Walsh permutation

The term Walsh permutation was chosen by the author for permutations that permute Walsh functions into other Walsh functions.
A Walsh permutation of degree n has length 2ⁿ, and corresponds to an n×n matrix in the Template:W of degree n over the Template:W.
This matrix will sometimes be called compression matrix, and its expression as a vector of n integers $\in \{1,...,2^n-1\}$ will be called compression vector.

There are Template:Oeis(n) Walsh permutations of degree n.

File:Walsh permutation wp(8,12,2,3) * wp(6,9,1,2).svg

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)

File:Walsh permutation wp(8,12,2,3) * wp(6,9,1,2) small.svg

Product of the compression matrices

Not all vectors $(v_1,...,v_n)$ with different elements $\in \{1,...,2^n-1\}$ correspond to Walsh permutations, as the following example shows:

No Walsh permutation wp(1,2,3,4)
Error creating thumbnail: This is not a permutation of 16 elements, so there is no Walsh permutation $wp(1,2,3,4)$. The corresponding compression matrix would be which has determinant 0.

Notation warning

This article and its subpages currently use two different ways to name and display Walsh permutations. In older files the direction of the compression vector is horizontal, and the permutation is vertical. In new files both compression vector and permutation are horizontal. The old files are gradually replaced. 3-bit Walsh permutation already uses new files.

File:Walsh permutation 135.svg wp(1, 3, 5) new = wp(7, 2, 4) old

Error creating thumbnail: Ambiguity of wp(1, 3, 5) new: horizontal (0, 1, 3, 2, 5, 4, 6, 7) old: vertical (0, 7, 2, 5, 4, 3, 6, 1)

File:Walsh permutation wp( 1,14, 4, 8).svg wp(1, 14, 4, 8) old = wp(1, 2, 6, 10) new

Schoute permutation

Error creating thumbnail:

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.

Template:Rdrdo Nimber multiplication

Error creating thumbnail:

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).

Template:Rdrdo Gray code permutation powers

Error creating thumbnail:

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.

See also Algebraic normal form.

Template:Rdrdo Sequency ordered Walsh matrix

Error creating thumbnail:

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.

Template:Rdrdo Bent functions

File:From bent function 0111 1000 1000 1000 to wp( 2, 1, 8, 4).svg

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).

Template:Rdrdo Magic squares

File:Multigrade operator XOR diamond.svg Multigrade operator XOR These matrices are found in the dual matrices of the magic squares.

File:Square 00-11-07-12---14-05-09-02---13-06-10-01---03-08-04-15.svg 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.

Template:Rdrdo Inversions

Error creating thumbnail: wp(13,11, 7,15)

File:Inversion set 16; Walsh (14, 13, 11, 7).svg wp(14,13,11, 7)

File:Inversion set 16; Walsh (4, 8, 1, 2).svg 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.

Template:Rdrdo 2-bit

Template:Multiple image The Template:Oeis(2) = 6 2-bit Walsh permutations form general linear group GL(2,2), which is also the symmetric group S₃.

Template:Rdrdo 3-bit

Template:Multiple image The Template:Oeis(3) = 168 3-bit Walsh permutations correspond to the collineations of the Fano plane.

Compression vectors, Permutations

4-bit

There are Template:Oeis(4) = 20160 4-bit Walsh permutations.

Compression vectors

Fixed points

The fixed points of an n-bit Walsh permutation are seals of arity n.