I could try enumerating orthogonal mates of F98aux31EFfDjHASPckbbhZCsweDB7qd3dcgExK6yTQ75FDD3, would that help? I need to modify my program. To enumerate all, it would take long time. Maybe for a boinc project?

This will help, but will not completely solve the problem.
There is a more interesting and solvable problem for BOINC - DB CF ODLS of order 9
https://boinc.tbrada.eu/forum_thread.php?id=3114

What did found is that there is 23040*8 isomorphisms of order 12 and 13, and 322560*8 isomorphisms of order 14.

See
https://oeis.org/A299784

It seems that a(n) = 2^m * m! * 4 for all n > 6.

Related:
https://oeis.org/A000316
https://oeis.org/A309283

Order 7:
x-fillings: 80
rules: 3
isotopes: 24*8 (192)
Order 8:
x-fillings: 4752
rules: 20
isotopes: 192*8 (1536)
Order 9:
x-fillings: 4752
rules: 20
isotopes: 192*8 (1536)
Order 10:
x-fillings: 440192
rules: 67
isotopes: 1920 *8 (15360)
Order 11:
x-fillings: 440192
rules: 67
isotopes: 1920 *8 (15360)
Order 12:
x-fillings: 59245120?
rules: 596?
isotopes: 23040*8 (184320)

The sequence 80, 4752, 440192 is A000316:
Count of x-fillings of DLS(n*2).
Two decks each have n kinds of cards, 2 of each kind. The first deck is laid out in order. The second deck is shuffled and laid out next to the first. A match occurs if a card from the second deck is next to a card of the same kind from the first deck. a(n) is the number of ways of achieving no matches.
Which is exactly what x-filling is. Again connecting Latin squares and card games.

Tomáš Brada
I posted the rules for SN DLS of order 12, 13 and 14
https://yadi.sk/d/6tkvilMwqzHDlA

Please check.

How many rules did you find for SN DLS of order 14?
I have found 278 rules so far. I have very few CF ODLS of this order.

I have found 911 rules so far.

How many rules did you find for SN DLS of order 14?
I have found 278 rules so far. I have very few CF ODLS of this order.

I have found 911 rules so far.

Please publish the corresponding 911 CFs.

I have found 1264 rules for SN ODLS of order 14 so far.
See
https://boinc.progger.info/odlk/forum_thread.php?id=162&postid=6340#6340

I have found 3483 rules for SN ODLS of order 14. The program was taking too long, I stopped it.
https://hr.tbrada.eu/tomas/rules14.txt

I have found 3483 rules for SN ODLS of order 14. The program was taking too long, I stopped it.
https://hr.tbrada.eu/tomas/rules14.txt

I need GFs ODLS (format 2) for verification.
Can you post?

I compared the first 20 rules, they do not match.
This is normal.

your

1 0 3 2 5 4 8 9 6 7 11 10 13 12
1 0 3 2 5 4 8 9 6 7 11 12 13 10
1 0 3 2 5 4 8 9 6 7 12 13 10 11
1 0 3 2 5 4 8 9 6 7 12 13 11 10
1 0 3 2 5 4 8 9 6 10 7 12 13 11
1 0 3 2 5 4 8 9 6 10 11 7 13 12
1 0 3 2 5 4 8 9 6 10 11 12 13 7
1 0 3 2 5 4 8 9 6 10 12 7 13 11
1 0 3 2 5 4 8 9 6 10 12 13 7 11
1 0 3 2 5 4 8 9 6 10 12 13 11 7
1 0 3 2 5 4 8 9 7 6 11 10 13 12
1 0 3 2 5 4 8 9 7 6 11 12 13 10
1 0 3 2 5 4 8 9 7 6 12 13 10 11
1 0 3 2 5 4 8 9 7 6 12 13 11 10
1 0 3 2 5 4 8 9 7 10 6 12 13 11
1 0 3 2 5 4 8 9 7 10 11 6 13 12
1 0 3 2 5 4 8 9 7 10 11 12 13 6
1 0 3 2 5 4 8 9 7 10 12 6 13 11
1 0 3 2 5 4 8 9 7 10 12 13 6 11
1 0 3 2 5 4 8 9 7 10 12 13 11 6
. . . . . . 

