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ProfileNatalia Makarova
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Message 1150 - Posted: 11 Jan 2021, 11:53:11 UTC

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(31,32)

Получаю

[32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring,
 32 x 32 dense matrix over Integer Ring]

Это интересно!
Не знаю, как составлять полную систему MOLS 32-го порядка.
Какое тут рассматривается кольцо?
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ProfileNatalia Makarova
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Message 1151 - Posted: 11 Jan 2021, 12:02:21 UTC
Last modified: 11 Jan 2021, 12:03:24 UTC

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(5,35)

Получаю

[35 x 35 dense matrix over Integer Ring,
 35 x 35 dense matrix over Integer Ring,
 35 x 35 dense matrix over Integer Ring,
 35 x 35 dense matrix over Integer Ring,
 35 x 35 dense matrix over Integer Ring]

Тоже весьма интересно!

Затем задаю команду
sage: designs.mutually_orthogonal_latin_squares(6,35)

Получаю
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-1-5bb645162bdc> in <module>
----> 1 designs.mutually_orthogonal_latin_squares(Integer(6),Integer(35))
      2 
      3 

/home/sc_serv/sage/local/lib/python3.8/site-packages/sage/misc/lazy_import.pyx in sage.misc.lazy_import.LazyImport.__call__ (build/cythonized/sage/misc/lazy_import.c:4027)()
    359             True
    360         """
--> 361         return self.get_object()(*args, **kwds)
    362 
    363     def __repr__(self):

/home/sc_serv/sage/local/lib/python3.8/site-packages/sage/combinat/designs/latin_squares.py in mutually_orthogonal_latin_squares(k, n, partitions, check)
    391 
    392     else:
--> 393         raise NotImplementedError("I don't know how to build {} MOLS of order {}".format(k, n))
    394 
    395     if check:

NotImplementedError: I don't know how to build 6 MOLS of order 35

То есть, группа MOLS 35-го порядка, состоящая из 5 ЛК, известна и получается из какого-то кольца (из какого?)
А вот группа MOLS 35-го порядка, состоящая из 6 ЛК, неизвестна, как я понимаю.
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Message 1152 - Posted: 11 Jan 2021, 12:07:26 UTC

Ещё интереснее!
Задаю команду
sage: designs.mutually_orthogonal_latin_squares(5,12)

Получаю

[ 0  1  2  3  4  5  6  7  8  9 10 11]
[ 1  0  3  2  5  4  7  6  9  8 11 10]
[ 2  3  0  1  6  7  4  5 10 11  8  9]
[ 3  2  1  0  7  6  5  4 11 10  9  8]
[ 8  9 10 11  0  1  2  3  4  5  6  7]
[ 9  8 11 10  1  0  3  2  5  4  7  6]
[10 11  8  9  2  3  0  1  6  7  4  5]
[11 10  9  8  3  2  1  0  7  6  5  4]
[ 4  5  6  7  8  9 10 11  0  1  2  3]
[ 5  4  7  6  9  8 11 10  1  0  3  2]
[ 6  7  4  5 10 11  8  9  2  3  0  1]
[ 7  6  5  4 11 10  9  8  3  2  1  0],

[ 0  1  2  3  4  5  6  7  8  9 10 11]
[ 6  7  4  5 10 11  8  9  2  3  0  1]
[ 4  5  6  7  8  9 10 11  0  1  2  3]
[ 2  3  0  1  6  7  4  5 10 11  8  9]
[10 11  8  9  2  3  0  1  6  7  4  5]
[ 5  4  7  6  9  8 11 10  1  0  3  2]
[ 7  6  5  4 11 10  9  8  3  2  1  0]
[ 8  9 10 11  0  1  2  3  4  5  6  7]
[ 9  8 11 10  1  0  3  2  5  4  7  6]
[ 3  2  1  0  7  6  5  4 11 10  9  8]
[ 1  0  3  2  5  4  7  6  9  8 11 10]
[11 10  9  8  3  2  1  0  7  6  5  4],

[ 0  1  2  3  4  5  6  7  8  9 10 11]
[ 3  2  1  0  7  6  5  4 11 10  9  8]
[11 10  9  8  3  2  1  0  7  6  5  4]
[ 6  7  4  5 10 11  8  9  2  3  0  1]
[ 4  5  6  7  8  9 10 11  0  1  2  3]
[ 7  6  5  4 11 10  9  8  3  2  1  0]
[ 9  8 11 10  1  0  3  2  5  4  7  6]
[ 1  0  3  2  5  4  7  6  9  8 11 10]
[ 5  4  7  6  9  8 11 10  1  0  3  2]
[ 2  3  0  1  6  7  4  5 10 11  8  9]
[10 11  8  9  2  3  0  1  6  7  4  5]
[ 8  9 10 11  0  1  2  3  4  5  6  7],

[ 0  1  2  3  4  5  6  7  8  9 10 11]
[ 9  8 11 10  1  0  3  2  5  4  7  6]
[ 8  9 10 11  0  1  2  3  4  5  6  7]
[ 1  0  3  2  5  4  7  6  9  8 11 10]
[ 5  4  7  6  9  8 11 10  1  0  3  2]
[ 3  2  1  0  7  6  5  4 11 10  9  8]
[ 2  3  0  1  6  7  4  5 10 11  8  9]
[ 4  5  6  7  8  9 10 11  0  1  2  3]
[11 10  9  8  3  2  1  0  7  6  5  4]
[ 6  7  4  5 10 11  8  9  2  3  0  1]
[ 7  6  5  4 11 10  9  8  3  2  1  0]
[10 11  8  9  2  3  0  1  6  7  4  5],

[ 0  1  2  3  4  5  6  7  8  9 10 11]
[10 11  8  9  2  3  0  1  6  7  4  5]
[ 5  4  7  6  9  8 11 10  1  0  3  2]
[ 7  6  5  4 11 10  9  8  3  2  1  0]
[ 9  8 11 10  1  0  3  2  5  4  7  6]
[11 10  9  8  3  2  1  0  7  6  5  4]
[ 3  2  1  0  7  6  5  4 11 10  9  8]
[ 6  7  4  5 10 11  8  9  2  3  0  1]
[ 2  3  0  1  6  7  4  5 10 11  8  9]
[ 1  0  3  2  5  4  7  6  9  8 11 10]
[ 8  9 10 11  0  1  2  3  4  5  6  7]
[ 4  5  6  7  8  9 10 11  0  1  2  3]

