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On correlation-immune and stable Boolean functions. (Russian) Zbl 1280.94128
The weight $$\operatorname{wt}(f)$$ of a Boolean function $$f(x)=f(x_1,\ldots,x_n)$$ is the number of binary tuples $$(\alpha_1,\ldots,\alpha_n)$$ such that $$f(\alpha_1,\ldots,\alpha_n)=1$$. The function $$f$$ is called balanced if $$\operatorname{wt}(f)=\operatorname{wt}(f\oplus 1)=2^{n-1}$$. A subfunction $$f'$$ of $$f$$ is a Boolean function obtained from $$f$$ by substituting constants 0 or 1 for some variables of $$f$$. The function $$f$$ is called correlation-immune of order $$m$$ $$(1\le m\le n)$$ if $$\operatorname{wt}(f')=\operatorname{wt}(f)/2^m$$ for each of its subfunctions $$f'$$ of $$n-m$$ variables. A balanced correlation-immune function of order $$m$$ is called $$m$$-stable. In other words, the function $$f$$ is $$m$$-stable if $$\operatorname{wt}(f')=2^{n-m-1}$$ for each of its subfunctions $$f'$$ of $$n-m$$ variables.
This paper reviews most of the author’s results obtained since 2001 concerning properties of $$m$$-stable correlation-immune functions.

##### MSC:
 94D10 Boolean functions 94C11 Switching theory, applications of Boolean algebras to circuits and networks 62H20 Measures of association (correlation, canonical correlation, etc.)