A new approach for the existence problem of minimal cubature formulas   based on the Larman-Rogers-Seidel theorem
  In this paper we consider the existence problem of cubature formulas of degree 4k+1 for spherically symmetric integrals for which the equality holds in the M\"oller lower bound. We prove that for sufficiently large dimensional minimal formulas, any two distinct points on some concentric sphere have inner products all of which are rational numbers. By applying this result we prove that for any d > 2 there exist no d-dimensional minimal formulas of degrees 13 and 21 for some special integral. 
Masatake Hirao, Hiroshi Nozaki, Masanori Sawa, Vesselin Vatchev
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