Hadamard Matrices - Revision history

Gfis: programs

2025-08-09T10:04:13Z

programs

← Older revision		Revision as of 10:04, 9 August 2025
Line 33:		Line 33:
	b11e8e		b11e8e
	be11e8		be11e8
	The [https://teherba.org/threejs/hapyramid.html 3D model] was developped with the great Javascript package [https://threejs.org/ threejs.org]. The matrices are represented by little cubes whose opacities resp. colors are given by the hexadecimal encoding described above. The matrices are then stacked as a pyramid.		The [https://teherba.org/threejs/hapyramid.html 3D model] was developped with the great Javascript package [https://threejs.org/ threejs.org]. The matrices are represented by little cubes whose opacities resp. colors are given by the hexadecimal encoding described above. The matrices are then stacked as a pyramid. The cubes inside the pyramid are omitted since from outside they cannot be seen anyway.

	The model turns slowly by itself. It can be rotated with the mouse cursor, and with the mouse wheel the user can zoom in or out.		The model turns slowly by itself. It can be rotated with the mouse cursor, and with the mouse wheel the user can zoom in or out.
Line 44:		Line 44:
	print(i, "========")		print(i, "========")
	print(H.str())		print(H.str())

			The model uses mainly the following programs on [https://github.com/gfis/fasces/tree/master/hadamard github.com/gfis/fasces]:
			* hapyramid.html
			* hapyramid_world.js
			* hapyramid_blocks.js (generated by hapy_blocks.pl)
			There is a <code>makefile</code> that controls the whole generation process.

Gfis: turning

2025-08-09T09:49:58Z

turning

← Older revision		Revision as of 09:49, 9 August 2025
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	Hadamard matrices have many applications, among them error-correcting codes.		Hadamard matrices have many applications, among them error-correcting codes.
	====The conjecture====		====The conjecture====
	[[wikipedia:Jacques_Hadamard\|Jacques Hadamard]] (1865 - 1963) himself proved that, apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4. In 1867 Sylvester gave a construction for all Hadamard matrices of order 2<sup>k</sup>, k >= 2. Paley published two construction methods for two special clases of orders in 1933. The smallest order that could not be constructed by a combination of Sylvester's and Paley's methods was 92. Several other authors published specialized constructions and raised the minimal unknown order to 428, and since 2005 that order is 668.		[[wikipedia:Jacques_Hadamard\|Jacques Hadamard]] (1865 - 1963) himself proved that, apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4. In 1867 Sylvester gave a construction for all Hadamard matrices of order 2<sup>k</sup>, k >= 2. Paley published two construction methods for two special clases of orders in 1933. The smallest order that could not be constructed by a combination of Sylvester's and Paley's methods was 92. Several other authors published specialized constructions and raised the minimal unknown order to 428, and since 2005 that order is 668.

	Mathematicians believe, but still cannot prove, that Hadamard matrices of all orders 4*n can be constructed.		Mathematicians believe, but still cannot prove, that Hadamard matrices of all orders 4*n can be constructed.
	====The [https://teherba.org/threejs/hapyramid.html 3D model] ====		====The [https://teherba.org/threejs/hapyramid.html 3D model] ====
	Since the 0/1-entries of the matrices quickly appear as random noise, I tried to visualize the inner structure of the matrices in a better way, and one result is a 3D model for the 11 matrices with order 4, 8, 12, ... 44.		Since the 0/1-entries of the matrices quickly appear as random noise, I tried to visualize the inner structure of the matrices in a better way, and one result is a 3D model for the 22 matrices with order 4, 8, 12, ... 88.

	As one can see in the examples above, there basic building blocks of 2x2 entries. In a first step, the two rows are encoded as binary values:		As one can see in the examples above, there basic building blocks of 2x2 entries. In a first step, the two rows are encoded as binary values:
Line 33:		Line 33:
	b11e8e		b11e8e
	be11e8		be11e8
	The [https://teherba.org/threejs/hapyramid.html 3D model] was developped with the great Javascript package [https://threejs.org/ threejs.org]. The 11 matrices are represented by little cubes whose opacities resp. colors are given by the hexadecimal encoding described above. The matrices are then stacked as a pyramid.		The [https://teherba.org/threejs/hapyramid.html 3D model] was developped with the great Javascript package [https://threejs.org/ threejs.org]. The matrices are represented by little cubes whose opacities resp. colors are given by the hexadecimal encoding described above. The matrices are then stacked as a pyramid.