my

 1  0  3  2  5  4  8  9  6  7 11 10 13 12
 1  0  3  2  5  4  8  9  6  7 12 13 10 11
 1  0  3  2  5  4  8  9  7 10  6 12 13 11
 1  0  3  2  5  4  8  9  7 10 12 13 11  6
 1  0  3  2  5  4  8  9 10 11  6 12 13  7
 1  0  3  2  5  4  8  9 10 11  7  6 13 12
 1  0  3  2  5  4  8  9 10 12  6 13 11  7
 1  0  3  2  5  4  8  9 10 12  7 13 11  6
 1  0  3  2  5  4  8 10  6 11  7  9 13 12
 1  0  3  2  5  4  8 10  6 12  7 13  9 11
 1  0  3  2  5  4  8 10  6 12 11  9 13  7
 1  0  3  2  5  4  8 10  7 11  6 12 13  9
 1  0  3  2  5  4  8 10  7 11 12  6 13  9
 1  0  3  2  5  4  8 10  7 11 12  9 13  6
 1  0  3  2  5  4  8 10  7 12  9 13 11  6
 1  0  3  2  5  4  8 10  9  6 12 13  7 11
 1  0  3  2  5  4  8 10  9 12 13  6 11  7
 1  0  3  2  5  4  8 10  9 12 13  7  6 11
 1  0  3  2  5  4  8 10  9 12 13  7 11  6
 1  0  3  2  5  4  8 10 11  6 12 13  7  9
. . . . . . 

Harry White found 5225 rules for order 14 as well as for order 15.
See
https://boinc.progger.info/odlk/forum_thread.php?id=162&postid=6345

Tomáš Brada
Please confirm the number of D-transversals for this DLK

H8YURdMXZnihqsPkArWe9CwrZqH62bHYQ3MY81AmSxkqDU4UM8WFe3ncZMbpZr5

I counted 31313088 D-transversals with your program ortogonb.exe.
Is it correct?

See
https://boinc.progger.info/odlk/forum_thread.php?id=162&postid=6926

I will get to it tomorrow.
Hurray!

I will get to it tomorrow.
Hurray!

Hurray!

I will get to it tomorrow.
Hurray!

Have you got tomorrow yet? :)
Are you joking?

Hurray! :)

A task

This DLS of order 12 has 24901 d-transversals (the maximum currently known).



See https://oeis.org/A287648

It is required to find all orthogonal DLSs for this DLS.
Program here
https://boinc.tbrada.eu/forum_thread.php?id=3104&postid=4166

Here's another program (by Belyshev)
https://yadi.sk/d/rW8gHaJgwLh3DA

This program runs in one thread.
Run the ortogon_u.exe program by double-clicking the left mouse button.
On the program request, enter the DLS order, it is 12.

PS. Code of square

DW71H14GBTJeK6fE1RcSBVdKdAhrHJYM

The code is needed for the first program (by Tomáš Brada).

You can interrupt the Belyshev's program (Ctrl + C) with some result.
Solutions will be written to the mates.txt file.
It takes a long time to complete the program.

In my article “Mutually Orthogonal Latin Squares (MOLS)”
http://www.natalimak1.narod.ru/grolk2.htm
showing four squares from MOLS of order 27.

These are two DLS

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
3 4 5 6 7 8 0 1 2 12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20
6 7 8 0 1 2 3 4 5 15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 0 1 2 3 4 5 6 7 8
12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20 3 4 5 6 7 8 0 1 2
15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23 6 7 8 0 1 2 3 4 5
18 19 20 21 22 23 24 25 26 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
21 22 23 24 25 26 18 19 20 3 4 5 6 7 8 0 1 2 12 13 14 15 16 17 9 10 11
24 25 26 18 19 20 21 22 23 6 7 8 0 1 2 3 4 5 15 16 17 9 10 11 12 13 14
13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18 4 5 3 7 8 6 1 2 0
16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21 7 8 6 1 2 0 4 5 3
10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24 1 2 0 4 5 3 7 8 6
22 23 21 25 26 24 19 20 18 4 5 3 7 8 6 1 2 0 13 14 12 16 17 15 10 11 9
25 26 24 19 20 18 22 23 21 7 8 6 1 2 0 4 5 3 16 17 15 10 11 9 13 14 12
19 20 18 22 23 21 25 26 24 1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15
4 5 3 7 8 6 1 2 0 13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18
7 8 6 1 2 0 4 5 3 16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21
1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24
26 24 25 20 18 19 23 21 22 8 6 7 2 0 1 5 3 4 17 15 16 11 9 10 14 12 13
20 18 19 23 21 22 26 24 25 2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16
23 21 22 26 24 25 20 18 19 5 3 4 8 6 7 2 0 1 14 12 13 17 15 16 11 9 10
8 6 7 2 0 1 5 3 4 17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22
2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25
5 3 4 8 6 7 2 0 1 14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19
17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22 8 6 7 2 0 1 5 3 4
11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25 2 0 1 5 3 4 8 6 7
14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19 5 3 4 8 6 7 2 0 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
4 5 3 7 8 6 1 2 0 13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18
8 6 7 2 0 1 5 3 4 17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22
12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20 3 4 5 6 7 8 0 1 2
16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21 7 8 6 1 2 0 4 5 3
11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25 2 0 1 5 3 4 8 6 7
24 25 26 18 19 20 21 22 23 6 7 8 0 1 2 3 4 5 15 16 17 9 10 11 12 13 14
19 20 18 22 23 21 25 26 24 1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15
23 21 22 26 24 25 20 18 19 5 3 4 8 6 7 2 0 1 14 12 13 17 15 16 11 9 10
22 23 21 25 26 24 19 20 18 4 5 3 7 8 6 1 2 0 13 14 12 16 17 15 10 11 9
26 24 25 20 18 19 23 21 22 8 6 7 2 0 1 5 3 4 17 15 16 11 9 10 14 12 13
18 19 20 21 22 23 24 25 26 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
7 8 6 1 2 0 4 5 3 16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21
2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25
3 4 5 6 7 8 0 1 2 12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20
10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24 1 2 0 4 5 3 7 8 6
14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19 5 3 4 8 6 7 2 0 1
15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23 6 7 8 0 1 2 3 4 5
17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22 8 6 7 2 0 1 5 3 4
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 0 1 2 3 4 5 6 7 8
13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18 4 5 3 7 8 6 1 2 0
20 18 19 23 21 22 26 24 25 2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16
21 22 23 24 25 26 18 19 20 3 4 5 6 7 8 0 1 2 12 13 14 15 16 17 9 10 11
25 26 24 19 20 18 22 23 21 7 8 6 1 2 0 4 5 3 16 17 15 10 11 9 13 14 12
5 3 4 8 6 7 2 0 1 14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19
6 7 8 0 1 2 3 4 5 15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23
1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24

