# Noble Boolean functions

 ANF   matrix   twins noble   gentle smallest Zhegalkin index

Noble Boolean functions are those who are their own Zhegalkin twins, i.e. the binary expression of their ANF is equal to their truth table.
They correspond to fixed points in the Zhegalkin permutation.
They are all even, i.e. the first digit of their truth table is false.
When a Boolean function is noble, its whole faction is noble.
Within the this project it is slightly misleading to apply the term noble to Boolean functions. It is a property of a truth table with a specific length.

3-ary nobles (Venn diagrams) assigned to juniors (tesseract vertices)

It is easily seen, that the left and right half of each row differ only in the last digit.
Those on the left/right have even/odd weight. They shall be called evil/odious.

There is a second way to partition the nobles in two halves:
Those in even/odd places of the triangle row have false/true entries in all 1-bit places of their truth table. They shall be called weak/strong.

So the nobles can be partitioned into four quadrants by depravity and strength.

illustration of quadrants for n = 3

(40 and 214 are relative complements. 40 and 168 differ only in the central vertex.)

The 16 noble 3-ary Boolean functions form 8 factions. So there are 2 royal factions.

Nobles that are evil and weak shall be called royal.
Each noble corresponds to a royal, and can easily be derived from it. (A royal corresponds to itself.)
When a Boolean function is royal, its whole faction is royal.
The function with the smallest Zhegalkin index in a faction shall be called king, and be used to represent it.
So all nobles of a given arity can effectively be represented by a rather short list of kings.

### group under exclusive or

With XOR as a group operation the n-ary noble and royal Boolean functions form a power of the cyclic group C2.

## patrons

The XOR of twins is noble, i.e. the XOR of a key and a value in ${\displaystyle \Pi _{n}}$ is an entry of ${\displaystyle \Phi _{n}}$.
For a Boolean function this means, that the XOR of its ANF and its truth table is a noble Boolean function, which shall be called its patron.
The patron of a noble is the contradiction.

For ${\displaystyle n\geq 1}$ the entries in ${\displaystyle \Xi _{n}}$ are repetitions of those in ${\displaystyle \Phi _{n}}$. E.g. ${\displaystyle \Xi _{2}}$ contains the entries ${\displaystyle \{0,6,8,14\}}$, each repeated four times.

The set of places where ${\displaystyle \Xi _{n}}$ has entries ${\displaystyle \Phi _{n,~k}}$ can be calculated by XORing ${\displaystyle \Pi _{n-1,~k}}$ with the entries of ${\displaystyle \Phi _{n}}$.