Add Construction of Hadamard matrices up to order 664

[MatteoCati]

With #34690, all the Hadamard matrices of order <=288 have been added. The next task is to implement matrices of order up to 664 (668 is the first order for which no construction is known). This will require implementing https://core.ac.uk/download/pdf/82189436.pdf, as well as adding some more data for the constructions that are already present.

Depends on #34690

CC: @dimpase

Component: combinatorics

Author: Matteo Cati

Branch/Commit: 1f624a4

Reviewer: Dima Pasechnik

Issue created by migration from https://trac.sagemath.org/ticket/34807

[MatteoCati]

Commit: 7e803fb

[MatteoCati]

Last 10 new commits:

8eb2285	Add spence construction to general construction function
7d9100b	Add construction for base sequences
3ac2cc2	Generalize construction of T-sequences to work with base sequences
733065b	Add Turyn type sequences smallcases
369f513	Add construction for small cases of base sequences
8ba531e	Use base sequences in general T-sequences construction
129351f	Add williamson type matrices of order 7
8adf237	Add williamson type matrices of order 9
5d329b3	Add construction of skew supplementary difference sets
7e803fb	Add construction of skew Hadamard matrices from skew SDS

[MatteoCati]

Branch: u/gh-MatteoCati/add_hadamard_matrices_up_to_664

[sagetrac-git]

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

cb9b0ca

fix documentation in difference_family

[sagetrac-git]

Changed commit from 7e803fb to cb9b0ca

[sagetrac-git]

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

b51cc14

fix seealso directive in T sequences docs

[sagetrac-git]

Changed commit from cb9b0ca to b51cc14

[dimpase]

comment:4

Wow! That's impressive! Could you say a bit what's in that almost 3000 lines you add here?

[sagetrac-git]

Changed commit from b51cc14 to 2cb617c

[sagetrac-git]

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

133e9d1	Fix failing test in base_sequences_smallcases
2cb617c	Fix typos in documentations

[MatteoCati]

comment:6

• I've implemented Spence construction in hadamard_matrix.py. This uses the supplementary difference sets created by supplementary_difference_set in difference_faimilies.py. In the same file, the relative difference sets used by Spence to construct supplementary difference sets are created by the function relative_difference_set.
• In t_sequences.py, I've added some new T-sequences. These are constructed from base sequences using the function T_sequences_construction_from_base_sequences (this function was already present, but was being used only for the case when p=1). Also, base sequences are constructed from Turyn type sequences using the (new) function base_sequences_construction (I've also added the function is_base_sequences_tuple, to check that the construction is correct). Turyn type sequences are stored directly in turyn_type_sequences_smallcases.
• Lastly, I've added in difference_family.py the function skew_supplementary_difference_set which creates skew SDS (as described by Djokovic in https://ajc.maths.uq.edu.au/pdf/10/ocr-ajc-v10-p259.pdf), which are used by GS_skew_hadamard_smallcases in hadamard_matrix.py to create new skew Hadamard matrices.

Also, note that the older commits (up until bd025a2) are coming from ticket #34690.

[dimpase]

comment:7

Does this mean that we also have new (for Sage) orders for skew Hadamard matrices?

[MatteoCati]

comment:8

Yes, I've added skew Hadamard matrices of order 268, 412, 452, 508, 604, 628, 652, 724, 964.

[sagetrac-git]

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

e231e5e	Add construction for small cases of base sequences
fcdcbb7	Use base sequences in general T-sequences construction
df544a0	Add williamson type matrices of order 7
35fffda	Add williamson type matrices of order 9
8097830	Add construction of skew supplementary difference sets
fbc27fc	Add construction of skew Hadamard matrices from skew SDS
7a9e002	fix documentation in difference_family
eb3474d	fix seealso directive in T sequences docs
3d6f3fb	Fix failing test in base_sequences_smallcases
1f624a4	Fix typos in documentations

[sagetrac-git]

Changed commit from 2cb617c to 1f624a4

[MatteoCati]

comment:10

Rebased over latest beta.

[dimpase]

comment:11

Replying to Matteo Cati:

Yes, I've added skew Hadamard matrices of order 268, 412, 452, 508, 604, 628, 652, 724, 964.

nice.

How about bringing Sage up to date w.r.t. skew Hadamard matrices too?
I actually don't know what the 1st unknown order is in this case (something that's probably mentioned in the literature, I didn't try too hard to find this),
but I noticed that Sage does not know about orders 76, 100, and 116, whereas examples are known, see e.g.

https://mathscinet.ams.org/mathscinet-getitem?mr=2416980

[dimpase]

Reviewer: Dima Pasechnik

[MatteoCati]

comment:14

Replying to Dima Pasechnik:

How about bringing Sage up to date w.r.t. skew Hadamard matrices too?

Okay, I will open a new ticket (#34848).

[vbraun]

Changed branch from u/gh-MatteoCati/add_hadamard_matrices_up_to_664 to 1f624a4