Zhegalkin twins

Zhegalkin twins are truth tables of the same length, that correspond to a column and its index in a finite Zhegalkin matrix. For a given arity, one can also describe two integers or Boolean functions as Zhegalkin twins. (For truth tables the arity is implied by their length.)

Calculation
These images show how one truth table is calculated from the other. (One could describe it as a of variadic  s .) Here the red patterns are meant as Boolean functions, while the green patterns are meant as their Zhegalkin indices.

The red pattern represents a truth table, which continues periodically. The green bit-pattern represents an integer, and continues with zeros.

The least significant green bit of the green pattern is on top. That of the red pattern is on the left. One can see, that the LSB s of both patterns are equal. Also that changing the green LSB will change all red bits.

Equivalence classes
See Smallest Zhegalkin index for details.

This example is just like the one above. But potential confusion comes from the fact, that the 4-ary function has a twin that is only 2-ary. The twin relationship is really one between truth tables of a given length, and not one between Boolean functions.

Selections by equal place
These images show half the columns of the 8×256 matrix. They are selected by the truth value of one row. (There are also images for other rows, e.g. false rows in the lower matrix) The columns of the upper long matrix are to be understood as Zhegalkin indeces, and the columns below as truth tables. The rows are linear Boolean functions, and the small matrices on the left show their Walsh indices. (If the row is a negated Walsh function, its first entry is true.) The file with even/evil functions shows the complements of the odd/odious functions. (Complements have opposite parity as well as opposite depravity.)