Группа MOLS 12-го порядка, состоящая из 5 ЛК!
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ProfileNatalia Makarova
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Message 1153 - Posted: 11 Jan 2021, 12:11:37 UTC

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(6,12)

Программа пишет
NotImplementedError: I don't know how to build 6 MOLS of order 12

Можно и дальше экспериментировать. Очень интересно!
11 лет назад я в своих статьях описывала группы MOLS разных порядков, какие мне тогда удалось найти в статьях.
Но не все удалось найти. Может быть, программа SageMath все их знает. Это здорово!
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Message 1154 - Posted: 11 Jan 2021, 12:24:48 UTC
Last modified: 11 Jan 2021, 12:44:07 UTC

Группу MOLS 15-го порядка из 4-х ЛК программа знает!
Задаю команду
sage: designs.mutually_orthogonal_latin_squares(4,15)

Получаю

[ 1  2  3  4  5  6  7  8  9 10 11 12 13 14  0]
[ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14]
[14  0  1  2  3  4  5  6  7  8  9 10 11 12 13]
[13 14  0  1  2  3  4  5  6  7  8  9 10 11 12]
[12 13 14  0  1  2  3  4  5  6  7  8  9 10 11]
[11 12 13 14  0  1  2  3  4  5  6  7  8  9 10]
[10 11 12 13 14  0  1  2  3  4  5  6  7  8  9]
[ 9 10 11 12 13 14  0  1  2  3  4  5  6  7  8]
[ 8  9 10 11 12 13 14  0  1  2  3  4  5  6  7]
[ 7  8  9 10 11 12 13 14  0  1  2  3  4  5  6]
[ 6  7  8  9 10 11 12 13 14  0  1  2  3  4  5]
[ 5  6  7  8  9 10 11 12 13 14  0  1  2  3  4]
[ 4  5  6  7  8  9 10 11 12 13 14  0  1  2  3]
[ 3  4  5  6  7  8  9 10 11 12 13 14  0  1  2]
[ 2  3  4  5  6  7  8  9 10 11 12 13 14  0  1],

[ 1  3  6  8 10 13  5  2  0 12  4  7  9 11 14]
[ 0  2  4  7  9 11 14  6  3  1 13  5  8 10 12]
[13  1  3  5  8 10 12  0  7  4  2 14  6  9 11]
[12 14  2  4  6  9 11 13  1  8  5  3  0  7 10]
[11 13  0  3  5  7 10 12 14  2  9  6  4  1  8]
[ 9 12 14  1  4  6  8 11 13  0  3 10  7  5  2]
[ 3 10 13  0  2  5  7  9 12 14  1  4 11  8  6]
[ 7  4 11 14  1  3  6  8 10 13  0  2  5 12  9]
[10  8  5 12  0  2  4  7  9 11 14  1  3  6 13]
[14 11  9  6 13  1  3  5  8 10 12  0  2  4  7]
[ 8  0 12 10  7 14  2  4  6  9 11 13  1  3  5]
[ 6  9  1 13 11  8  0  3  5  7 10 12 14  2  4]
[ 5  7 10  2 14 12  9  1  4  6  8 11 13  0  3]
[ 4  6  8 11  3  0 13 10  2  5  7  9 12 14  1]
[ 2  5  7  9 12  4  1 14 11  3  6  8 10 13  0],

[ 1  7  4  0 11  8 14  5 12  3  9  6  2 13 10]
[11  2  8  5  1 12  9  0  6 13  4 10  7  3 14]
[ 0 12  3  9  6  2 13 10  1  7 14  5 11  8  4]
[ 5  1 13  4 10  7  3 14 11  2  8  0  6 12  9]
[10  6  2 14  5 11  8  4  0 12  3  9  1  7 13]
[14 11  7  3  0  6 12  9  5  1 13  4 10  2  8]
[ 9  0 12  8  4  1  7 13 10  6  2 14  5 11  3]
[ 4 10  1 13  9  5  2  8 14 11  7  3  0  6 12]
[13  5 11  2 14 10  6  3  9  0 12  8  4  1  7]
[ 8 14  6 12  3  0 11  7  4 10  1 13  9  5  2]
[ 3  9  0  7 13  4  1 12  8  5 11  2 14 10  6]
[ 7  4 10  1  8 14  5  2 13  9  6 12  3  0 11]
[12  8  5 11  2  9  0  6  3 14 10  7 13  4  1]
[ 2 13  9  6 12  3 10  1  7  4  0 11  8 14  5]
[ 6  3 14 10  7 13  4 11  2  8  5  1 12  9  0],

[ 1 11  7  2 12  3  8 13  4  9 14  5  0 10  6]
[ 7  2 12  8  3 13  4  9 14  5 10  0  6  1 11]
[12  8  3 13  9  4 14  5 10  0  6 11  1  7  2]
[ 3 13  9  4 14 10  5  0  6 11  1  7 12  2  8]
[ 9  4 14 10  5  0 11  6  1  7 12  2  8 13  3]
[ 4 10  5  0 11  6  1 12  7  2  8 13  3  9 14]
[ 0  5 11  6  1 12  7  2 13  8  3  9 14  4 10]
[11  1  6 12  7  2 13  8  3 14  9  4 10  0  5]
[ 6 12  2  7 13  8  3 14  9  4  0 10  5 11  1]
[ 2  7 13  3  8 14  9  4  0 10  5  1 11  6 12]
[13  3  8 14  4  9  0 10  5  1 11  6  2 12  7]
[ 8 14  4  9  0  5 10  1 11  6  2 12  7  3 13]
[14  9  0  5 10  1  6 11  2 12  7  3 13  8  4]
[ 5  0 10  1  6 11  2  7 12  3 13  8  4 14  9]
[10  6  1 11  2  7 12  3  8 13  4 14  9  5  0]

Отлично!