			The model turns slowly by itself. It can be rotated with the mouse cursor, and with the mouse wheel the user can zoom in or out.
	==== Technical details ====		==== Technical details ====
	The article https://arxiv.org/html/2411.18897v1 describes in detail how to compute the matrices with SageMath. Online you can run https://sagecell.sagemath.org/ with the following program:		The article https://arxiv.org/html/2411.18897v1 describes in detail how to compute the matrices with SageMath. Online you can run https://sagecell.sagemath.org/ with the following program:

Gfis: sup

2025-08-09T08:41:27Z

sup

← Older revision		Revision as of 08:41, 9 August 2025
Line 17:		Line 17:
	Hadamard matrices have many applications, among them error-correcting codes.		Hadamard matrices have many applications, among them error-correcting codes.
	====The conjecture====		====The conjecture====
	[[wikipedia:Jacques_Hadamard\|Jacques Hadamard]] (1865 - 1963) himself proved that, apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4. In 1867 Sylvester gave a construction for all Hadamard matrices of order 2<sup>k, k >= 2. Paley published two construction methods for two special clases of orders in 1933. The smallest order that could not be constructed by a combination of Sylvester's and Paley's methods was 92. Several other authors published specialized constructions and raised the minimal unknown order to 428, and since 2005 that order is 668.		[[wikipedia:Jacques_Hadamard\|Jacques Hadamard]] (1865 - 1963) himself proved that, apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4. In 1867 Sylvester gave a construction for all Hadamard matrices of order 2<sup>k</sup>, k >= 2. Paley published two construction methods for two special clases of orders in 1933. The smallest order that could not be constructed by a combination of Sylvester's and Paley's methods was 92. Several other authors published specialized constructions and raised the minimal unknown order to 428, and since 2005 that order is 668.

	Mathematicians believe, but still cannot prove, that Hadamard matrices of all orders 4*n can be constructed.		Mathematicians believe, but still cannot prove, that Hadamard matrices of all orders 4*n can be constructed.

Gfis: /* The 3D model */

2025-08-09T08:40:48Z

The 3D model

← Older revision		Revision as of 08:40, 9 August 2025
Line 17:		Line 17:
	Hadamard matrices have many applications, among them error-correcting codes.		Hadamard matrices have many applications, among them error-correcting codes.
	====The conjecture====		====The conjecture====
	[[wikipedia:Jacques_Hadamard\|Jacques Hadamard]] (1865 - 1963) himself proved that, apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4. In 1867 Sylvester gave a construction for all Hadamard matrices of order 2<sup>k, k >= 2. Paley published two construction methods for two special clases of orders in 1933. The smallest order that could not be constructed by a combination of Sylvester's and Paley's methods was 92. Several other authors published specialized constructions and raised the minimal unknown order to 428, and since 2005 that order is ~~'''~~668~~'''~~.		[[wikipedia:Jacques_Hadamard\|Jacques Hadamard]] (1865 - 1963) himself proved that, apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4. In 1867 Sylvester gave a construction for all Hadamard matrices of order 2<sup>k, k >= 2. Paley published two construction methods for two special clases of orders in 1933. The smallest order that could not be constructed by a combination of Sylvester's and Paley's methods was 92. Several other authors published specialized constructions and raised the minimal unknown order to 428, and since 2005 that order is 668.

	Mathematicians believe, but still cannot prove, that Hadamard matrices of all orders 4*n can be constructed.		Mathematicians believe, but still cannot prove, that Hadamard matrices of all orders 4*n can be constructed.
	====The [https://teherba.org/threejs/hapyramid.html 3D model] ====		====The [https://teherba.org/threejs/hapyramid.html 3D model] ====
	Since the 0/1-entries of the matrices quickly appear as random noise, I tried to visualize the inner structure of the matrices in a better way, and one result is a 3D model for the 11 matrices with 4, 8, 12, ... 44 ~~= 4*n for n = 1..11~~.		Since the 0/1-entries of the matrices quickly appear as random noise, I tried to visualize the inner structure of the matrices in a better way, and one result is a 3D model for the 11 matrices with order 4, 8, 12, ... 44.

