Published in Research Communications of the conference held in the memory of Paul Erdos Budapest, Hungary, July 4-11, 1999, ed. A. Sali, M. Simonovits and V.T. Ss, Jnos Bolyai Math. Soc., Budapest, 1999, pp. 10-14.
Introduction. Let denote the unit sphere in
real Euclidean -dimensional space
is the inner product of vectors
. For a set
least two points is called a spherical -code, if for any
their inner product
satisfies to the condition This condition means that an
angular distance (angle) between any pair of distinct points in
is greater than or equal to :
The basic problem for spherical codes can be formulated as follows:
for given and find a spherical -code
of largest possible size . We denote
this maximal size by This problem can be consider in the
inverse form: for given and find a spherical
with largest possible minimal
, which we denote by
In the articles , , Levenstein
obtained universal upper bound for the quantity
Levenstein's result can be reformulated for estimating
(cf. ); in particular,
The arrangement of points on with the minimal
was constructed by Hardin, Sloane and
Smith . Thus, the following estimations
Delsarte method. The method which arose in investigations of Delsarte  on upper bounds for packings in some metric spaces is often used for estimating from above the quantities This method was developed and successful applied in the works , , , , ,  and in of many other articles; the rich bibliography on the subject is contained in monograph .
Let be the system of ultraspherical (Gegenbauer) polynomials which are orthogonal on interval with respect to weight function and normed by condition The mentioned approach is founded on a positive-definite property of these polynomials (if we will consider them as kernels on on the Cartesian product ).
denote by a set
consisting of all continuous functions on which are
represented on by series
and The fact that
is non-empty for all
proved in . Let us put
The following statement is contained in , .
Theorem A. Let Then
Formulation of results. Further we consider four-dimensional
case () only. In this case index is equal to
and the polynomials
are Chebyshev polynomials
of the second kind:
Theorem 1. In the equation has the unique solution: where is thirteenth on the increase real root of polynomial
Consider the polynomial
Theorem 2. In the equation has the unique solution: where is twelfth on the increase real root of polynomial
Corollary. Among arbitrary resp.
points located on
two points with the angle between them strictly less than
Let us remind also (see (1), (2)) that the answer to the question: "whether is the least angular distance between pairs of points of arbitrary configuration of points on strictly less than degrees?" should give a solution of the kissing number problem in
The theorem 1 implies that in order to find the number it is necessary to use other methods. We hope that application of results and ideas of Paul Erdös, his coauthors and pupils, in particular, concerning properties of distribution of distances of given number points on sphere , will help to decide problems of such kind.