Кстати, в моей статье была построена другая группа MOLS 15-го порядка, состоящая из четырёх ЛК (по алгоритму, который мне удалось найти в какой-то статье).
Из этой группы MOLS я получила несложным преобразованием группу MODLS 15-го порядка, состоящую из четырёх ДЛК.
Смотрите тему
https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=115
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Message 1155 - Posted: 11 Jan 2021, 12:57:31 UTC

Группу MOLS 18-го порядка из трёх ЛК программа тоже знает!
Задаю команду
sage: designs.mutually_orthogonal_latin_squares(3,18)

Получаю

[ 1  6  4  9  7 10 12 14  3  2 13  0 17  8 11 15  5 16]
[ 4  2  7  5  1  8 11 13 15 17  3 14 10  0  9 12 16  6]
[16  5  3  8  6  2  9 12 14  7  0  4 15 11 10  1 13 17]
[15 17  6  4  9  7  3  1 13  0  8 10  5 16 12 11  2 14]
[14 16  0  7  5  1  8  4  2 15 10  9 11  6 17 13 12  3]
[ 3 15 17 10  8  6  2  9  5  4 16 11  1 12  7  0 14 13]
[ 6  4 16  0 11  9  7  3  1 14  5 17 12  2 13  8 10 15]
[ 2  7  5 17 10 12  1  8  4 16 15  6  0 13  3 14  9 11]
[ 5  3  8  6  0 11 13  2  9 12 17 16  7 10 14  4 15  1]
[11 13  1 16  2 14  6 15  0 10 12  7  3 17  8  5  4  9]
[10 12 14  2 17  3 15  7 16  1 11 13  8  4  0  9  6  5]
[17 11 13 15  3  0  4 16  8  6  2 12 14  9  5 10  1  7]
[ 9  0 12 14 16  4 10  5 17  8  7  3 13 15  1  6 11  2]
[ 0  1 10 13 15 17  5 11  6  3  9  8  4 14 16  2  7 12]
[ 7 10  2 11 14 16  0  6 12 13  4  1  9  5 15 17  3  8]
[13  8 11  3 12 15 17 10  7  9 14  5  2  1  6 16  0  4]
[ 8 14  9 12  4 13 16  0 11  5  1 15  6  3  2  7 17 10]
[12  9 15  1 13  5 14 17 10 11  6  2 16  7  4  3  8  0],

[ 1  4 16 15 14  3  6  2  5 11 10 17  9  0  7 13  8 12]
[ 6  2  5 17 16 15  4  7  3 13 12 11  0  1 10  8 14  9]
[ 4  7  3  6  0 17 16  5  8  1 14 13 12 10  2 11  9 15]
[ 9  5  8  4  7 10  0 17  6 16  2 15 14 13 11  3 12  1]
[ 7  1  6  9  5  8 11 10  0  2 17  3 16 15 14 12  4 13]
[10  8  2  7  1  6  9 12 11 14  3  0  4 17 16 15 13  5]
[12 11  9  3  8  2  7  1 13  6 15  4 10  5  0 17 16 14]
[14 13 12  1  4  9  3  8  2 15  7 16  5 11  6 10  0 17]
[ 3 15 14 13  2  5  1  4  9  0 16  8 17  6 12  7 11 10]
[ 2 17  7  0 15  4 14 16 12 10  1  6  8  3 13  9  5 11]
[13  3  0  8 10 16  5 15 17 12 11  2  7  9  4 14  1  6]
[ 0 14  4 10  9 11 17  6 16  7 13 12  3  8  1  5 15  2]
[17 10 15  5 11  1 12  0  7  3  8 14 13  4  9  2  6 16]
[ 8  0 11 16  6 12  2 13 10 17  4  9 15 14  5  1  3  7]
[11  9 10 12 17  7 13  3 14  8  0  5  1 16 15  6  2  4]
[15 12  1 11 13  0  8 14  4  5  9 10  6  2 17 16  7  3]
[ 5 16 13  2 12 14 10  9 15  4  6  1 11  7  3  0 17  8]
[16  6 17 14  3 13 15 11  1  9  5  7  2 12  8  4 10  0],

[ 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17  0]
[17  1 10  0 12  5  6  3  4  7 16  9  8 13 14 15  2 11]
[12 11  1 16  7  8  5  6  0  9  3  2 17  4 13 14 15 10]
[ 7  8  9  1  2  3  4  5  6 16 17  0 10 11 12 13 14 15]
[ 6  3  4 17  1 10  0 12  5 15  2 11  7 16  9  8 13 14]
[ 5  6  0 12 11  1 16  7  8 14 15 10  9  3  2 17  4 13]
[ 4  5  6  7  8  9  1  2  3 13 14 15 16 17  0 10 11 12]
[ 0 12  5  6  3  4 17  1 10  8 13 14 15  2 11  7 16  9]
[16  7  8  5  6  0 12 11  1 17  4 13 14 15 10  9  3  2]
[11  4 16  9 15 14 13  0  2  1 10  8  3  5  6 12  7 17]
[10  9 12  2 17 15 14 13  7 11  1 16  4  0  5  6  8  3]
[ 3 16 17  8 10 11 15 14 13  0  9  1  2 12  7  5  6  4]
[13  0  2 11  4 16  9 15 14 12  7 17  1 10  8  3  5  6]
[14 13  7 10  9 12  2 17 15  6  8  3 11  1 16  4  0  5]
[15 14 13  3 16 17  8 10 11  5  6  4  0  9  1  2 12  7]
[ 9 15 14 13  0  2 11  4 16  3  5  6 12  7 17  1 10  8]
[ 2 17 15 14 13  7 10  9 12  4  0  5  6  8  3 11  1 16]
[ 8 10 11 15 14 13  3 16 17  2 12  7  5  6  4  0  9  1]

В своё время я получила из известной группы MOLS 18-го порядка ортогональную пару ДЛК 18-го порядка; третий ЛК системы в ДЛК не превратился.
Может быть, из показанной сейчас группы MOLS 18-го порядка можно получить группу MODLS из трёх ДЛК?
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Message 1156 - Posted: 11 Jan 2021, 13:04:12 UTC

Программа знает и группу MOLS 18-го порядка из четырёх ЛК!
Задаю команду
sage: designs.mutually_orthogonal_latin_squares(4,18)

