ISSN 0032-9460, Problems of Information Transmission, 2013, Vol. 49, No. 1, pp. 32–39.
Pleiades Publishing, Inc., 2013.
Original Russian Text
A.Yu. Vasil’eva, 2013, published in Problemy Peredachi Informatsii, 2013, Vol. 49, No. 1, pp. 37–45.
Local Distributions and Reconstruction
of Hypercube Eigenfunctions
A. Yu. Vasil’eva
Sobolev Institute of Mathematics,
Siberian Branch, Russian Academy of Sciences, Novosibirsk
Received March 7, 2012; in ﬁnal form, October 31, 2012
Abstract—We study eigenfunctions of a binary n-dimensional hypercube. We obtain a for-
mula relating local distributions of such a function in a pair of orthogonal faces. Based on
this, we prove that under certain conditions an eigenfunction can be reconstructed partially or
completely given its values on a sphere.
Objects of our study are eigenfunctions of the binary n-dimensional hypercube (n-cube) F
precisely, eigenfunctions of the adjacency matrix of its graph. It is well known that eigenvalues
of the n-cube graph are integers of the form λ = n − 2i, i =0, 1,...,n. The corresponding
eigenfunctions are deﬁned by
f(y)=λf(x), x ∈ F
where by S
(x) we denote the sphere of radius 1 centered at a vertex x. Eigenfunctions are a
convenient tool for representing objects in a hypercube such as, for instance, codes and colorings.
In the present paper we study distributions of an arbitrary such function on faces and investigate
the possibility to reconstruct it from its values on a sphere. Results of the paper were ﬁrst presented
in . Previously, similar questions for various objects in a Boolean cube have been considered
An orthogonal basis of all real-valued functions deﬁned on a hypercube can be given as follows:
→ R | a ∈ F
, ∀ x ∈ F
, a ∈ F
, is an eigenfunction with eigenvalue n − 2wt(a). Therefore, the set
→ R | wt(a)=i} (2)
formsabasisoftheith eigenspace with eigenvalue λ = n − 2i, i =0, 1,...,n.Thiseigenspace
consists of all function f such that their Fourier coeﬃcients
f(x), a ∈ F
Supported in part by the Ministry of Education and Science of Russian Federation, project no. 8227,
and Target Program of the Siberian Branch of the Russian Academy of Sciences (2012–2014), integration
project no. 14.