	As one can see in the examples above, there basic building blocks of 2x2 entries. In a first step, the two rows are encoded as binary values:		As one can see in the examples above, there basic building blocks of 2x2 entries. In a first step, the two rows are encoded as binary values:
	a b -> a b c d 1 1 -> 1 1 1 0		a b -> a b c d 1 1 -> 1 1 1 0 -> e
	c d 1 0		c d 1 0
	The resulting 4-bit pattern is encoded as a hexadecimal digit 0-9a-f. Now it is much easier to recognize the structure in the 4 example matrices above:		The resulting 4-bit pattern is encoded as a hexadecimal digit 0-9a-f. Now it is much easier to recognize the structure in the 4 example matrices above:
Line 33:		Line 33:
	b11e8e		b11e8e
	be11e8		be11e8
	The [https://teherba.org/threejs/hapyramid.html 3D model] was developped with the great Javascript package [https://threejs.org/ threejs.org]. The ~~first~~ 11 matrices are represented by little cubes whose opacities resp. colors are given by the hexadecimal encoding described above. The matrices are then stacked as a pyramid.		The [https://teherba.org/threejs/hapyramid.html 3D model] was developped with the great Javascript package [https://threejs.org/ threejs.org]. The 11 matrices are represented by little cubes whose opacities resp. colors are given by the hexadecimal encoding described above. The matrices are then stacked as a pyramid.
	==== Technical details ====		==== Technical details ====
	The article https://arxiv.org/html/2411.18897v1 describes in detail how to compute the matrices with SageMath. Online you can run https://sagecell.sagemath.org/ with the following program:		The article https://arxiv.org/html/2411.18897v1 describes in detail how to compute the matrices with SageMath. Online you can run https://sagecell.sagemath.org/ with the following program:

Gfis: Hex encoding

2025-08-09T08:33:36Z

Hex encoding

← Older revision		Revision as of 08:33, 9 August 2025
Line 24:		Line 24:

	As one can see in the examples above, there basic building blocks of 2x2 entries. In a first step, the two rows are encoded as binary values:		As one can see in the examples above, there basic building blocks of 2x2 entries. In a first step, the two rows are encoded as binary values:
			a b -> a b c d 1 1 -> 1 1 1 0
	a b -> a b c d 1 1 -> 1 1 1 0		c d 1 0
			The resulting 4-bit pattern is encoded as a hexadecimal digit 0-9a-f. Now it is much easier to recognize the structure in the 4 example matrices above:
	c d 1 0		e ee eeee eddddd
			e1 e1e1 b8e11e
			ee11 be8e11
			e11e b1e8e1
			b11e8e
			be11e8
			The [https://teherba.org/threejs/hapyramid.html 3D model] was developped with the great Javascript package [https://threejs.org/ threejs.org]. The first 11 matrices are represented by little cubes whose opacities resp. colors are given by the hexadecimal encoding described above. The matrices are then stacked as a pyramid.
	==== Technical details ====		==== Technical details ====
	The article https://arxiv.org/html/2411.18897v1 describes in detail how to compute the matrices with SageMath. Online you can run https://sagecell.sagemath.org/ with the following program:		The article https://arxiv.org/html/2411.18897v1 describes in detail how to compute the matrices with SageMath. Online you can run https://sagecell.sagemath.org/ with the following program:

Gfis: 3D model

2025-08-09T08:17:53Z

3D model

← Older revision		Revision as of 08:17, 9 August 2025
Line 1:		Line 1:
	In the last chapter of Ian Stewart's book "The Great Mathematical Problems" (ISBN 978-1846683374), there is a section about [[wikipedia:Hadamard_matrix\|Hadamard matrices]]. These are square arrays whose entries are either 1 or 0 (Hadamard's original definition used −1 of 0). Each row in the matrix, when compared with some different row, must have exactly one half of the entries matching and one half non-matching~~. Apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4~~. The first ~~few~~ matrices are:		In the last chapter of Ian Stewart's book "The Great Mathematical Problems" (ISBN 978-1846683374), there is a section about [[wikipedia:Hadamard_matrix\|Hadamard matrices]]. These are square arrays whose entries are either 1 or 0 (Hadamard's original definition used −1 of 0). Each row in the matrix, when compared with some different row, must have exactly one half of the entries matching and one half non-matching. The first 4 matrices are:

	2x2 4x4 8x8 12x12		2x2 4x4 8x8 12x12
Line 17:		Line 17:
	Hadamard matrices have many applications, among them error-correcting codes.		Hadamard matrices have many applications, among them error-correcting codes.
	====The conjecture====		====The conjecture====
	[~~https~~:~~//en.wikipedia.org/wiki/~~Jacques_Hadamard Jacques Hadamard (1865 - 1963) himself proved that, apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4. In 1867 Sylvester gave a construction for all Hadamard matrices of order 2<sup>k, k >= 2. Paley published two construction methods for two special clases of orders in 1933. The smallest order that could not be constructed by a combination of Sylvester's and Paley's methods was 92. Several other authors published specialized constructions and raised the minimal unknown order to 428, and since 2005 that order is 668.		[[wikipedia:Jacques_Hadamard\|Jacques Hadamard]] (1865 - 1963) himself proved that, apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4. In 1867 Sylvester gave a construction for all Hadamard matrices of order 2<sup>k, k >= 2. Paley published two construction methods for two special clases of orders in 1933. The smallest order that could not be constructed by a combination of Sylvester's and Paley's methods was 92. Several other authors published specialized constructions and raised the minimal unknown order to 428, and since 2005 that order is '''668'''.