Получаю

[ 0  3  6 16 10 13  1  4  7 11 14 17 15  9 12  2  5  8]
[ 7  1  4 14 17 11  8  2  5 15  9 12 13 16 10  6  0  3]
[ 5  8  2  9 12 15  3  6  0 13 16 10 11 14 17  4  7  1]
[ 4  7  1  3  6  0 10 13 16  5  8  2 14 17 11  9 12 15]
[ 2  5  8  1  4  7 17 11 14  0  3  6  9 12 15 16 10 13]
[ 6  0  3  8  2  5 12 15  9  7  1  4 16 10 13 14 17 11]
[13 16 10  7  1  4  6  0  3 12 15  9  8  2  5 17 11 14]
[11 14 17  5  8  2  4  7  1 10 13 16  3  6  0 12 15  9]
[15  9 12  0  3  6  2  5  8 17 11 14  1  4  7 10 13 16]
[14 17 11  6  0  3  9 12 15  8  2  5 10 13 16  1  4  7]
[ 9 12 15  4  7  1 16 10 13  3  6  0 17 11 14  8  2  5]
[16 10 13  2  5  8 14 17 11  1  4  7 12 15  9  3  6  0]
[12 15  9 17 11 14  0  3  6  4  7  1  2  5  8 13 16 10]
[10 13 16 12 15  9  7  1  4  2  5  8  6  0  3 11 14 17]
[17 11 14 10 13 16  5  8  2  6  0  3  4  7  1 15  9 12]
[ 3  6  0 15  9 12 11 14 17 16 10 13  7  1  4  5  8  2]
[ 1  4  7 13 16 10 15  9 12 14 17 11  5  8  2  0  3  6]
[ 8  2  5 11 14 17 13 16 10  9 12 15  0  3  6  7  1  4],

[ 0  7  5 12 10 17 16 14  9  2  6  4 13 11 15  1  8  3]
[ 3  1  8 15 13 11 10 17 12  5  0  7 16 14  9  4  2  6]
[ 6  4  2  9 16 14 13 11 15  8  3  1 10 17 12  7  5  0]
[10 17 12  3  1  8 15 13 11  4  2  6  5  0  7 16 14  9]
[13 11 15  6  4  2  9 16 14  7  5  0  8  3  1 10 17 12]
[16 14  9  0  7  5 12 10 17  1  8  3  2  6  4 13 11 15]
[ 9 16 14 13 11 15  6  4  2 10 17 12  7  5  0  8  3  1]
[12 10 17 16 14  9  0  7  5 13 11 15  1  8  3  2  6  4]
[15 13 11 10 17 12  3  1  8 16 14  9  4  2  6  5  0  7]
[11 15 13  7  5  0  2  6  4  9 16 14  3  1  8 17 12 10]
[14  9 16  1  8  3  5  0  7 12 10 17  6  4  2 11 15 13]
[17 12 10  4  2  6  8  3  1 15 13 11  0  7  5 14  9 16]
[ 5  0  7 14  9 16  1  8  3 11 15 13 12 10 17  6  4  2]
[ 8  3  1 17 12 10  4  2  6 14  9 16 15 13 11  0  7  5]
[ 2  6  4 11 15 13  7  5  0 17 12 10  9 16 14  3  1  8]
[ 4  2  6  8  3  1 17 12 10  0  7  5 14  9 16 15 13 11]
[ 7  5  0  2  6  4 11 15 13  3  1  8 17 12 10  9 16 14]
[ 1  8  3  5  0  7 14  9 16  6  4  2 11 15 13 12 10 17],

[ 0  8  4 14 10 15 11 16 12  9 17 13  7  3  2  5  1  6]
[ 5  1  6 16 12 11 13  9 17 14 10 15  0  8  4  7  3  2]
[ 7  3  2  9 17 13 15 14 10 16 12 11  5  1  6  0  8  4]
[14 10 15  3  2  7 17 13  9  8  4  0 12 11 16  1  6  5]
[16 12 11  8  4  0 10 15 14  1  6  5 17 13  9  3  2  7]
[ 9 17 13  1  6  5 12 11 16  3  2  7 10 15 14  8  4  0]
[11 16 12 17 13  9  6  5  1  4  0  8  2  7  3 15 14 10]
[13  9 17 10 15 14  2  7  3  6  5  1  4  0  8 11 16 12]
[15 14 10 12 11 16  4  0  8  2  7  3  6  5  1 13  9 17]
[ 6  5  1 11 16 12  7  3  2  0  8  4  9 17 13 14 10 15]
[ 2  7  3 13  9 17  0  8  4  5  1  6 14 10 15 16 12 11]
[ 4  0  8 15 14 10  5  1  6  7  3  2 16 12 11  9 17 13]
[ 1  6  5  0  8  4 14 10 15 17 13  9  3  2  7 12 11 16]
[ 3  2  7  5  1  6 16 12 11 10 15 14  8  4  0 17 13  9]
[ 8  4  0  7  3  2  9 17 13 12 11 16  1  6  5 10 15 14]
[17 13  9  4  0  8  3  2  7 15 14 10 11 16 12  6  5  1]
[10 15 14  6  5  1  8  4  0 11 16 12 13  9 17  2  7  3]
[12 11 16  2  7  3  1  6  5 13  9 17 15 14 10  4  0  8],

[ 0  5  7  4  6  2 16  9 14  3  8  1 17 10 12 13 15 11]
[ 8  1  3  0  5  7 12 17 10  2  4  6 13 15 11  9 14 16]
[ 4  6  2  8  1  3 11 13 15  7  0  5  9 14 16 17 10 12]
[10 12 17  3  8  1  7  0  5 16  9 14  6  2  4 11 13 15]
[15 11 13  2  4  6  3  8  1 12 17 10  5  7  0 16  9 14]
[14 16  9  7  0  5  2  4  6 11 13 15  1  3  8 12 17 10]
[ 1  3  8 13 15 11  6  2  4 14 16  9 10 12 17  0  5  7]
[ 6  2  4  9 14 16  5  7  0 10 12 17 15 11 13  8  1  3]
[ 5  7  0 17 10 12  1  3  8 15 11 13 14 16  9  4  6  2]
[ 9 14 16 15 11 13  0  5  7  4  6  2 12 17 10  3  8  1]
[17 10 12 14 16  9  8  1  3  0  5  7 11 13 15  2  4  6]
[13 15 11 10 12 17  4  6  2  8  1  3 16  9 14  7  0  5]
[ 3  8  1 12 17 10  9 14 16  6  2  4  7  0  5 15 11 13]
[ 2  4  6 11 13 15 17 10 12  5  7  0  3  8  1 14 16  9]
[ 7  0  5 16  9 14 13 15 11  1  3  8  2  4  6 10 12 17]
[12 17 10  6  2  4 15 11 13  9 14 16  0  5  7  1  3  8]
[11 13 15  5  7  0 14 16  9 17 10 12  8  1  3  6  2  4]
[16  9 14  1  3  8 10 12 17 13 15 11  4  6  2  5  7  0]

