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Computation of a centered function through its values on middle layers of the Boolean cube. (Russian) Zbl 1067.94018
The notion of a \(K\)-centered function is introduced as a generalization of the notion of a perfect code. A real-valued function defined on the Boolean \(n\)-dimensional cube \(E^n\) is called \(K\)-centered if the sum of its values over every sphere of radius 1 equals \(K\). The characteristic function of a perfect code is a 1-centered function on \(E^n\), for an appropriate \(n\), with values from \(\{0,1\}\) and vice versa. It is proven that the values of a centered function on all vertices of the \(n\)-cube are uniquely determined by its values on a middle layer of the \(n\)-cube. An explicit formula is presented.

MSC:
94B25 Combinatorial codes
06E30 Boolean functions
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