Linear and noble Boolean functions

arity	$n$	1	2	3	4	5
linear	$2^{n+1}$	4	8	16	32	64
noble	$2^{2^{n-1}}$	2	4	16	256	65536

Among the truth tables for a given arity, the linears and the nobles are important subsets.

Each linear can be assigned a patron, which is noble. Each noble can be assigned a prefect, which is linear.

For arity 3 they form a bijection. For higher arities the nobles outnumber the linears (i.e. the patrons of the linears are a subset of the nobles).

overview

2-ary
	File:2-ary nobles in matrix.svg nobles

3-ary
	Error creating thumbnail: nobles

3-ary

nobles in tesseract
	dihedral symmetry	mirror symmetry

halves
even Walsh functions	odd complements of Walsh functions

all

4-ary

linear to patron (noble)

There are 32 linears, which form 10 factions. (Even and odd faction for each Walsh weight 0...4.) Their patrons are 32 nobles, which also form 10 factions (among the 44 noble factions).

Their juniors are the 3-ary Boolean functions with consul 0.   (This is not the case for arities 2 and 3, but probably for all arities ≥ 4.)

Template:Collapsible START The faction is determined by Walsh weight and parity (or quadrant), and represented by variants of the quadrant colors.
A good sort order is first by quadrant, and then by Walsh weight. Template:Linear to patron 4-ary list Template:Collapsible START Template:Linear to patron 4-ary sequences Template:Collapsible END Template:Collapsible END

Template:Collapsible START The numbers next to the images correspond to the truth tables.
The linears are in the left columns, their patrons in the middle, and their twins on the right.   (So the Zhegalkin indices of the linear functions are in the columns on the right.) Template:Linear to patron 4-ary factions Template:Collapsible END

The factions are shown in ten different colors, which are variants of the quadrant colors.   (There are three shades of red and green, and two shades of blue and yellow.)
truth tables	Zhegalkin indices

The following images show that pairs of complementary factions are in the same principality.

king	0	3808	26752
king index	0	14	104

Template:Colorbox   The patrons of the 8 even Walsh functions (quadrant 0) are the red entries in these 3 principalities.
Template:Colorbox   The patrons of their complements (quadrant 3) are the green entries.

truth tables

Zhegalkin indices

king	5760	10920
king index	22	42

Template:Colorbox   The patrons of the 8 odious Walsh functions (quadrant 2) are the yellow entries in these 2 principalities.
Template:Colorbox   The patrons of their complements (quadrant 1) are the blue entries.

truth tables

Zhegalkin indices

noble to prefect (linear)

There are 256 nobles. A prefect is one of the 32 linears. Every linear is the prefect of 8 nobles.

Template:Collapsible START This table is an extension of the one shown above (from linear to patron). The leftmost noble column is equal to that of patrons.
But here the assignment goes in the other direction. Each of the nobles on the right has the prefect on the left. E.g. the nobles 3870 and 2600 have prefect Ж 17.

The nobles are represented by the integer values of their truth tables, which are also their Zhegalkin indices.
The small numbers to their right are the juniors 0...255.
They correspond to 3-ary Boolean functions, so among them are also linears and nobles. The linears are highlighted in bold, and the nobles with a box.

The background colors of the nobles denote the noble quadrants.
The tiny red numbers are the king indices. Together with the noble quadrants they denote the noble factions.   (The eleven king indices are 0, 2, 6, 8, 14, 22, 26, 42, 44, 104, 110.)

Template:Collapsible START The right side of the following table shows the same information as the table above.
Entries of one color in one matrix are the eight juniors in a row of the table above.   (Linears are highlighted in bold, and the nobles with a crown.)

The left side of the table shows the prefects of all 3-ary Boolean functions. (And the great prefects as sort keys.)
It can be seen, that the Template:W of the 4-ary prefects are a refinement of the 3-ary prefects. This motivates the term subprefect.

A more detailed version of this table can be seen here. Template:Collapsible END