И из пяти ЛК тоже знает группу MOLS 18-го порядка!
Задаю команду
sage: designs.mutually_orthogonal_latin_squares(5,18)

Получаю

[ 0  3  6 16 10 13  1  4  7 11 14 17 15  9 12  2  5  8]
[ 7  1  4 14 17 11  8  2  5 15  9 12 13 16 10  6  0  3]
[ 5  8  2  9 12 15  3  6  0 13 16 10 11 14 17  4  7  1]
[ 4  7  1  3  6  0 10 13 16  5  8  2 14 17 11  9 12 15]
[ 2  5  8  1  4  7 17 11 14  0  3  6  9 12 15 16 10 13]
[ 6  0  3  8  2  5 12 15  9  7  1  4 16 10 13 14 17 11]
[13 16 10  7  1  4  6  0  3 12 15  9  8  2  5 17 11 14]
[11 14 17  5  8  2  4  7  1 10 13 16  3  6  0 12 15  9]
[15  9 12  0  3  6  2  5  8 17 11 14  1  4  7 10 13 16]
[14 17 11  6  0  3  9 12 15  8  2  5 10 13 16  1  4  7]
[ 9 12 15  4  7  1 16 10 13  3  6  0 17 11 14  8  2  5]
[16 10 13  2  5  8 14 17 11  1  4  7 12 15  9  3  6  0]
[12 15  9 17 11 14  0  3  6  4  7  1  2  5  8 13 16 10]
[10 13 16 12 15  9  7  1  4  2  5  8  6  0  3 11 14 17]
[17 11 14 10 13 16  5  8  2  6  0  3  4  7  1 15  9 12]
[ 3  6  0 15  9 12 11 14 17 16 10 13  7  1  4  5  8  2]
[ 1  4  7 13 16 10 15  9 12 14 17 11  5  8  2  0  3  6]
[ 8  2  5 11 14 17 13 16 10  9 12 15  0  3  6  7  1  4],

[ 0  7  5 12 10 17 16 14  9  2  6  4 13 11 15  1  8  3]
[ 3  1  8 15 13 11 10 17 12  5  0  7 16 14  9  4  2  6]
[ 6  4  2  9 16 14 13 11 15  8  3  1 10 17 12  7  5  0]
[10 17 12  3  1  8 15 13 11  4  2  6  5  0  7 16 14  9]
[13 11 15  6  4  2  9 16 14  7  5  0  8  3  1 10 17 12]
[16 14  9  0  7  5 12 10 17  1  8  3  2  6  4 13 11 15]
[ 9 16 14 13 11 15  6  4  2 10 17 12  7  5  0  8  3  1]
[12 10 17 16 14  9  0  7  5 13 11 15  1  8  3  2  6  4]
[15 13 11 10 17 12  3  1  8 16 14  9  4  2  6  5  0  7]
[11 15 13  7  5  0  2  6  4  9 16 14  3  1  8 17 12 10]
[14  9 16  1  8  3  5  0  7 12 10 17  6  4  2 11 15 13]
[17 12 10  4  2  6  8  3  1 15 13 11  0  7  5 14  9 16]
[ 5  0  7 14  9 16  1  8  3 11 15 13 12 10 17  6  4  2]
[ 8  3  1 17 12 10  4  2  6 14  9 16 15 13 11  0  7  5]
[ 2  6  4 11 15 13  7  5  0 17 12 10  9 16 14  3  1  8]
[ 4  2  6  8  3  1 17 12 10  0  7  5 14  9 16 15 13 11]
[ 7  5  0  2  6  4 11 15 13  3  1  8 17 12 10  9 16 14]
[ 1  8  3  5  0  7 14  9 16  6  4  2 11 15 13 12 10 17],

[ 0  8  4 14 10 15 11 16 12  9 17 13  7  3  2  5  1  6]
[ 5  1  6 16 12 11 13  9 17 14 10 15  0  8  4  7  3  2]
[ 7  3  2  9 17 13 15 14 10 16 12 11  5  1  6  0  8  4]
[14 10 15  3  2  7 17 13  9  8  4  0 12 11 16  1  6  5]
[16 12 11  8  4  0 10 15 14  1  6  5 17 13  9  3  2  7]
[ 9 17 13  1  6  5 12 11 16  3  2  7 10 15 14  8  4  0]
[11 16 12 17 13  9  6  5  1  4  0  8  2  7  3 15 14 10]
[13  9 17 10 15 14  2  7  3  6  5  1  4  0  8 11 16 12]
[15 14 10 12 11 16  4  0  8  2  7  3  6  5  1 13  9 17]
[ 6  5  1 11 16 12  7  3  2  0  8  4  9 17 13 14 10 15]
[ 2  7  3 13  9 17  0  8  4  5  1  6 14 10 15 16 12 11]
[ 4  0  8 15 14 10  5  1  6  7  3  2 16 12 11  9 17 13]
[ 1  6  5  0  8  4 14 10 15 17 13  9  3  2  7 12 11 16]
[ 3  2  7  5  1  6 16 12 11 10 15 14  8  4  0 17 13  9]
[ 8  4  0  7  3  2  9 17 13 12 11 16  1  6  5 10 15 14]
[17 13  9  4  0  8  3  2  7 15 14 10 11 16 12  6  5  1]
[10 15 14  6  5  1  8  4  0 11 16 12 13  9 17  2  7  3]
[12 11 16  2  7  3  1  6  5 13  9 17 15 14 10  4  0  8],

