This file illustrates the nonlinearity of the 4-ary Boolean function
represented by the binary row vector on top. (white 0, red 1)

Below there are the binary Walsh matrix of order 16 and its complement.
Their rows are the 32 linear 4-ary Boolean functions.

The dots in the rows show in which bits they differ from the Boolean function.
The numbers on the right are the number of dots in each row
and thus the Hamming distance of the row and the Boolean function.

The lowest and biggest numbers are bold.

The lowest number is the Boolean functions nonlinearity,
i.e. the lowest number of bits in which it differs from a linear function.

(When the nonlinearity is 0 the function is linear,
when it's 6 the function is bent.)

This is a Bent function.

Its Walsh spectrum is ${\displaystyle (6,-2,-2,2,~~~-2,-2,2,-2,~~~-2,2,2,2,~~~-2,2,-2,-2)}$.

Its binary Walsh spectrum is the null vector.

It belongs to this big equivalence class (top row in matrix number 0),
containing 12*16 = 192 functions, which are probably all bent as well.