Reconstruction theorems for centered functions and perfect codes

The article addresses the centered functions and perfect codes in the space of all binary $n$-tuples. We prove that all values of a centered function in a ball of radius $k\le(n+1)/2$ are uniquely defined from its radial sums with respect to the vertices of the corresponding sphere. We present some theorems of full and partial reconstruction of a centered function from part of its values and derive a new property of the symmetry groups of centered functions.