	Mathematicians believe, but still cannot prove that Hadamard matrices of all orders 4*n can be constructed.		Mathematicians believe, but still cannot prove, that Hadamard matrices of all orders 4*n can be constructed.
	====The 3D model ====		====The [https://teherba.org/threejs/hapyramid.html 3D model] ====
	~~That inspired me~~ to ~~develop~~ a 3D model for the 11 matrices with 4, 8, 12, ... 44 = 4*n for n = 1..11.		Since the 0/1-entries of the matrices quickly appear as random noise, I tried to visualize the inner structure of the matrices in a better way, and one result is a 3D model for the 11 matrices with 4, 8, 12, ... 44 = 4*n for n = 1..11.

			As one can see in the examples above, there basic building blocks of 2x2 entries. In a first step, the two rows are encoded as binary values:

			a b -> a b c d 1 1 -> 1 1 1 0

			c d 1 0

	==== Technical details ====		==== Technical details ====
Line 27:		Line 33:

	<code>from sage.combinat.matrices.hadamard_matrix import hadamard_matrix, skew_hadamard_matrix</code>		<code>from sage.combinat.matrices.hadamard_matrix import hadamard_matrix, skew_hadamard_matrix</code>
	for i in range(1,12):		for i in range(1,11):
	H = hadamard_matrix(4*i)		H = hadamard_matrix(4*i)
	print(i, "========")		print(i, "========")
	print(H.str())		print(H.str())
	~~The output can be reduced with:~~
	~~perl -pe "s{\-1}{\.}g; s{[\[\]\\|\+\-]}{}g; s{\A\s+\Z}{};" infile > outfile~~
	~~===3D Design===~~
	Since the edge lengths increase by 4, solid blocks of bxwxh = 1x1x2 will be used to represent a one in a matrix plane, and empty space for a zero (resp. -1). The matrix planes will be stacked as a pyramid, with the 44x44 matrix at the bottom and the 4x4 at the top. Maybe the 2x2 matrix becomes the real top, maybe with blocks 1x1x1?

Gfis: History

2025-08-09T08:02:18Z

History

← Older revision		Revision as of 08:02, 9 August 2025
Line 1:		Line 1:
	In the last chapter of Ian Stewart's book "The Great Mathematical Problems" (ISBN 978-1846683374), there is a section about [[wikipedia:Hadamard_matrix\|Hadamard matrices]]. These are square arrays whose entries are either 1 or 0 (Hadamard's original definition used −1 of 0). Each row in the matrix, when compared with some different row, must have exactly one half of the entries matching and one half non-matching. Apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4.		In the last chapter of Ian Stewart's book "The Great Mathematical Problems" (ISBN 978-1846683374), there is a section about [[wikipedia:Hadamard_matrix\|Hadamard matrices]]. These are square arrays whose entries are either 1 or 0 (Hadamard's original definition used −1 of 0). Each row in the matrix, when compared with some different row, must have exactly one half of the entries matching and one half non-matching. Apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4. The first few matrices are:

	~~<code>~~1 1 1 1 1 1~~</code>~~		2x2 4x4 8x8 12x12
			1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
			1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1
			1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 1 1
			1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 1 0 0 1 0 1 1 0
			1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0
			1 0 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1
			1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 0 1 1 0 0
			1 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 0 1
			1 0 0 0 0 0 1 1 1 0 1 1
			1 1 0 1 0 1 1 0 0 0 1 0
			1 0 1 1 0 0 0 0 1 1 1 0
			1 1 1 0 0 1 0 1 1 0 0 0

	~~<code>1 0 1 0 1 0</code>~~		Hadamard matrices have many applications, among them error-correcting codes.
			====The conjecture====
	~~<code>1 1 0 0<~~/~~code>~~		[https://en.wikipedia.org/wiki/Jacques_Hadamard Jacques Hadamard (1865 - 1963) himself proved that, apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4. In 1867 Sylvester gave a construction for all Hadamard matrices of order 2<sup>k, k >= 2. Paley published two construction methods for two special clases of orders in 1933. The smallest order that could not be constructed by a combination of Sylvester's and Paley's methods was 92. Several other authors published specialized constructions and raised the minimal unknown order to 428, and since 2005 that order is 668.

