Linear Boolean functions



Linear Boolean functions are Walsh functions and their negations. A Walsh function is a variadic (a.k.a. ). The set of arguments in that XOR can easily be expressed as an integer, which shall be called Walsh index.

The natural way to denote a linear function is as Walsh index and parity. Walsh functions have parity 0. Their negations have parity 1. Sometimes a more arcane notation is useful, namely as leader and quadrant. The leader is the of the Walsh index. The quadrant is the first and last digit of the truth table expressed in an integer. The tables below show the Walsh indices on gray and the quadrants on colored background. The Walsh indices of functions with quadrants 0 and 3 are s 0, 3, 5, 6...  For quadrants 1 and 2 they are s 1, 2, 4, 7...

The columns marked with Ж show the Zhegalkin indices. Those of the Walsh functions are sequence. Those of the negations are bigger by 1.

See also Linear and noble Boolean functions.

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