ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 2, pp. 151–157.
Pleiades Publishing, Inc., 2009.
Original Russian Text
A.Yu. Vasil’eva, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 2, pp. 84–90.
Local and Interweight Spectra of Completely
Regular Codes and of Perfect Colorings
A. Yu. Vasil’eva
Sobolev Institute of Mathematics, Siberian Branch of the RAS, Novosibirsk
Received December 29, 2008
Abstract—We introduce notions of local and interweight spectra of an arbitrary coloring of
a Boolean cube, which generalize the notion of a weight spectrum. The main objects of our
research are colorings that are called perfect. We establish an interrelation of local spectra of
such a coloring in two orthogonal faces of a Boolean cube and study properties of the interweight
spectrum. Based on this, we prove a new metric property of perfect colorings, namely, their
strong distance invariance. As a consequence, we obtain an analogous property of an arbitrary
completely regular code, which, together with his neighborhoods, forms a perfect coloring.
The main objects of our study are perfect colorings of a Boolean cube, i.e., colorings in which
the color composition of a neighborhood of each vertex is uniquely determined by the color of
this vertex. By analogy with codes, for colorings we introduce the notions of a local spectrum
(Section 3), which gives color compositions of neighborhoods of a ﬁxed vertex in a face, and of an
interweight spectrum (Section 4), which characterizes the distribution (over layers of the cube) of
pairs of vertices of any given colors at any given distance apart from each other.
In Section 3 we prove mutual determinacy of local spectra of a perfect coloring in a pair of
orthogonal faces of a cube (Theorem 1). In Section 4 we formulate a result (Theorem 3) on the
unique determinacy of the interweight spectrum of a perfect coloring by the color of the zero vertex.
This theorem allows us to speak about strong distance invariance of perfect colorings (Corollary 1).
A similar property is also found for completely regular codes (Corollary 2), which by deﬁnition
generate a perfect coloring of all vertices of a cube if they are colored according to the distance to
the code. A proof of Theorem 3 is the content of Section 5.
We denote the set of all binary n-tuples by E
and call it a Boolean cube; its elements are
referred to as vertices. In this space we consider the Hamming metric, where the distance ρ(x, y)
between two vertices x and y equals the number of positions in which they diﬀer. The Hamming
weight wt(x)ofavertexx is the number of nonzero positions of x; we denote the set of all vertices
of weight k by W
and call it the kth layer of the Boolean cube.
The set of all vertices of an n-cube that coincide in ﬁxed n − k coordinates is called a k-dimen-
sional face of the n-cube. Faces γ and
γ are said to be otrthogonal if the set of coordinates in
which all vertices of
γ coincide is the complement of the set of coordinates in which all vertices
Supported in part by the Russian Foundation for Basic Research, project no. 07-01-00248-a.