An important problem in discrete geometry and computational mathematics is the estimation of the minimum number of nodes $N(s)$ of a quadrature formula (weighted $s$-design) of the form $\frac{1}{|\mathbb{S}^{2}|}\int_{\mathbb{S}^{2}}f(x) dx=\sum_{\nu=1}^{N}\lambda_{\nu}f(x_{\nu})$ with positive weights, exact for all spherical polynomials of degree at most $s$. P. Delsarte, J.M Goethals, and J.J. Seidel (1977) to estimate $N(s)$ from below formulated an extremal problem $A_{s}$ for expansions in terms of orthogonal Gegenbauer (Legendre for $\mathbb{S}^{2}$) polynomials with restrictions on the sign of the Fourier–Gegenbauer coefficients. Using a version of this problem $A_{s,n}$ on polynomials of degree $n=s$, they proved the classical estimate for tight designs. This estimate is sharp and gives a solution to $A_{s}$ only in exceptional cases ($s=0,1,2,3,5$ for $\mathbb{S}^{2}$). For general dimensions, there are cases when $A_{s,n}>A_{s,s}$ for $n>s$, which leads to better estimates for $N(s)$. In particular, N.N. Andreev (2000) proved in this way the minimality of an $11$-design on the sphere $\mathbb{S}^{3}$. Related Delsarte problems are also formulated for estimating the cardinality of spherical codes. In this direction, V.V. Arestov and A.G. Babenko (1997), based on the methods of infinite-dimensional linear programming, solved an analog of the $A_{s}$ problem for the case of spherical $0.5$-codes on the sphere $\mathbb{S}^{3}$ (the kissing number problem). Then this method was developed in the works of D.V. Shtrom, N.A. Kuklin.
A.V. Bondarenko and D.V. Gorbachev (2012) showed that $N(4)=10$. This fact follows from the estimate $A_{4,7}>9$, previously obtained by P. Boyvalenkov and S. Nikova (1998), and the existence of weighted 4-designs of 10 nodes. Nevertheless, it is of interest to solve the problem $A_ {4}$ exactly, aiming to transfer the method of calculating $A_ {s}$ to the general dimensions and orders of designs. In this paper, it is proved that
$$
A_{4}=A_{4,22}=9.31033\ldots
$$
For this, the Arestov–Babenko–Kuklin method is adapted and the problem is reduced to the construction of a special quadrature formula for $[-1,1]$, consistent with the form of the assumed extremal function (polynomial). The proposed method is based on the use of nonlinear programming, in particular, semidefinite programming, and the solution of a polynomial system of equations arising from a quadrature formula. To prove the existence of an analytical solution of such a system in the neighborhood of the numerical solution, interval Krawczyk's method from HomotopyContinuation.jl is used.
