From the above image: "/gds_rip/" —
Related geek lore:
Wednesday, June 28, 2023
Tuesday, June 27, 2023
The Representation of Minus One…
Continued from February 6, 2014.
This flashback to 2014 was prompted by the following search history —
Related logic —
Related material — "Boolean Functions" in this journal.
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Wednesday, June 14, 2023
From Mysticism to Mathematics…
Continued from October 6, 2022 —
A paper from an August 2017 Melbourne conference
on artificial intelligence —
See as well a Log24 search for Boolean functions.
A check on the date of the above paper's presentation —
From this journal on that date —
Happy 10th birthday to the hashtag.
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Monday, February 27, 2023
Friday, October 7, 2022
From Mysticism to Mathematics… And Back Again
See the previous post as well as posts now tagged Soul and Spirit.
|
Soul
|
Spirit
|
The mirror has two faces (at least).
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Thursday, October 6, 2022
From Mysticism to Mathematics
[Klein, 1983] S. Klein.
"Analogy and Mysticism and the Structure of Culture
(and Comments & Reply)"
Current Anthropology , 24 (2):151–180, 1983.
The citation above is from a 2017 paper —
"Analogy-preserving Functions:
A Way to Extend Boolean Samples,"
by M. Couceiro, N. Hug, H. Prade, G, Richard.
26th International Joint Conference on Artificial Intelligence
(IJCAI 2017), Aug. 2017, Melbourne, Australia. pp.1-7, ff.
That 2017 paper discusses Boolean functions .
Some more-recent remarks on these functions
as pure mathematics —
"On the Number of Affine Equivalence Classes
of Boolean Functions," by Xiang-dong Hou,
arXiv:2007.12308v2 [math.CO]. Rev. Aug. 18, 2021.
See also other posts now tagged Analogy and Mysticism.
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Thursday, September 29, 2022
The 4×6 Problem*
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?
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Wednesday, September 28, 2022
Bitspace Note
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.
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Wednesday, September 21, 2022
Outside the White Cube
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Tuesday, September 20, 2022
Raiders of the Lost Space… Continues.
From "Raiders of the Lost Space," Sept. 11, 2022 —
A related technique appears in a 1989 paper by Cheng and Sloane
that I saw for the first time today:
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Motif-Space Updates
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
|
| 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 —
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Monday, September 19, 2022
From a Literature Search for Binary [24, 8, 8] Codes
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:
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Friday, September 16, 2022
Symmetric Generation
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.
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Wednesday, September 14, 2022
A Linear Code with 4×6 Symmetry
The exercise of 9/11 continues . . .
As noted in an update at the end of the 9/11 post,
these 24 motifs, along with 3 bricks and 4 half-arrays,
generate a linear code of 12 dimensions. I have not
yet checked the code's minimum weight.
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Tuesday, September 13, 2022
A Helpful Survey of the Literature
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."
"Remember, remember the fifth of November"
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Cinema News
From a search in this journal for Godard —
"I perceived . . . cinema is that which is between things,
not things [themselves] but between one and another."
— Jean-Luc Godard, "Introduction à une véritable histoire
du cinéma," Albatros , Paris, 1980, p. 145
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“We Got This Covered”
The previous post's quotation of the word "leitmotif" suggests a review:
See as well Sunday's post "Raiders of the Lost Space."
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Monday, September 12, 2022
“Hard Boiled” (Action Movie Title, Hong Kong, 1992)
The "all-time great actioner" of the above news story is "Hard Boiled,"
a 1992 Hong Kong action film by John Woo. Related art —
Revised version of the
New Yorker cover of 5/21/07
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Sunday, September 11, 2022
Raiders of the Lost Space
From 1981 —
From today —
Update —
A Magma check of the motif-generated space shows that
its dimension is only 8, not 12 as with the MOG space.
Four more basis vectors can be added to the 24 motifs to
bring the generated space up to 12 dimensions: the left
brick, the middle brick, the top half (2×6), the left half (4×3).
I have not yet checked the minimum weight in the resulting
12-dimensional 4×6 bit-space.
— SHC 4 PM ET, Sept. 12, 2022.
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Monday, November 5, 2012
For the Hollow Men
"Remember, remember the Fifth of November."
Very well. See a post of Nov. 5, 2005, and the related posts
Shadows, Cuber, Cube Partitions, and Cube Review.
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Saturday, November 5, 2005
Saturday November 5, 2005
Contrapuntal Themes
in a Shadowland
(See previous entry.)
Douglas Hofstadter on his magnum opus:
"… I realized that to me, Gödel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book."
Hofstadter's cover
Here are three patterns,
"shadows" of a sort,
derived from a different
"central object":
For details, see
Solomon's Cube.
Related material:
The reference to a
"permutation fugue"
(pdf) in an article on
Gödel, Escher, Bach.
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