[ 0  5  7  4  6  2 16  9 14  3  8  1 17 10 12 13 15 11]
[ 8  1  3  0  5  7 12 17 10  2  4  6 13 15 11  9 14 16]
[ 4  6  2  8  1  3 11 13 15  7  0  5  9 14 16 17 10 12]
[10 12 17  3  8  1  7  0  5 16  9 14  6  2  4 11 13 15]
[15 11 13  2  4  6  3  8  1 12 17 10  5  7  0 16  9 14]
[14 16  9  7  0  5  2  4  6 11 13 15  1  3  8 12 17 10]
[ 1  3  8 13 15 11  6  2  4 14 16  9 10 12 17  0  5  7]
[ 6  2  4  9 14 16  5  7  0 10 12 17 15 11 13  8  1  3]
[ 5  7  0 17 10 12  1  3  8 15 11 13 14 16  9  4  6  2]
[ 9 14 16 15 11 13  0  5  7  4  6  2 12 17 10  3  8  1]
[17 10 12 14 16  9  8  1  3  0  5  7 11 13 15  2  4  6]
[13 15 11 10 12 17  4  6  2  8  1  3 16  9 14  7  0  5]
[ 3  8  1 12 17 10  9 14 16  6  2  4  7  0  5 15 11 13]
[ 2  4  6 11 13 15 17 10 12  5  7  0  3  8  1 14 16  9]
[ 7  0  5 16  9 14 13 15 11  1  3  8  2  4  6 10 12 17]
[12 17 10  6  2  4 15 11 13  9 14 16  0  5  7  1  3  8]
[11 13 15  5  7  0 14 16  9 17 10 12  8  1  3  6  2  4]
[16  9 14  1  3  8 10 12 17 13 15 11  4  6  2  5  7  0],

[ 0  2  1  9 11 10  3  5  4  8  7  6 17 16 15 14 13 12]
[ 2  1  0 11 10  9  5  4  3  7  6  8 16 15 17 13 12 14]
[ 1  0  2 10  9 11  4  3  5  6  8  7 15 17 16 12 14 13]
[ 6  8  7  3  5  4 12 14 13 17 16 15  2  1  0 11 10  9]
[ 8  7  6  5  4  3 14 13 12 16 15 17  1  0  2 10  9 11]
[ 7  6  8  4  3  5 13 12 14 15 17 16  0  2  1  9 11 10]
[15 17 16  0  2  1  6  8  7 14 13 12 11 10  9  5  4  3]
[17 16 15  2  1  0  8  7  6 13 12 14 10  9 11  4  3  5]
[16 15 17  1  0  2  7  6  8 12 14 13  9 11 10  3  5  4]
[ 4  3  5 14 13 12  9 11 10  2  1  0  6  8  7 16 15 17]
[ 3  5  4 13 12 14 11 10  9  1  0  2  8  7  6 15 17 16]
[ 5  4  3 12 14 13 10  9 11  0  2  1  7  6  8 17 16 15]
[12 14 13  7  6  8 17 16 15 10  9 11  5  4  3  0  2  1]
[14 13 12  6  8  7 16 15 17  9 11 10  4  3  5  2  1  0]
[13 12 14  8  7  6 15 17 16 11 10  9  3  5  4  1  0  2]
[11 10  9 15 17 16  1  0  2  3  5  4 13 12 14  8  7  6]
[10  9 11 17 16 15  0  2  1  5  4  3 12 14 13  7  6  8]
[ 9 11 10 16 15 17  2  1  0  4  3  5 14 13 12  6  8  7]
2 10 9 11 4 3 5 6 8 7 15 17 16 12 14 13]
[ 6 8 7 3 5 4 12 14 13 17 16 15 2 1 0 11 10 9]
[ 8 7 6 5 4 3 14 13 12 16 15 17 1 0 2 10 9 11]
[ 7 6 8 4 3 5 13 12 14 15 17 16 0 2 1 9 11 10]
[15 17 16 0 2 1 6 8 7 14 13 12 11 10 9 5 4 3]
[17 16 15 2 1 0 8 7 6 13 12 14 10 9 11 4 3 5]
[16 15 17 1 0 2 7 6 8 12 14 13 9 11 10 3 5 4]
[ 4 3 5 14 13 12 9 11 10 2 1 0 6 8 7 16 15 17]
[ 3 5 4 13 12 14 11 10 9 1 0 2 8 7 6 15 17 16]
[ 5 4 3 12 14 13 10 9 11 0 2 1 7 6 8 17 16 15]
[12 14 13 7 6 8 17 16 15 10 9 11 5 4 3 0 2 1]
[14 13 12 6 8 7 16 15 17 9 11 10 4 3 5 2 1 0]
[13 12 14 8 7 6 15 17 16 11 10 9 3 5 4 1 0 2]
[11 10 9 15 17 16 1 0 2 3 5 4 13 12 14 8 7 6]
[10 9 11 17 16 15 0 2 1 5 4 3 12 14 13 7 6 8]
[ 9 11 10 16 15 17 2 1 0 4 3 5 14 13 12 6 8 7][/code]
А из шести ЛК уже не знает группу MOLS 18-го порядка
[code]NotImplementedError: I don't know how to build 6 MOLS of order 18[/code]
Супер!
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Message 1157 - Posted: 11 Jan 2021, 13:34:26 UTC
Last modified: 11 Jan 2021, 13:34:48 UTC

Группу из двух ортогональных ЛК 10-го порядка программа знает :)
sage: designs.mutually_orthogonal_latin_squares(2,10)

[1 8 9 0 2 4 6 3 5 7]  [1 7 6 5 0 9 8 2 3 4]
[7 2 8 9 0 3 5 4 6 1]  [8 2 1 7 6 0 9 3 4 5]
[6 1 3 8 9 0 4 5 7 2]  [9 8 3 2 1 7 0 4 5 6]
[5 7 2 4 8 9 0 6 1 3]  [0 9 8 4 3 2 1 5 6 7]
[0 6 1 3 5 8 9 7 2 4]  [2 0 9 8 5 4 3 6 7 1]
[9 0 7 2 4 6 8 1 3 5]  [4 3 0 9 8 6 5 7 1 2]
[8 9 0 1 3 5 7 2 4 6]  [6 5 4 0 9 8 7 1 2 3]
[2 3 4 5 6 7 1 8 9 0]  [3 4 5 6 7 1 2 8 0 9]
[3 4 5 6 7 1 2 0 8 9]  [5 6 7 1 2 3 4 0 9 8]
[4 5 6 7 1 2 3 9 0 8], [7 1 2 3 4 5 6 9 8 0]

