of the Line (March 2, 2011)—

"… construct the Golay code by taking the 24 points
to be the points of the projective line F23 ∪ {}…."

— Robert A. Wilson

A simpler projective line— a Galois geometry
model of the line F2 ∪ {}—

Image- The Three-Point Line: A Finite Projective Geometry

Here we may consider  to be modeled*
by the third square above— the Galois window .

* Update of about 1 AM Jan. 25, 2012—
  This infinity-modeling is of course a poetic conceit,
  not to be taken too seriously. For a serious 
  discussion of points at infinity and finite fields,
  see (for instance) Daniel Bump's "The Group GL(2)."

Saturday, September 3, 2011

The Galois Tesseract (continued)

A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).

Here is some supporting material—

http://www.log24.com/log/pix11B/110903-Carmichael-Conway-Curtis.jpg

The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.

The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.

The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.

The affine structure appears in the 1979 abstract mentioned above—

IMAGE- An AMS abstract from 1979 showing how the affine group AGL(4,2) of 322,560 transformations acts on a 4x4 square

The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978

IMAGE- Projective-space structure and Latin-square orthogonality in a set of 35 square arrays

See also a 1987 article by R. T. Curtis—

Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Update of Sept. 4— This post is now a page at finitegeometry.org.

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.

Monday, January 5, 2009

Monday January 5, 2009

Filed under: General,Geometry — Tags: , , , , — m759 @ 9:00 pm

A Wealth of
Algebraic Structure

A 4x4 array (part of chessboard)

A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4×4 square is now available online ($20):

Further elementary techniques using the miracle octad generator
, by R. T. Curtis. Abstract:

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

 

(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353, doi:10.1017/S0013091500004600.

(Published online by Cambridge University Press 19 Dec 2008.)

In the above article, Curtis explains how two-thirds of his 4×6 MOG array may be viewed as the 4×4 model of the four-dimensional affine space over GF(2).  (His earlier 1974 paper (below) defining the MOG discussed the 4×4 structure in a purely combinatorial, not geometric, way.)

For further details, see The Miracle Octad Generator as well as Geometry of the 4×4 Square and Curtis’s original 1974 article, which is now also available online ($20):

A new combinatorial approach to M24, by R. T. Curtis. Abstract:

“In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.”

 

(Received June 15 1974)

Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075.

(Published online by Cambridge University Press 24 Oct 2008.)

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Click for details.

Thursday, March 6, 2008

Thursday March 6, 2008

Filed under: General,Geometry — Tags: , — m759 @ 12:00 pm

This note is prompted by the March 4 death of Richard D. Anderson, writer on geometry, President (1981-82) of the Mathematical Association of America (MAA), and member of the MAA's Icosahedron Society.

Royal Road

"The historical road
from the Platonic solids
to the finite simple groups
is well known."

— Steven H. Cullinane,
November 2000,
Symmetry from Plato to
the Four-Color Conjecture

Euclid is said to have remarked that "there is no royal road to geometry." The road to the end of the four-color conjecture may, however, be viewed as a royal road from geometry to the wasteland of mathematical recreations.* (See, for instance, Ch. VIII, "Map-Colouring Problems," in Mathematical Recreations and Essays, by W. W. Rouse Ball and H. S. M. Coxeter.) That road ended in 1976 at the AMS-MAA summer meeting in Toronto– home of H. S. M. Coxeter, a.k.a. "the king of geometry."

See also Log24, May 21, 2007.

A different road– from Plato to the finite simple groups– is, as I noted in November 2000, well known. But new roadside attractions continue to appear. One such attraction is the role played by a Platonic solid– the icosahedron– in design theory, coding theory, and the construction of the sporadic simple group M24.

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

— "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics, Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))

This Steiner system is closely connected to M24 and to the extended binary Golay code. Brouwer gives an elegant construction of that code (and therefore of  M24):

"Let N be the adjacency matrix of the icosahedron (points: 12 vertices, adjacent: joined by an edge). Then the rows of the 12×24 matrix (I  J-N) generate the extended binary Golay code." [Here I is the identity matrix and J is the matrix of all 1's.]

