Linear and noble Boolean functions

Studies of Boolean functions
arity 1 2 3 4 5
linear 4 8 16 32 64
noble 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

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

3-ary

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


4-ary

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linear to patron (noble)

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There are 32 nobles, 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.

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

noble to prefect (linear)

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