Need find the rest of the DLS.
This can be done in the Maple mathematical software package.
The command MOLS(3,3,26) may need to be used.

Can someone do this?

Releasing programs for DLK(N).
Source on: https://github.com/tomasbrod/tbboinc
Download windows build ec0d025846:
https://boinc.tbrada.eu/download/ndlk-2101a.zip
Download windows build b1b4dc24f04:
https://boinc.tbrada.eu/download/ndlk-2101b.zip

dlkconv.exe

dlkconv.exe: (input-spec) [input] [(output-spec) [output]
** Diagonal Latin Square converter and normalizer **
input-spec: Endoded/compressed (e), Alphanumeric (a), stdin (s) File (f).
 Encoded input can be passed immediately on command line.
 Othervise a NxN matrix of decimal numbers, or Alphanumeric (a) must be
 passed via stdin or file. Multiple orders (N) may be mixed on input.
 Empty lines and (one) trailing spaces are ignored.
output-spec: (e)(a)(f), stdout (s), Compact (c), Diagonal (d)
 LS normalized on first row are written to stdout by default.
 Space between output squares are ommited in Compact (c) mode.
 With (d), output matrix is normalized on main diagonal.
Example dlkconv.exe e 7yPG4 dc -- decode 7yPG4 and normalize diagonal
Author: Tomas Brada (GPL)

ortogonb.exe

ortogonb.exe: [-co] (Input) >output
** Search for Orthogonal mates of Diagonal Latin Square **
 -c print number of transverses and exit
 -o output orthogonal mates like they are found
Input: encoded diagonal latin square (see dlkconv.exe)
Output: metadata and orthogonal squares in encoded format
Returns 0 if orthogonal mates found, 1 if not. Multi Thread.
Example ortogonb.exe BJMzz45jMmo1FfZtCjN
Author: Tomas Brada (GPL)

ortogonbw.exe

ortogonbw.exe: (Input) (Path) >output
** Search for Orthogonal mates of Diagonal Latin Square **
Input: encoded diagonal latin square (see dlkconv.exe)
Output: metadata and orthogonal squares in encoded format
Prints to output immediately as square is found. Single thread.
Path: sequence of space-separated numbers narrowing the problem space
Explanation: L(level) c(column) choosen-row / count-rows
Specify number as Path from 1 up to count-rows to narrow down the Search,
specify multiple numbers to narrow down further.
Example: ortogonbw.exe DeGkNZGc43qaHWnNUsjfvuQNbREDKc4 3 120
Author: Tomas Brada (GPL)

kanonb.exe

kanonb.exe: <input >output
** Diagonal Latin Square Kanonizer **
Input and Output in Encoded form (see dlkconv)
Uses cache files kanonb_cache_NN.dat in the current directory.

ortowalk.exe

ortowalk.exe: FileName
** Explore the full graph of DLK orthogonality **
Insert the initial CF-DLK into FileName.todo
then run "ortowalk.exe FileName" (without the .todo)
Author: Tomas Brada (GPL)

rules.exe

rules.exe: (N) >output
** Diagonal Latin Square Rule Enumerator **
Enumerates all rules of CF-DLK of order N
Uses cache files kanonb_cache_NN.dat in the current directory.
Author: Tomas Brada (GPL)