Walsh permutation; nimber multiplication; powers of wp(15, 5,11,13)

Walsh permutation; nimber multiplication

Walsh permutation wp(15,5,11,13), related to the nimber multiplication table, has one fixed point and a 15-cycle.
Thus it has 15 different powers, which form a cyclic group.

Cycle graph of the cyclic group Z15

The 15 permutations in a matrix, below the dual matrix The black numbers are the exponents. The grey numbers from 1 to 15 on the left refer to the Walsh functions defined by the pattern of odd numbers. The orange numbers in the dual matrix read f.r.t.l. give the compression vectors.

This is the group's Cayley table, showing the multiplication of the compression matrices:

Cayley table of the cyclic group Z₁₅

Compression vectors:
 1   2   4   8
15   5  11  13
12   4   7   9
 6  11   1   2
14   7  15   5
 3   1  12   4
10  15   6  11
 8  12  14   7
13   6   3   1
 9  14  10  15
 2   3   8  12
 5  10  13   6
 4   8   9  14
11  13   2   3
 7   9   5  10

Permutations:
0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
0  15   5  10  11   4  14   1  13   2   8   7   6   9   3  12
0  12   4   8   7  11   3  15   9   5  13   1  14   2  10   6
0   6  11  13   1   7  10  12   2   4   9  15   3   5   8  14
0  14   7   9  15   1   8   6   5  11   2  12  10   4  13   3
0   3   1   2  12  15  13  14   4   7   5   6   8  11   9  10
0  10  15   5   6  12   9   3  11   1   4  14  13   7   2   8
0   8  12   4  14   6   2  10   7  15  11   3   9   1   5  13
0  13   6  11   3  14   5   8   1  12   7  10   2  15   4   9
0   9  14   7  10   3   4  13  15   6   1   8   5  12  11   2
0   2   3   1   8  10  11   9  12  14  15  13   4   6   7   5
0   5  10  15  13   8   7   2   6   3  12   9  11  14   1   4
0   4   8  12   9  13   1   5  14  10   6   2   7   3  15  11
0  11  13   6   2   9  15   4   3   8  14   5   1  10  12   7
0   7   9  14   5   2  12  11  10  13   3   4  15   8   6   1

These 15 Walsh permutations are also closed under addition (bit-XOR) together with the neutral element wp(zeros)=zeros.
This is the corresponding Cayley table, where 0 stands for the netral element and all other n for wp(15, 5,11,13)^(n-1):

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

The matrix is symmetric and the permutations formed by the rows are not Walsh.