The search for the sphere of the best cubature formulae invariant under octahedral group of rotations

A new optimality criterion of the cubature formula, invariant under any symmetry group for a sphere is proposed. An essential difference of this criterion from others consists in using the main term of the cubature formula error. The work of the new criterion is demonstrated on an example of the cubature formulae invariant under the octahedral group of rotations. The table which contains the main characteristics of all the best today cubature formulae of the octahedral group of rotations up to the 35th algebraic order of accuracy is given. The weights and the coordinates of the new cubature formulae of the 26th and the 27th orders of accurapy are given to 16 significant digits.