Op. cit., p. 719

Related material:

Finite Geometry of
the Square and Cube

and
Jewel in the Crown

"There is a pleasantly discursive
treatment of Pontius Pilate's
unanswered question
'What is truth?'"
— H. S. M. Coxeter, 1987,
introduction to Trudeau's
"story theory" of truth

Those who prefer stories to truth
may consult the Log24 entries
 of March 1, 2, 3, 4, and 5.

They may also consult
the poet Rubén Darío:

Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.


* For a road out of this wasteland, back to geometry, see The Kaleidoscope Puzzle and Reflection Groups in Finite Geometry.

Thursday, May 31, 2007

Thursday May 31, 2007

Filed under: General,Geometry — Tags: — m759 @ 8:06 pm

Blitz by anonymous
New Delhi user

From Wikipedia on 31 May, 2007:

Shown below is a list of 25 alterations to Wikipedia math articles made today by user 122.163.102.246.

All of the alterations involve removal of links placed by user Cullinane (myself).

The 122.163… IP address is from an internet service provider in New Delhi, India.

The New Delhi anonymous user was apparently inspired by an earlier blitz by Wikipedia administrator Charles Matthews. (See User talk: Cullinane.)

Related material:

Ashay Dharwadker and Usenet Postings
and Talk: Four color theorem/Archive 2.
See also some recent comments from 122.163…
at Talk: Four color theorem.

May 31, 2007, alterations by
user 122.163.102.246:

  1. 17:17 Orthogonality (rm spam)
  2. 17:16 Symmetry group (rm spam)
  3. 17:14 Boolean algebra (rm spam)
  4. 17:12 Permutation (rm spam)
  5. 17:10 Boolean logic (rm spam)
  6. 17:08 Gestalt psychology (rm spam)
  7. 17:05 Tesseract (rm spam)
  8. 17:02 Square (geometry) (rm spam)
  9. 17:00 Fano plane (rm spam)
  10. 16:55 Binary Golay code (rm spam)
  11. 16:53 Finite group (rm spam)
  12. 16:52 Quaternion group (rm spam)
  13. 16:50 Logical connective (rm spam)
  14. 16:48 Mathieu group (rm spam)
  15. 16:45 Tutte–Coxeter graph (rm spam)
  16. 16:42 Steiner system (rm spam)
  17. 16:40 Kaleidoscope (rm spam)
  18. 16:38 Efforts to Create A Glass Bead Game (rm spam)
  19. 16:36 Block design (rm spam)
  20. 16:35 Walsh function (rm spam)
  21. 16:24 Latin square (rm spam)
  22. 16:21 Finite geometry (rm spam)
  23. 16:17 PSL(2,7) (rm spam)
  24. 16:14 Translation plane (rm spam)
  25. 16:13 Block design test (rm spam)

The deletions should please Charles Matthews and fans of Ashay Dharwadker’s work as a four-color theorem enthusiast and as editor of the Open Directory sections on combinatorics and on graph theory.

There seems little point in protesting the deletions while Wikipedia still allows any anonymous user to change their articles.

Cullinane 23:28, 31 May 2007 (UTC)

Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — Tags: , , , — m759 @ 9:26 am

Serious

"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."

Charles Matthews at Wikipedia, Oct. 2, 2006

"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

— G. H. Hardy, A Mathematician's Apology

Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:

Affine group‎
Reflection group‎
Symmetry in mathematics‎
Incidence structure‎
Invariant (mathematics)‎
Symmetry‎
Finite geometry‎
Group action‎
History of geometry‎

This would appear to be a fairly large complex of mathematical ideas.

See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:

Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines,  sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.

Sunday, June 15, 2003

Sunday June 15, 2003

Filed under: General,Geometry — Tags: , , — m759 @ 3:00 pm

Readings for Trinity Sunday

  1. Triune knot:
    Problems in Combinatorial Group Theory, 7 and 8, in light of the remark in Section 8.3 of Lattice Polygons and the Number 12 
  2. Cardinal Newman:
    Sermon 24
  3. Simon Nickerson:
    24=8×3.

For more on the structure
discussed by Nickerson, see

Raiders of the Lost Matrix:

For theology in general, see

Jews Telling Stories.

Confession in 'The Seventh Seal'

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