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 ≤ (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.In the study of mathematical objects, the question often arises of the exact determination of an object of a given class by some of its properties or parts; i.e., the question of the reconstruction of the object. In this paper we consider the question for the class of centered functions on the space of the binary tuples of fixed length; i.e., the functions such that the sum of the values of a function over any radius-1 balls is constant. In particular, this class includes some important discrete objects such as perfect binary codes with distance 3 (i.e., 1-error-correcting codes).It is known [1-3] that a centered function is uniquely defined by the values at the vertices of weight (n + 1)/2. Moreover, all values of a centered function in the ball of radius k ≤ (n + 1)/2 (where n is the dimension of the space) are uniquely defined by its values at the vertices of the corresponding sphere [4]. In this paper, we prove (Theorem 3) that all values of a centered function in a ball of radius k ≤ (n +1)/2 (where n is the dimension of the space) are uniquely defined from its radial sums with respect to the vertices of the corresponding sphere; i.e., the sums of values over the minimal faces each of which contains one vertex of the sphere and its center. Furthermore, we adduce the theorem on unique reconstruction of all values of a centered function in a ball from its values in the corresponding sphere (Theorem 2) and the theorem on unique reconstruction of a centered function from its values in a sphere of radius k = (n + 1)/2 (Theorem 1). As a corollary, we get a new property of the symmetry groups of centered functions (Theorem 4) and, consequently, of perfect codes.

We study the infinite graph of n-dimensional rectangular grid that doesn't appear distance regular and the distance regular colorings of this graph, which are defined as the distance colorings with respect to completely regular codes. It is proved that the elements of the parameter matrix of an arbitrary distance regular coloring form two monotonic sequences. It is shown that every irreducible distance regular coloring of the n-dimensional rectangular grid has at most 2n + 1 colors.

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