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

Walsh permutation, 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.

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

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.