	<~~code~~>~~1 0 0 1</code~~>

			Mathematicians believe, but still cannot prove that Hadamard matrices of all orders 4*n can be constructed.
			====The 3D model ====
	That inspired me to develop a 3D model for the 11 matrices with 4, 8, 12, ... 44 = 4*n for n = 1..11.		That inspired me to develop a 3D model for the 11 matrices with 4, 8, 12, ... 44 = 4*n for n = 1..11.

	~~The model has the s~~

	==== Technical details ====		==== Technical details ====

Gfis at 07:40, 9 August 2025

2025-08-09T07:40:08Z

@@ Line 1: / Line 1: @@
-In Ian Stewart's book "Die letzten Rätsel der Mathematik" (ISBN 978-3499616945), at the end there is a section about Hadamard matrices. That inspired me to develop a 3D model for the first 12 matrices (of edge length 2, 4, 8, 12, ... 44).
+In the last chapter of Ian Stewart's book "The Great Mathematical Problems" (ISBN 978-1846683374), there is a section about [[wikipedia:Hadamard_matrix|Hadamard matrices]]. These are square arrays whose entries are either 1 or 0 (Hadamard's original definition used −1 of 0). Each row in the matrix, when compared with some different row, must have exactly one half of the entries matching and one half non-matching. Apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4.
-===Raw data===
+<code>1 1        1 1 1 1</code>
 The article https://arxiv.org/html/2411.18897v1 describes in detail how to compute the matrices with SageMath. Online you can run https://sagecell.sagemath.org/ with the following program:

Gfis: Gfis moved page Hadamard matrices to Hadamard Matrices: Misspelled title

2025-08-09T07:06:53Z

Gfis moved page Hadamard matrices to Hadamard Matrices: Misspelled title

← Older revision		Revision as of 07:06, 9 August 2025
(No difference)

Gfis: 3D Design

2025-07-19T20:44:57Z

3D Design

← Older revision		Revision as of 20:44, 19 July 2025
Line 1:		Line 1:
	In Ian Stewart's book "Die letzten Rätsel der Mathematik" (ISBN 978-3499616945), at the end there is a section about Hadamard matrices. That inspired me to develop a 3D model for the first 12 matrices (of edge length 2, 4, 8, 12, ... 44).		In Ian Stewart's book "Die letzten Rätsel der Mathematik" (ISBN 978-3499616945), at the end there is a section about Hadamard matrices. That inspired me to develop a 3D model for the first 12 matrices (of edge length 2, 4, 8, 12, ... 44).

			===Raw data===
	The article https://arxiv.org/html/2411.18897v1 describes in detail how to compute the matrices with SageMath. Online you can run https://sagecell.sagemath.org/ with the following program:		The article https://arxiv.org/html/2411.18897v1 describes in detail how to compute the matrices with SageMath. Online you can run https://sagecell.sagemath.org/ with the following program:

	<code>from sage.combinat.matrices.hadamard_matrix import hadamard_matrix, skew_hadamard_matrix</code>		<code>from sage.combinat.matrices.hadamard_matrix import hadamard_matrix, skew_hadamard_matrix</code>
			for i in range(1,12):
	~~<code>~~for i in range(1,12):~~</code>~~		H = hadamard_matrix(4*i)
			print(i, "========")
	~~<code>~~ H = hadamard_matrix(4*i)~~</code>~~		print(H.str())

	~~<code>~~ print(i, "========")~~</code>~~

	~~<code>~~print(H.str())~~</code>~~

	The output can be reduced with:		The output can be reduced with:
			perl -pe "s{\-1}{\.}g; s{[\[\]\\|\+\-]}{}g; s{\A\s+\Z}{};" infile > outfile
	~~<code>~~perl -pe "s{\-1}{\.}g; s{[\[\]\\|\+\-]}{}g; s{\A\s+\Z}{};" infile > outfile~~</code>~~		===3D Design===
			Since the edge lengths increase by 4, solid blocks of bxwxh = 1x1x2 will be used to represent a one in a matrix plane, and empty space for a zero (resp. -1). The matrix planes will be stacked as a pyramid, with the 44x44 matrix at the bottom and the 4x4 at the top. Maybe the 2x2 matrix becomes the real top, maybe with blocks 1x1x1?