А вот группу MOLS 10-го порядка из трёх ЛК не знает :)
sage: designs.mutually_orthogonal_latin_squares(3,10)

Ответ
NotImplementedError: I don't know how to build 3 MOLS of order 10

Я тоже не знаю такую группу :)
Её, похоже, пока никто не знает.
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Message 1158 - Posted: 11 Jan 2021, 13:39:04 UTC

Эх, а строить группы MODLS программа SageMath не умеет :(
Задаю команду
sage: designs.mutually_orthogonal_diagonal_latin_squares(2,4)

Получаю в ответ
AttributeError: module 'sage.combinat.designs.design_catalog' has no attribute 'mutually_orthogonal_diagonal_latin_squares'

Жаль!
Надо бы подсказать разработчикам программы.
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Message 1159 - Posted: 11 Jan 2021, 13:51:15 UTC
Last modified: 11 Jan 2021, 13:51:44 UTC

В моей статье
"ГРУППЫ ВЗАИМНО ОРТОГОНАЛЬНЫХ ЛАТИНСКИХ КВАДРАТОВ
Mutually Orthogonal Latin squares (MOLS)"
http://www.natalimak1.narod.ru/grolk.htm
приведена следующая таблица

n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Q(n) 1 2 3 4 1 6 7 8 2 10 5 12 3 4 15 16 3 18 4

где n - порядок ЛК, а Q(n) - максимальное количество (известное на тот момент) ЛК в группе MOLS порядка n.
Как видим, результат для порядка 18 изменился, теперь Q(18)=5.

Сейчас ещё для порядка 20 проверю.
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Message 1160 - Posted: 11 Jan 2021, 14:18:47 UTC

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(4,20)

Получаю
[20 x 20 dense matrix over Integer Ring,
 20 x 20 dense matrix over Integer Ring,
 20 x 20 dense matrix over Integer Ring,
 20 x 20 dense matrix over Integer Ring]

Кольцо!

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(5,20)

Получаю в ответ
NotImplementedError: I don't know how to build 5 MOLS of order 20

Не знает :)
Я тоже не знаю :)
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Message 1161 - Posted: 11 Jan 2021, 14:57:36 UTC
Last modified: 12 Jan 2021, 7:24:54 UTC

Смотрим последовательность OEIS https://oeis.org/A001438
A001438 Maximal number of mutually orthogonal Latin squares (or MOLS) of order n.
1, 2, 3, 4, 1, 6, 7, 8 (list; graph; refs; listen; history; text; internal format)

Здесь приведены значения Q(n) до порядка 9 включительно.

Для следующих порядков имеем на данный момент
Q(10)>=2, Q(11)=10, Q(12)>=5, Q(13)=12, Q(14)>=4, Q(15)>=4, Q(16)=15, Q(17)=16, Q(18)>=5, Q(19)=18, Q(20)>=4.
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Message 1162 - Posted: 11 Jan 2021, 15:02:44 UTC
Last modified: 11 Jan 2021, 15:03:43 UTC

Для порядка 14, кажется, ещё не показала группу MOLS.
Задаю команду
sage: designs.mutually_orthogonal_latin_squares(3,14)

Получаю
[ 1  9  8  7  6 10  4  2  0 11 13  5 12  3]
[ 5  2 10  9  1  7 11  4  3  8 12  0  6 13]
[12  6  3 11 10  2  1  0  5  4  9 13  8  7]
[ 2 13  7  4 12 11  3  1  8  6  5 10  0  9]
[ 4  3  0  1  5 13 12 10  2  9  7  6 11  8]
[13  5  4  8  2  6  0  9 11  3 10  1  7 12]
[ 8  0  6  5  9  3  7 13 10 12  4 11  2  1]
[ 3 11 13 10  4  0  5  8 12  7  2  9  1  6]
[ 6  4 12  0 11  5  8  7  9 13  1  3 10  2]
[ 9  7  5 13  8 12  6  3  1 10  0  2  4 11]
[ 7 10  1  6  0  9 13 12  4  2 11  8  3  5]
[ 0  1 11  2  7  8 10  6 13  5  3 12  9  4]
[11  8  2 12  3  1  9  5  7  0  6  4 13 10]
[10 12  9  3 13  4  2 11  6  1  8  7  5  0],

[ 1  5 12  2  4 13  8  3  6  9  7  0 11 10]
[ 9  2  6 13  3  5  0 11  4  7 10  1  8 12]
[ 8 10  3  7  0  4  6 13 12  5  1 11  2  9]
[ 7  9 11  4  1  8  5 10  0 13  6  2 12  3]
[ 6  1 10 12  5  2  9  4 11  8  0  7  3 13]
[10  7  2 11 13  6  3  0  5 12  9  8  1  4]
[ 4 11  1  3 12  0  7  5  8  6 13 10  9  2]
[ 2  4  0  1 10  9 13  8  7  3 12  6  5 11]
[ 0  3  5  8  2 11 10 12  9  1  4 13  7  6]
[11  8  4  6  9  3 12  7 13 10  2  5  0  1]
[13 12  9  5  7 10  4  2  1  0 11  3  6  8]
[ 5  0 13 10  6  1 11  9  3  2  8 12  4  7]
[12  6  8  0 11  7  2  1 10  4  3  9 13  5]
[ 3 13  7  9  8 12  1  6  2 11  5  4 10  0],

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

О-о-о!
Есть группа MOLS 14-го порядка из 4-х ЛК

[ 1  9  8  7  6 10  4  2  0 11 13  5 12  3]
[ 5  2 10  9  1  7 11  4  3  8 12  0  6 13]
[12  6  3 11 10  2  1  0  5  4  9 13  8  7]
[ 2 13  7  4 12 11  3  1  8  6  5 10  0  9]
[ 4  3  0  1  5 13 12 10  2  9  7  6 11  8]
[13  5  4  8  2  6  0  9 11  3 10  1  7 12]
[ 8  0  6  5  9  3  7 13 10 12  4 11  2  1]
[ 3 11 13 10  4  0  5  8 12  7  2  9  1  6]
[ 6  4 12  0 11  5  8  7  9 13  1  3 10  2]
[ 9  7  5 13  8 12  6  3  1 10  0  2  4 11]
[ 7 10  1  6  0  9 13 12  4  2 11  8  3  5]
[ 0  1 11  2  7  8 10  6 13  5  3 12  9  4]
[11  8  2 12  3  1  9  5  7  0  6  4 13 10]
[10 12  9  3 13  4  2 11  6  1  8  7  5  0],

