Walsh permutation

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The term n-bit Walsh permutation was chosen by the author for permutations of 2n elements, that permute Walsh functions into other Walsh functions.

Walsh permutations of 2n elements can be represented by an n-element compression vector
of integers referring to the numbering of Walsh functions (displayed in orange in the images).
It corresponds to the n×n compression matrix. It's columns show the elements of the compression vector in binary representation.
The multiplication of the compression matrices (with F2 operations) corresponds to the composition of the permutations.
As the compression matrices are the elements of the general linear group GL(n,2) the Walsh permutations form a group isomorphic to GL(n,2).

There are Sloane'sA002884(n) n-bit Walsh permutations.

Product of two Walsh permutations: wp( 8,12, 2, 3) * wp( 6, 9, 1, 2) = wp(14,11, 8,12)
(the product of two Walsh permutations related to bent functions)
Product of the compression matrices

Not all vectors with different elements correspond to Walsh permutations, as the following example shows:

Notation warning[edit | edit source]

In this article and the related material it was chosen to base the compression vector on the columns of the compression matrix. But it would probably have been better to base it on the rows. Then the bit permutations and the small permutations corresponding to their compression vectors would be homomorphic. With the current convention they form an antihomomorphism. This might be corrected in the future.

With the row based convention the result of the example multiplication above would be wp(2, 3, 9, 15) instead of wp(14, 11, 8, 12).

Rdrdo.svg Bit permutation[edit | edit source]

Walsh permutation wp( 2, 1, 8, 4).svg

When a simple permutation of n elements is applied on the binary digits of numbers from 0 to 2n-1 the result is a permutation of the numbers from 0 to 2n-1.
The example on the right corresponds to the simple permutation that swaps the first two and the last two digits of the 4-bit binary numbers.
Probably the most important bit permutation is the bit-reversal permutation.

Rdrdo.svg Nimber multiplication[edit | edit source]

Nimbers 0...15 multiplication.svg

The rows of each nimber multiplication table are Walsh permutations (except row 0).
They are not only closed under multiplication (function composition), but even under addition (bitwise XOR).

Rdrdo.svg Gray code permutation powers[edit | edit source]

Cayley table with compression matrices

Powers of the Gray code permutation have very simple compression matrices and vectors.
In each vector all entries follow from the first one, and the first entries follow from the rows of the Sierpinski triangle.

Rdrdo.svg Sequency ordered Walsh matrix[edit | edit source]

Natural and sequency ordered Walsh 16.svg

The permutation that changes the natural ordered into the sequency ordered Walsh matrix is the product of the Gray code permutation and the bit-reversal permutation.

Rdrdo.svg Bent functions[edit | edit source]

From 0111 1000 1000 1000 to wp(2,1,8,4)

Each bent function corresponds to a Walsh permutation.
This can be seen when all rows of its sec matrix are XORed with the function itself (second step in the example on the right).

Rdrdo.svg Magic squares[edit | edit source]

Multigrade operator XOR
These matrices are found in the dual matrices of the magic squares.
The Melencolia square, corresponding to wp(11, 7,14,13)

There are 24*9=216 Walsh permutations that correspond to magic squares of order 4.
One my say that only 6 of them are essentially different.

Rdrdo.svg Inversions[edit | edit source]

wp(13,11, 7,15)
wp(14,13,11, 7)
wp( 4, 8, 1, 2)

Some inversion sets of Walsh permutations are very regular.
E.g. there are 2n-1 n-bit Walsh permutations with horizontally striped inversion sets (like the left one of the examples).
The pattern of the stripes is that of Walsh functions.

2-bit[edit | edit source]

The 2-bit Walsh permutations form the general linear group GL(2,2) which is also the symmetric group S3.

GL(2,2) = SL(2,2)
2-bit Walsh permutations
Permutations of 3 elements
Symmetric group 3; Cayley table (left); subgroup of S4 (elements 0,2,6,8,12,14).svg

Rdrdo.svg 3-bit[edit | edit source]

wp(5,3,7) as a permutation of the Fano plane
Connection between permutation and matrix patterns
wp(5,3,7) has permutation pattern 73, matrix pattern 74.b, and its inversion set is striped.

The Sloane'sA002884(3) = 168 3-bit Walsh permutations correspond to the collineations of the Fano plane.

Compression vectors, Permutations

4-bit[edit | edit source]

There are Sloane'sA002884(4) = 20160 4-bit Walsh permutations.

Compression vectors

Fixed points[edit | edit source]

The fixed points of an n-bit Walsh permutation correspond to a subgroup of Z2n. (Compare Subgroups of nimber addition)

Sona number 12 (compare matrix 12 in this list)