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
linears   (truth tables and Zhegalkin indices)	nobles

3-ary
linears   (truth tables and Zhegalkin indices)	nobles

4-ary
linears   (truth tables) linears   (Zhegalkin indices) nobles

3-ary

nobles in tesseract
	dihedral symmetry	mirror symmetry

quadrants
0   (even, evil)	3   (odd, odious)
2   (even, odious)	1   (odd, evil)

halves
even Walsh functions	odd complements of Walsh functions

all

4-ary

linear to patron (noble)

The patrons of the 32 linears are 32 nobles. Both linears and nobles belong to ten factions.
Their indices are the 3-ary Boolean functions with consul 0.

3-ary noble indices (juniors)
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

Evil/odious Walsh functions are in three/two factions. So are their complements.
The following images show that pairs of complementary factions are in the same principality.

quadrants 0 and 3
king 0 3808 26752 king index 0 14 104        The patrons of the 8 even Walsh functions (quadrant 0) are the red entries in these 3 principalities.        The patrons of their complements (quadrant 3) are the green entries. truth tables Zhegalkin indices

quadrants 1 and 2
king 5760 10920 king index 22 42        The patrons of the 8 odious Walsh functions (quadrant 2) are the yellow entries in these 2 principalities.        The patrons of their complements (quadrant 1) are the blue entries. truth tables Zhegalkin indices

noble to prefect (linear)

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