[ 1  5 12  2  4 13  8  3  6  9  7  0 11 10]
[ 9  2  6 13  3  5  0 11  4  7 10  1  8 12]
[ 8 10  3  7  0  4  6 13 12  5  1 11  2  9]
[ 7  9 11  4  1  8  5 10  0 13  6  2 12  3]
[ 6  1 10 12  5  2  9  4 11  8  0  7  3 13]
[10  7  2 11 13  6  3  0  5 12  9  8  1  4]
[ 4 11  1  3 12  0  7  5  8  6 13 10  9  2]
[ 2  4  0  1 10  9 13  8  7  3 12  6  5 11]
[ 0  3  5  8  2 11 10 12  9  1  4 13  7  6]
[11  8  4  6  9  3 12  7 13 10  2  5  0  1]
[13 12  9  5  7 10  4  2  1  0 11  3  6  8]
[ 5  0 13 10  6  1 11  9  3  2  8 12  4  7]
[12  6  8  0 11  7  2  1 10  4  3  9 13  5]
[ 3 13  7  9  8 12  1  6  2 11  5  4 10  0],

[ 1  2  3  4  5  6  7  8  9 10 11 12 13  0]
[ 6 13 10  9  3 12  8  2 11  1  7  0  4  5]
[11  8  5  7  1  9  4 12 10  0 13  6  2  3]
[ 3  0 11 12  6 13  1  9  4  7  2  5  8 10]
[13  9  2  0  4  8  5  7  1  3  6 10 12 11]
[12  5  1 10  2  3 11  0  6 13  9  8  7  4]
[ 0  4 12 13  7 10  9  3  2  8  5  1 11  6]
[ 7  3 13  8 10  1  0  6 12  2  4 11  5  9]
[ 2  6  9  5 11  7 13  1  8  4 10  3  0 12]
[ 5 10  8  1 12  0  6  4 13 11  3  7  9  2]
[ 8 12  7 11 13  4  2 10  3  5  0  9  6  1]
[10 11  4  6  0  5  3 13  7  9 12  2  1  8]
[ 9  7  0  3  8  2 12 11  5  6  1  4 10 13]
[ 4  1  6  2  9 11 10  5  0 12  8 13  3  7],

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

А вот из 5 ЛК пока нет.
NotImplementedError: I don't know how to build 5 MOLS of order 14
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Message 1163 - Posted: 11 Jan 2021, 15:12:06 UTC
Last modified: 11 Jan 2021, 15:50:01 UTC

Для порядка 21
sage: designs.mutually_orthogonal_latin_squares(5,21)

Получаю
[21 x 21 dense matrix over Integer Ring,
 21 x 21 dense matrix over Integer Ring,
 21 x 21 dense matrix over Integer Ring,
 21 x 21 dense matrix over Integer Ring,
 21 x 21 dense matrix over Integer Ring]

Кольцо!
Группа MOLS 21-го порядка из 6 ЛК пока неизвестна
NotImplementedError: I don't know how to build 6 MOLS of order 21
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Message 1164 - Posted: 11 Jan 2021, 15:53:06 UTC

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(3,22)

Получаю
[22 x 22 dense matrix over Integer Ring,
 22 x 22 dense matrix over Integer Ring,
 22 x 22 dense matrix over Integer Ring]

Кольцо!

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(4,22)

Получаю
NotImplementedError: I don't know how to build 4 MOLS of order 22
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Message 1165 - Posted: 11 Jan 2021, 15:56:27 UTC

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(7,24)

Получаю
[24 x 24 dense matrix over Integer Ring,
 24 x 24 dense matrix over Integer Ring,
 24 x 24 dense matrix over Integer Ring,
 24 x 24 dense matrix over Integer Ring,
 24 x 24 dense matrix over Integer Ring,
 24 x 24 dense matrix over Integer Ring,
 24 x 24 dense matrix over Integer Ring]

Кольцо!

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(8,24)

получаю
NotImplementedError: I don't know how to build 8 MOLS of order 24
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Message 1166 - Posted: 11 Jan 2021, 15:59:54 UTC

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(4,26)

Получаю
[26 x 26 dense matrix over Integer Ring,
 26 x 26 dense matrix over Integer Ring,
 26 x 26 dense matrix over Integer Ring,
 26 x 26 dense matrix over Integer Ring]

Кольцо!

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(5,26)

получаю
NotImplementedError: I don't know how to build 5 MOLS of order 26
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Message 1167 - Posted: 11 Jan 2021, 16:03:09 UTC

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(5,28)

получаю
[28 x 28 dense matrix over Integer Ring,
 28 x 28 dense matrix over Integer Ring,
 28 x 28 dense matrix over Integer Ring,
 28 x 28 dense matrix over Integer Ring,
 28 x 28 dense matrix over Integer Ring]

Кольцо!

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(6,28)

получаю
NotImplementedError: I don't know how to build 6 MOLS of order 28
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Message 1168 - Posted: 11 Jan 2021, 16:08:24 UTC

Ну, и ещё для порядка 30.
Задаю команду
sage: designs.mutually_orthogonal_latin_squares(4,30)

получаю
[30 x 30 dense matrix over Integer Ring,
 30 x 30 dense matrix over Integer Ring,
 30 x 30 dense matrix over Integer Ring,
 30 x 30 dense matrix over Integer Ring]

Кольцо!

Задаю команду
sage: designs.mutually_orthogonal_latin_squares(5,30)

получаю
NotImplementedError: I don't know how to build 5 MOLS of order 30

Пока хватит :)
Хотелось бы узнать, что такое "dense matrix over Integer Ring", что Гугл переводит так "плотная матрица над целым кольцом".
И как эти квадратики получить, ежели они существуют?
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Message 1169 - Posted: 11 Jan 2021, 17:47:42 UTC

Проверка всех ЛК полной системы MOLS 27-го порядка программой Harry White GetOrthogonal

Order? 27

Enter the name of the squares file: inp
..output file inpPairs.txt
..output file inpPairNos.txt
squares 26 orthogonal pairs 325

Всё в порядке: ЛК образуют 325 ортогональных пар.
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