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.

Let denote the greatest number of
nonoverlapping equal balls that touch another ball of the same
radius in Due to A.M. Odlyzko and N.J.A. Sloane
[11] the problem of determination of the number
(kissing number problem in ) was reduced to
investigation of minimal distances of arrangements of points
on unit sphere in In this presentation we
demonstrate possibilities of the classical Delsarte method for
estimating the upper bounds of minimal distances of arrangements of
and points on .

**Introduction.** Let denote the unit sphere in
real Euclidean -dimensional space
where
is the inner product of vectors
(points)
. For a set
containing at
least two points is called *a spherical -code,* if for any
two points
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

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
code
with largest possible minimal
angle
, which we denote by
i.e.

The indicated problem and the kissing number problem are connected by equality (cf. [3])

The known example (cf. [3, Ch.1, §2]) of spherical -code in is the set consisting from vertexes of the regular convex polyhedron with Schläfli symbol ; therefore Odlyzko and Sloane [11] have proved that does not exceed ; thus,

In the articles [9], [10], Levenstein
obtained universal upper bound for the quantity
Levenstein's result can be reformulated for estimating
(cf. [2]); in particular,

For concrete cases stronger estimations were presented by Boyvalenkov, Danev and Bumova [2]; for example, (in our notations) they obtained the inequality

The arrangement of points on with the minimal
angle
was constructed by Hardin, Sloane and
Smith [7]. Thus, the following estimations
are fulfilled:

**Delsarte method.** The method which arose in investigations of
Delsarte [4] 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 [5],
[8], [11],
[9], [10],
[2] and in of many other articles; the
rich bibliography on the subject is contained in monograph
[3].

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 ).

For
denote by a set
consisting of all continuous functions on which are
non-positive on
and
represented on by series
with non-negative
coefficients
and The fact that
is non-empty for all
was
proved in [8]. Let us put

The following statement is contained in [5], [8].

**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:

Using technique worked out by authors in [1] we find exact values of a parameter for which function takes values and . For formulating the result, we need the following polynomial:

**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
there exist
two points with the angle between them strictly less than
resp.
. Thus,*

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 [6], will help to decide problems of such kind.