Implement new orders of skew Hadamard matrices

[MatteoCati]

In #34807, new hadamard matrices up to order 664has been constructed (668 is the smallest unresolved case). the next step is to add more constructions for skew hadamrd matrices.

According to https://arxiv.org/pdf/2301.02751.pdf, the first unresolved case is 356.

Depends on #34807
Depends on #34892

CC: @dimpase

Component: combinatorics

Author: Matteo Cati

Branch/Commit: u/gh-MatteoCati/add_new_skew_hadamard_matrices @ 67b74bf

Reviewer: Dima Pasechnik

Issue created by migration from https://trac.sagemath.org/ticket/34848

[MatteoCati]

Branch: u/gh-MatteoCati/add_new_skew_hadamard_matrices

[MatteoCati]

Commit: 553a8d4

[MatteoCati]

Last 10 new commits:

67e9000	Add Spence skew hadamard matrix construction
4f7b28f	Add skew SDS of size 37 and 43
bc4d841	Add more skew SDS
868dcd9	Refactor GS_skew_hadamard_smallcases
4dedee0	Add skew SDS for matrix of order 388
eac0e72	Add skew hadamard matrix of order 188
2b5db0e	Add skew SDS of order 109, 145, 247
dd8b477	Fix whitespaces
4b852c2	Add skew Hadmard matrix order 324
553a8d4	Add 324 construction to skew hadamard function

[dimpase]

Attachment: hm.pdf.gz

reference for skew-Hadamard matrix of order 116

[dimpase]

comment:2

my OCR attempt didn't work on the most interesting part of the scan, so here is the original scan in pdf.

[sagetrac-git]

Changed commit from 553a8d4 to ea4a67c

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

e316c8d	Fix documentation for SDS
9ae928f	Add skew Hadamard matrix construction from good matrices
77f5b10	Add function for smallcases of good matrices
be38d08	Add skew matrices from good matrices to general function
5f8eddc	Add skew Hadamard matrix of order 236
ff1de29	Add skew Hadamard matrix of order 276
ea4a67c	Add Whiteman construction for skew Hadamard matrix of order 340

[MatteoCati]

comment:4

Added three new constructions:

• Whiteman contruction (used for matrix of order 340)
• Construction from good matrices (for order 100 and 116)
• Spence construction for skew hadamard matrices of order n=2(q+1) where q = 5 (mod 8)

Also, more data has been added for skew Supplementary Difference Sets, as well as for the Williamson-Goethals-Seidel construction. The Hadamard matrix of order 324 is implemented using skew SDS, but it is done in a separate function since the sets are created in a different way from all the others.

Up to order 668, all the skew Hadamard matrices can now be constructed with the exception of orders 356, 404, 428, 476, 596, 612, 668 (for which no construction is known), and 292 (for which the construction still needs to be implemented).

[sagetrac-git]

Changed commit from ea4a67c to 0e95f98

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

0e95f98

Fix good matrices construction

[sagetrac-git]

Changed commit from 0e95f98 to 5dd0439

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

5dd0439

Fix tests of construction of skew Hadamard matrices from good matrices

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. Last 10 new commits:

d4e35a9	Add function for smallcases of good matrices
98487bf	Add skew matrices from good matrices to general function
5c3e94d	Add skew Hadamard matrix of order 236
004cc4f	Add skew Hadamard matrix of order 276
e64051e	Add Whiteman construction for skew Hadamard matrix of order 340
d51a157	Fix good matrices construction
4ef8680	Fix tests of construction of skew Hadamard matrices from good matrices
29a6d86	Fix trailing whitespaces in difference_family.py
2116084	Fix trailing whitespaces in hadamard_matrix.py
5912ae4	Fix code style for hadamard_matrix.py

[sagetrac-git]

Changed commit from 5dd0439 to 5912ae4

[sagetrac-git]

Changed commit from 5912ae4 to 63101bc

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

c41e4d7	fixing the linter again
4c30a09	Merge branch 'fixing_linter' into add_new_skew_hadaard_matrices
63101bc	Add skew sds for skew Hadamard matrix of order 292

[MatteoCati]

comment:9

All known skew Hadamard matrices of order < 668 can now be constructed.
The orders for which no skew Hadamard matrix is currently known are: 356, 404, 428, 476, 596, 612, 668. (see https://arxiv.org/pdf/2301.02751.pdf).

[MatteoCati]

Changed dependencies from #34807 to #34807, #34892

[dimpase]

comment:10

the size 292 case ​63101bc should mention somewhere that it's from a personal communication by D. Djokovic.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

02b174f

Add reference for skew SDS of size 73

[sagetrac-git]

Changed commit from 63101bc to 02b174f

[dimpase]

comment:12

For the ease of testing, can we have is_skew_hadamard() function?

As well, an option skew= for the function which normalises Hadamard matrices?
(the result should be skew, and have all-1s first row - if started from a skew matrix)

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

0fd4540	normalise_hadamard can make matrix skew-normalized
f99daba	add function is_skew_hadamard_matrix

[sagetrac-git]

Changed commit from 02b174f to f99daba

[MatteoCati]

comment:14

I've added the skew option to normalise_hadamard, and modified is_hadamard_matrix so that if both skew and normalize are true, the function works correctly.

I've also added the is_skew_hadamard_matrix function.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

67b74bf

Fix is_skew_hadamard_matrix documentation

[sagetrac-git]

Changed commit from f99daba to 67b74bf

[dimpase]

Reviewer: Dima Pasechnik

[dimpase]

fixed by #34985