Geodesic
Delsarte problem for 4-designs on the unit 3-sphere
Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 157-161.

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The extremal Delsarte problem $A(d,s)$ for spherical $s$-designs allows us to estimate from below the minimum number of nodes $N(d,s)$ of a weighted quadrature formula on the sphere $\mathbb{S}^{d}$. We prove that $$ A(3,4)=14.560317967882\ldots. $$ Hence $N(3,4)\ge 15$. Our open conjecture is that $N(3,4)=16$.
Keywords: unit sphere, spherical design, quadrature formula, Delsarte problem. 
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     title = {Delsarte problem for 4-designs on the unit 3-sphere},
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D. V. Gorbachev; I. A. Martyanov. Delsarte problem for 4-designs on the unit 3-sphere. Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 157-161. http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a13/

[1] Martyanov I.A., “Reshenie zadachi Delsarta dlya 4-dizainov na sfere $S^{2}$”, Chebyshevskii sbornik, 22:3 (2021), 154–165 | DOI | MR | Zbl