Perfectly Balanced Boolean Functions and Golić Conjecture

Abstract

Golić conjecture ([3]) states that the necessary condition for a function to be perfectly balanced for any choice of a tapping sequence is linearity of a function in the first or in the last essential variable. In the current paper we prove Golić conjecture.

On ρ-balanced Boolean functions

“…This generalises the result obtained in [2], which breaks the criterion for perfect balancedness found in [1].…”

“…The proof of this assertion in [1] was fundamentally incomplete, and it was indeed disproved in [2].…”

Perfectly balanced k-valued functions and the Golić condition

“…In [5], Golić's conjecture was completely proved. Thus, it is impossible to design a coding device on the base of a shift register and a filter function (over the alphabet f0; 1g) which preserves true randomness of a bit sequence under an arbitrary choice of the entry point (see [2]).…”

$\rho$-уравновешенные булевы функции

“…Данное утверждение является обобщением представленного в [2] результата, опровергающего полученный Голичем в [1] критерий совершенной уравнове-шенности.…”