Order 4 magic squares; Walsh permutations

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Order 4 magic squares Rdrup.svg
Walsh permutation Rdrup.svg
Multigrade operator XOR
These matrices are found in the dual matrices of the magic squares.
The magic square corresponding to wp( 7,11,13,14), below the dual matrix

There are 24*9=216 Walsh permutations that correspond to magic squares of order 4.
They have these 9 compression vectors and all their permutations:

 5   7  10  11
 5   7  10  14
 5   7  11  13
 5   7  13  14
 5  10  11  13
 5  10  13  14
 7  10  11  14
 7  11  13  14
10  11  13  14


The following table shows the compression vectors in the RevCoLex order of the permutations.
Half of those of type IV, V, VI are omitted, because they are reflections. Half of those of type I, II, III are reflections, but they are shown.


I Square 0101 0101 1010 1010.svg    (p=6, x=9) II Square 0011 1100 0011 1100.svg    (p=9, x=6) III Square 0110 1001 1001 0110.svg    (p=15, x=15) IV Square 0000 1111 0000 1111.svg    (p=1, x=4) V Square 0000 0000 1111 1111.svg    (p=2, x=8) VI Square 0000 1111 1111 0000.svg    (p=3, x=12)
11  13   5  10
11   5  13  10
13  11   5  10
 5  11  13  10
13   5  11  10
 5  13  11  10
11  13  10   5
11   5  10  13
13  11  10   5
 5  11  10  13
13   5  10  11
 5  13  10  11
11  10  13   5
11  10   5  13
13  10  11   5
 5  10  11  13
13  10   5  11
 5  10  13  11
10  11  13   5
10  11   5  13
10  13  11   5
10   5  11  13
10  13   5  11
10   5  13  11
 7   5  14  10
 7   5  10  14
 5   7  14  10
 5   7  10  14
 7  14   5  10
 7  10   5  14
 5  14   7  10
 5  10   7  14
14   7   5  10
10   7   5  14
14   5   7  10
10   5   7  14
 7  14  10   5
 7  10  14   5
 5  14  10   7
 5  10  14   7
14   7  10   5
10   7  14   5
14   5  10   7
10   5  14   7
14  10   7   5
10  14   7   5
14  10   5   7
10  14   5   7
 7  11  13  14
11   7  13  14
 7  13  11  14
11  13   7  14
13   7  11  14
13  11   7  14
 7  11  14  13
11   7  14  13
 7  13  14  11
11  13  14   7
13   7  14  11
13  11  14   7
 7  14  11  13
11  14   7  13
 7  14  13  11
11  14  13   7
13  14   7  11
13  14  11   7
14   7  11  13
14  11   7  13
14   7  13  11
14  11  13   7
14  13   7  11
14  13  11   7
 7  11  13   5
 7  11   5  13
11   7  13   5
11   7   5  13
 7  13  11   5
 7   5  11  13
11  13   7   5
11   5   7  13
13   7  11   5
 5   7  11  13
13  11   7   5
 5  11   7  13
 7  13   5  11
 7   5  13  11
11  13   5   7
11   5  13   7
13   7   5  11
 5   7  13  11
13  11   5   7
 5  11  13   7
13   5   7  11
 5  13   7  11
13   5  11   7
 5  13  11   7
 7  11  14  10
11   7  14  10
 7  11  10  14
11   7  10  14
 7  14  11  10
11  14   7  10
 7  10  11  14
11  10   7  14
14   7  11  10
14  11   7  10
10   7  11  14
10  11   7  14
 7  14  10  11
 7  10  14  11
11  14  10   7
11  10  14   7
14   7  10  11
10   7  14  11
14  11  10   7
10  11  14   7
14  10   7  11
10  14   7  11
14  10  11   7
10  14  11   7
13   5  14  10
 5  13  14  10
13   5  10  14
 5  13  10  14
13  14   5  10
 5  14  13  10
13  10   5  14
 5  10  13  14
14  13   5  10
14   5  13  10
10  13   5  14
10   5  13  14
13  14  10   5
 5  14  10  13
13  10  14   5
 5  10  14  13
14  13  10   5
14   5  10  13
10  13  14   5
10   5  14  13
14  10  13   5
14  10   5  13
10  14  13   5
10  14   5  13
Square 00-07-09-14---06-01-15-08---11-12-02-05---13-10-04-03.svg

wp(11,13, 5,10)
Square 00-06-11-13---07-01-12-10---09-15-02-04---14-08-05-03.svg

wp(14, 7, 5,10)
Square 00-07-14-09---13-10-03-04---11-12-05-02---06-01-08-15.svg

wp(13,11,14, 7)
Square 00-15-06-09---13-02-11-04---03-12-05-10---14-01-08-07.svg

wp(13,11, 7, 5)
Square 00-06-15-09---03-05-12-10---13-11-02-04---14-08-01-07.svg

wp(14, 7,11,10)
Square 00-06-09-15---14-08-07-01---13-11-04-02---03-05-10-12.svg

wp(10, 5,13,14)

Sign1101.svg   Sign1011.svg   Sign1010.svg   Sign0101.svg

Sign0111.svg   Sign1110.svg   Sign1010.svg   Sign0101.svg

Sign1011.svg   Sign1101.svg   Sign0111.svg   Sign1110.svg

Sign1011.svg   Sign1101.svg   Sign1110.svg   Sign1010.svg

Sign0111.svg   Sign1110.svg   Sign1101.svg   Sign0101.svg

Sign0101.svg   Sign1010.svg   Sign1011.svg   Sign0111.svg



There are 24*81=1944 Walsh permutations that correspond to semi-magic squares of order 4 (where the diagonals do not necessarily sum up to 30).
They have these 81 compression vectors and all their permutations:

 5   6   7   9
 5   6   7  10
 5   6   7  11
 5   6   7  13
 5   6   7  14
 5   6   7  15
 5   6   9  11
 5   6   9  13
 5   6   9  14
 5   6  10  11
 5   6  10  13
 5   6  10  14
 5   6  11  15
 5   6  13  15
 5   6  14  15
 5   7   9  10
 5   7   9  13
 5   7   9  15
 5   7  10  11
 5   7  10  14
 5   7  11  13
 5   7  11  15
 5   7  13  14
 5   7  14  15
 5   9  10  11
 5   9  10  13
 5   9  10  14
 5   9  11  13
 5   9  11  15
 5   9  13  14
 5   9  13  15
 5   9  14  15
 5  10  11  13
 5  10  13  14
 5  11  13  15
 5  13  14  15
 6   7   9  10
 6   7   9  11
 6   7   9  13
 6   7  10  14
 6   7  10  15
 6   7  11  14
 6   7  11  15
 6   7  13  14
 6   7  13  15
 6   9  10  11
 6   9  10  13
 6   9  10  14
 6   9  11  14
 6   9  13  14
 6  10  11  14
 6  10  11  15
 6  10  13  14
 6  10  13  15
 6  10  14  15
 6  11  14  15
 6  13  14  15
 7   9  10  11
 7   9  10  15
 7   9  11  13
 7   9  11  15
 7   9  13  15
 7  10  11  14
 7  10  11  15
 7  10  14  15
 7  11  13  14
 7  11  13  15
 7  11  14  15
 7  13  14  15
 9  10  11  13
 9  10  11  14
 9  10  11  15
 9  10  13  15
 9  10  14  15
 9  11  13  14
 9  11  14  15
 9  13  14  15
10  11  13  14
10  11  13  15
10  13  14  15
11  13  14  15